research topic on mathematics

251+ Math Research Topics [2024 Updated]

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

How Do You Write A Math Research Topic?

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

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

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

251+ Math Research Topics: Beginners To Advanced

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

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

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

Related Posts

Step by Step Guide on The Best Way to Finance Car

The Best Way on How to Get Fund For Business to Grow it Efficiently

• Write my thesis
• Thesis writers
• Buy thesis papers
• Bachelor thesis
• Master's thesis
• Thesis editing services
• Thesis proofreading services
• Buy a thesis online
• Write my dissertation
• Dissertation proposal help
• Pay for dissertation
• Custom dissertation
• Dissertation help online
• Buy dissertation online
• Cheap dissertation
• Dissertation editing services
• Write my research paper
• Buy research paper online
• Pay for research paper
• Research paper help
• Order research paper
• Custom research paper
• Cheap research paper
• Research papers for sale
• Thesis subjects
• How It Works

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?

Need Help With Research Paper?

We offer the absolute best high school and college research paper writing service on the Internet. When you need any kind of research paper help, our experienced ENL writers and professional editors are here to help. With years of experience under their belts, our experts can get your research paper done in as little as 3 hours.

Getting cheap online help with research papers has never been easier. College students should just get in touch with us and tell us what they need. We will assign them our most affordable and experienced math writer in minutes, even during the night. We are the best-rated online writing company on the Internet because we always deliver high-quality academic content at the most competitive prices. Give us a try today!

Leave a Reply Cancel reply

Research Areas

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

Articles on Mathematics

Displaying 1 - 20 of 551 articles.

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

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

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

John Eglin , University of Montana

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

Lisa De Bortoli , Australian Council for Educational Research

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

Saverio Eric Spagnolie , University of Wisconsin-Madison

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

Debbie Passey , The University of Melbourne

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

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

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

Neil Saunders , City, University of London

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

Samar Khatiwala , University of Oxford

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

Nalini Joshi , University of Sydney

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

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

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

Manil Suri , University of Maryland, Baltimore County

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

Elena Prieto-Rodriguez , University of Newcastle

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

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

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

Amanda Bao , Rochester Institute of Technology

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

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

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

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

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

Zachary del Rosario , Olin College of Engineering

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

Colin Foster , Loughborough University

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

Related Topics

• Mathematics education
• Maths education
• Quick reads

Top contributors

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

• X (Twitter)
• Unfollow topic Follow topic

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

• View all journals

Mathematics and computing articles from across Nature Portfolio

Mathematics and computing is the study and analysis of abstract concepts, such as numbers and patterns. Mathematics is the language of choice for scientifically describing and modelling the universe and everything that happens within it. Computing enables these mathematical problems to be solved in an efficient and automated way.

The importance of spatial heterogeneity in disease transmission

Spatial heterogeneity in disease transmission rates and in mixing patterns between regions makes predicting epidemic trajectories hard. Quantifying the mixing rates within and between spatial regions can improve predictions.

• Emily Paige Harvey
• Dion R. J. O’Neale

Meta’s AI translation model embraces overlooked languages

More than 7,000 languages are in use throughout the world, but popular translation tools cannot deal with most of them. A translation model that was tested on under-represented languages takes a key step towards a solution.

• David I. Adelani

Related Subjects

• Applied mathematics
• Computational science
• Computer science
• Information technology
• Pure mathematics
• Scientific data

Latest Research and Reviews

Human mobility description by physical analogy of electric circuit network based on GPS data

• Zhihua Zhong
• Hideki Takayasu
• Misako Takayasu

Multistage feature fusion knowledge distillation

• Chuanyun Xu

Learning cooking algorithm for solving global optimization problems

• Prabhujit Mohapatra

Exact analytical soliton solutions of the M-fractional Akbota equation

• Muath Awadalla
• Aigul Taishiyeva
• Ahmet Bekir

Machine learning based urban sprawl assessment using integrated multi-hazard and environmental-economic impact

• Anjar Dimara Sakti
• Albertus Deliar
• Ketut Wikantika

Remotely sensed BC columns over rapidly changing Western China show significant decreases in mass and inconsistent changes in number, size, and mixing properties due to policy actions

• Jason Blake Cohen

News and Comment

Computational morphology and morphogenesis for empowering soft-matter engineering

Morphing soft matter, which is capable of changing its shape and function in response to stimuli, has wide-ranging applications in robotics, medicine and biology. Recently, computational models have accelerated its development. Here, we highlight advances and challenges in developing computational techniques, and explore the potential applications enabled by such models.

Research needs both academia and industry

There are growing concerns about industry’s influence and dominance in computation-based physics research. Why is this so?

Advancing computational sustainability in higher education

• Mayank Kejriwal
• Victoria Petryshyn

Behavioral health and generative AI: a perspective on future of therapies and patient care

There have been considerable advancements in artificial intelligence (AI), specifically with generative AI (GAI) models. GAI is a class of algorithms designed to create new data, such as text, images, and audio, that resembles the data on which they have been trained. These models have been recently investigated in medicine, yet the opportunity and utility of GAI in behavioral health are relatively underexplored. In this commentary, we explore the potential uses of GAI in the field of behavioral health, specifically focusing on image generation. We propose the application of GAI for creating personalized and contextually relevant therapeutic interventions and emphasize the need to integrate human feedback into the AI-assisted therapeutics and decision-making process. We report the use of GAI with a case study of behavioral therapy on emotional recognition and management with a three-step process. We illustrate image generation-specific GAI to recognize, express, and manage emotions, featuring personalized content and interactive experiences. Furthermore, we highlighted limitations, challenges, and considerations, including the elements of human emotions, the need for human-AI collaboration, transparency and accountability, potential bias, security, privacy and ethical issues, and operational considerations. Our commentary serves as a guide for practitioners and developers to envision the future of behavioral therapies and consider the benefits and limitations of GAI in improving behavioral health practices and patient outcomes.

• Emre Sezgin

Quick links

• Explore articles by subject
• Guide to authors
• Editorial policies

• Welcome from the Chair
• Michalik Distinguished Lecture Series
• Open Faculty Positions
• Advising & Support
• Calculus Curriculum
• Degree Programs/Requirements
• Extracurricular Activities
• Math Placement Assessment
• Math Assistance Center/Posvar Computing Lab
• Research/Career Opportunities
• Admissions & Financial Aid
• Degree Programs
• Graduate Employment
• Graduate Handbook
• Information for Incoming Graduate Students
• Organizations
• Research Opportunities
• Teaching Opportunities

Research Areas

• Graduate Research
• Undergraduate Research
• Mathematics Research Center
• Technical Reports
• Publications
• Gallery of Research Images
• Faculty Admin
• Adjunct Faculty
• Part-Time Faculty
• Emeritus Faculty
• Post-Doctoral Associates
• Graduate Students
• Stay in Touch
• Newsletter Archive
• Upcoming Events
• Past Events
• Prospective Students

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.

• Services Paper editing services Paper proofreading Business papers Philosophy papers Write my paper Term papers for sale Term paper help Academic term papers Buy research papers College writing services Paper writing help Student papers Original term papers Research paper help Nursing papers for sale Psychology papers Economics papers Medical papers Blog

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.

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

Terms & Conditions Loyalty Program Privacy Policy Money-Back Policy

Copyright © 2013-2024 MyPaperDone.com

Exploring Best Math Research Topics That Push the Boundaries

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

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

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

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

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

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

What is math research?

Table of Contents

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

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

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

Math Research Topics

A few examples of math research topics:

Number theory

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

The Riemann hypothesis

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

The Birch and Swinnerton-Dyer conjecture

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

The Langlands program

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

The classification of finite simple groups

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

The study of cryptography

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

The study of prime numbers

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

The study of algebraic number theory

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

The study of combinatoric number theory

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

The study of computational number theory

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

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

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

Algebraic topology

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

Geometric topology

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

Differential topology

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

Knot theory

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

Low-dimensional topology

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

Topological quantum field theory

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

Topological data analysis

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

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

Differential geometry research topics

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

Riemannian geometry

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

Complex geometry

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

Geometric analysis

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

Mathematical physics

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

Computer graphics

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

Medical imaging

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

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

Algebraic geometry research topics

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

Algebraic curves

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

Algebraic surfaces

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

Algebraic threefolds

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

Algebraic varieties

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

Algebraic groups

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

Moduli spaces

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

Arithmetic geometry

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

Complex algebraic geometry

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

Algebraic combinatorics

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

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

Mathematical physics research topics

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

Quantum mechanics

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

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

Statistical mechanics

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

Chaos theory

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

Mathematical finance

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

Computational physics

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

Mathematical biology

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

Mathematical chemistry

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

Mathematical engineering

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

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

Mathematical biology research topics

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

Modeling of biological systems

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

Computational biology

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

Biostatistics

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

Mathematical epidemiology

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

Mathematical ecology

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

Mathematical neuroscience

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

Mathematical genetics

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

Mathematical evolution

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

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

Mathematical finance research topics

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

Asset pricing

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

Portfolio optimization

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

Derivative pricing

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

Risk management

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

Market microstructure

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

Financial econometrics

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

Computational finance

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

Mathematical finance and machine learning

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

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

Numerical analysis research topics

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

Error analysis

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

Stability analysis

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

Convergence analysis

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

Adaptive methods

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

Parallel methods

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

Heterogeneous computing

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

Nonlinear problems

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

Optimization

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

Scientific computing

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

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

Computational chemistry

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

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

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

Probability research topics

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

Foundations of probability

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

Stochastic processes

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

Random graphs

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

Markov chains

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

Queueing theory

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

Optimal stopping theory

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

Information theory

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

Computational probability

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

Applied probability

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

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

Statistics research topics

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

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

Machine learning

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

Data mining

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

Bayesian statistics

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

Nonparametric statistics

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

Multivariate statistics

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

Time series analysis

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

Survival analysis

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

Quality control

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

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

How to find math research topics

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

Talk to your professors and advisors

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

Read math journals and conferences

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

Attend math conferences and workshops

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

Think about your own interests and passions

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

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

How to get started with math research

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

Find a mentor

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

Do your research

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

Develop a research plan

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

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

Start small

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

Get feedback

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

Don’t be afraid to ask for help

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

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

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

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

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

Frequently Asked Question

How do i choose a math research topic.

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

Can I pursue research in math as an undergraduate student?

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

What are some emerging areas in math research?

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

How can I conduct math research effectively?

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

Can math research have real-world applications?

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

What resources can I use for math research?

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

Similar Articles

How To Do Homework Fast – 11 Tips To Do Homework Fast

Homework is one of the most important parts that have to be done by students. It has been around for…

How to Write an Assignment Introduction – 6 Best Tips

In essence, the writing tasks in academic tenure students are an integral part of any curriculum. Whether in high school,…

Leave a Comment Cancel Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed .

Advertisement

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 )

Cite this article

You have full access to this open access article

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

30k Accesses

83 Citations

17 Altmetric

Explore all metrics

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.

Similar content being viewed by others

Learning from Research, Advancing the Field

The Narcissism of Mathematics Education

Educational Research on Learning and Teaching Mathematics

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.

Akkerman, S. F., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research , 81 (2), 132–169. https://doi.org/10.3102/0034654311404435

Article   Google Scholar  

Arendt, H. (1958/1998). The human condition (2nd ed.). University of Chicago Press.

Backes, B., & Cowan, J. (2019). Is the pen mightier than the keyboard? The effect of online testing on measured student achievement. Economics of Education Review , 68 , 89–103. https://doi.org/10.1016/j.econedurev.2018.12.007

Bakkenes, I., Vermunt, J. D., & Wubbels, T. (2010). Teacher learning in the context of educational innovation: Learning activities and learning outcomes of experienced teachers. Learning and Instruction , 20 (6), 533–548. https://doi.org/10.1016/j.learninstruc.2009.09.001

Bakker, A. (2019). What is worth publishing? A response to Niss. For the Learning of Mathematics , 39 (3), 43–45.

Google Scholar  

Bakker, A., & Gravemeijer, K. P. (2006). An historical phenomenology of mean and median. Educational Studies in Mathematics , 62 (2), 149–168. https://doi.org/10.1007/s10649-006-7099-8

Bakx, A., Bakker, A., Koopman, M., & Beijaard, D. (2016). Boundary crossing by science teacher researchers in a PhD program. Teaching and Teacher Education , 60 , 76–87. https://doi.org/10.1016/j.tate.2016.08.003

Battey, D. (2013). Access to mathematics: “A possessive investment in whiteness”. Curriculum Inquiry , 43 (3), 332–359.

Bawa, P. (2020). Learning in the age of SARS-COV-2: A quantitative study of learners’ performance in the age of emergency remote teaching. Computers and Education Open , 1 , 100016. https://doi.org/10.1016/j.caeo.2020.100016

Beckers, D., & Beckers, A. (2019). ‘Newton was heel exact wetenschappelijk – ook in zijn chemische werk’. Nederlandse wetenschapsgeschiedenis in niet-wetenschapshistorische tijdschriften, 1977–2017. Studium , 12 (4), 185–197. https://doi.org/10.18352/studium.10203

Bessot, A., & Ridgway, J. (Eds.). (2000). Education for mathematics in the workplace . Springer.

Bickerton, R. T., & Sangwin, C. (2020). Practical online assessment of mathematical proof. arXiv preprint:2006.01581 . https://arxiv.org/pdf/2006.01581.pdf .

Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education . Springer.

Bini, G., Robutti, O., & Bikner-Ahsbahs, A. (2020). Maths in the time of social media: Conceptualizing the Internet phenomenon of mathematical memes. International Journal of Mathematical Education in Science and Technology , 1–40. https://doi.org/10.1080/0020739x.2020.1807069

Bosch, M., Dreyfus, T., Primi, C., & Shiel, G. (2017, February). Solid findings in mathematics education: What are they and what are they good for? CERME 10 . Ireland: Dublin https://hal.archives-ouvertes.fr/hal-01849607

Bowker, G. C., & Star, S. L. (2000). Sorting things out: Classification and its consequences . MIT Press. https://doi.org/10.7551/mitpress/6352.001.0001

Burkhardt, H. (2019). Improving policy and practice. Educational Designer , 3 (12) http://www.educationaldesigner.org/ed/volume3/issue12/article46/

Cai, J., & Hwang, S. (2019). Constructing and employing theoretical frameworks in (mathematics) education research. For the Learning of Mathematics , 39 (3), 44–47.

Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and U.S. elementary mathematics textbooks. International Journal of Science and Mathematics Education , 15 (8), 1521–1540. https://doi.org/10.1007/s10763-016-9758-2

Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics , 105 , 287–301. https://doi.org/10.1007/s10649-020-10008-x

Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., … Hiebert, J. (2020). Improving the impact of research on practice: Capitalizing on technological advances for research. Journal for Research in Mathematics Education , 51 (5), 518–529 https://pubs.nctm.org/view/journals/jrme/51/5/article-p518.xml

Chronaki, A. (2019). Affective bodying of mathematics, children and difference: Choreographing ‘sad affects’ as affirmative politics in early mathematics teacher education. ZDM-Mathematics Education , 51 (2), 319–330. https://doi.org/10.1007/s11858-019-01045-9

Civil, M., & Bernier, E. (2006). Exploring images of parental participation in mathematics education: Challenges and possibilities. Mathematical Thinking and Learning , 8 (3), 309–330. https://doi.org/10.1207/s15327833mtl0803_6

Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education , 40 ( 1 ), 40–68 https://pubs.nctm.org/view/journals/jrme/40/1/article-p40.xml

Darragh, L. (2016). Identity research in mathematics education. Educational Studies in Mathematics , 93 (1), 19–33. https://doi.org/10.1007/s10649-016-9696-5

de Abreu, G., Bishop, A., & Presmeg, N. C. (Eds.). (2006). Transitions between contexts of mathematical practices . Kluwer.

de Freitas, E., Ferrara, F., & Ferrari, G. (2019). The coordinated movements of collaborative mathematical tasks: The role of affect in transindividual sympathy. ZDM-Mathematics Education , 51 (2), 305–318. https://doi.org/10.1007/s11858-018-1007-4

Deng, Z. (2018). Contemporary curriculum theorizing: Crisis and resolution. Journal of Curriculum Studies , 50 (6), 691–710. https://doi.org/10.1080/00220272.2018.1537376

Dobie, T. E., & Sherin, B. (2021). The language of mathematics teaching: A text mining approach to explore the zeitgeist of US mathematics education. Educational Studies in Mathematics .  https://doi.org/10.1007/s10649-020-10019-8

Eames, C., & Eames, R. (1977). Powers of Ten [Film]. YouTube. https://www.youtube.com/watch?v=0fKBhvDjuy0

Engelbrecht, J., Borba, M. C., Llinares, S., & Kaiser, G. (2020). Will 2020 be remembered as the year in which education was changed? ZDM-Mathematics Education , 52 (5), 821–824. https://doi.org/10.1007/s11858-020-01185-3

English, L. (2008). Setting an agenda for international research in mathematics education. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 3–19). Routledge.

Ernest, P. (2020). Unpicking the meaning of the deceptive mathematics behind the COVID alert levels. Philosophy of Mathematics Education Journal , 36 http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/pome36/index.html

Freudenthal, H. (1986). Didactical phenomenology of mathematical structures . Springer.

Gilmore, C., Göbel, S. M., & Inglis, M. (2018). An introduction to mathematical cognition . Routledge.

Goos, M., & Beswick, K. (Eds.). (2021). The learning and development of mathematics teacher educators: International perspectives and challenges . Springer. https://doi.org/10.1007/978-3-030-62408-8

Gorard, S. (Ed.). (2020). Getting evidence into education. Evaluating the routes to policy and practice . Routledge.

Gravemeijer, K., Stephan, M., Julie, C., Lin, F.-L., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education , 15 (1), 105–123. https://doi.org/10.1007/s10763-017-9814-6

Hannula, M. S. (2019). Young learners’ mathematics-related affect: A commentary on concepts, methods, and developmental trends. Educational Studies in Mathematics , 100 (3), 309–316. https://doi.org/10.1007/s10649-018-9865-9

Hilt, L. T. (2015). Included as excluded and excluded as included: Minority language pupils in Norwegian inclusion policy. International Journal of Inclusive Education , 19 (2), 165–182.

Hodgen, J., Taylor, B., Jacques, L., Tereshchenko, A., Kwok, R., & Cockerill, M. (2020). Remote mathematics teaching during COVID-19: Intentions, practices and equity . UCL Institute of Education https://discovery.ucl.ac.uk/id/eprint/10110311/

Horn, I. S. (2017). Motivated: Designing math classrooms where students want to join in . Heinemann.

Hoyles, C., Noss, R., & Pozzi, S. (2001). Proportional reasoning in nursing practice. Journal for Research in Mathematics Education , 32 (1), 4–27. https://doi.org/10.2307/749619

Ito, M., Martin, C., Pfister, R. C., Rafalow, M. H., Salen, K., & Wortman, A. (2018). Affinity online: How connection and shared interest fuel learning . NYU Press.

Jackson, K. (2011). Approaching participation in school-based mathematics as a cross-setting phenomenon. The Journal of the Learning Sciences , 20 (1), 111–150. https://doi.org/10.1080/10508406.2011.528319

Jansen, A., Herbel-Eisenmann, B., & Smith III, J. P. (2012). Detecting students’ experiences of discontinuities between middle school and high school mathematics programs: Learning during boundary crossing. Mathematical Thinking and Learning , 14 (4), 285–309. https://doi.org/10.1080/10986065.2012.717379

Johnson, L. F., Smith, R. S., Smythe, J. T., & Varon, R. K. (2009). Challenge-based learning: An approach for our time (pp. 1–38). The New Media Consortium https://www.learntechlib.org/p/182083

Jullien, F. (2018). Living off landscape: Or the unthought-of in reason . Rowman & Littlefield.

Kazima, M. (2019). What is proven to work in successful countries should be implemented in other countries: The case of Malawi and Zambia. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 73–78). PME.

Kim, H. (2019). Ask again, “why should we implement what works in successful countries?” In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 79–82). PME.

Kolovou, A., Van Den Heuvel-Panhuizen, M., & Bakker, A. (2009). Non-routine problem solving tasks in primary school mathematics textbooks—a needle in a haystack. Mediterranean Journal for Research in Mathematics Education , 8 (2), 29–66.

Kwon, O. N., Han, C., Lee, C., Lee, K., Kim, K., Jo, G., & Yoon, G. (2021). Graphs in the COVID-19 news: A mathematics audit of newspapers in Korea. Educational Studies in Mathematics . https://doi.org/10.1007/s10649-021-10029-0

Lefebvre, H. (2004). Rhythmanalysis: Space, time and everyday life (Original 1992; Translation by S. Elden & G. Moore) . Bloomsbury Academic. https://doi.org/10.5040/9781472547385 .

Li, Y. (2019). Should what works in successful countries be implemented in other countries? In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 67–72). PME.

Martin, D., Gholson, M., & Leonard, J. (2010). Mathematics as gatekeeper: Power and privilege in the production of power. Journal of Urban Mathematics Education , 3 (2), 12–24.

Masschelein, J., & Simons, M. (2019). Bringing more ‘school’ into our educational institutions. Reclaiming school as pedagogic form. In A. Bikner-Ahsbahs & M. Peters (Eds.), Unterrichtsentwicklung macht Schule (pp. 11–26) . Springer. https://doi.org/10.1007/978-3-658-20487-7_2

Meeter, M., Bele, T., den Hartogh, C., Bakker, T., de Vries, R. E., & Plak, S. (2020). College students’ motivation and study results after COVID-19 stay-at-home orders. https://psyarxiv.com .

Nemirovsky, R., Kelton, M. L., & Civil, M. (2017). Toward a vibrant and socially significant informal mathematics education. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 968–979). National Council of Teachers of Mathematics.

Niss, M. (2019). The very multi-faceted nature of mathematics education research. For the Learning of Mathematics , 39 (2), 2–7.

OECD. (2020). Back to the Future of Education: Four OECD Scenarios for Schooling. Educational Research and Innovation . OECD Publishing. https://doi.org/10.1787/20769679

Potari, D., Psycharis, G., Sakonidis, C., & Zachariades, T. (2019). Collaborative design of a reform-oriented mathematics curriculum: Contradictions and boundaries across teaching, research, and policy. Educational Studies in Mathematics , 102 (3), 417–434. https://doi.org/10.1007/s10649-018-9834-3

Proulx, J., & Maheux, J. F. (2019). Effect sizes, epistemological issues, and identity of mathematics education research: A commentary on editorial 102(1). Educational Studies in Mathematics , 102 (2), 299–302. https://doi.org/10.1007/s10649-019-09913-7

Roos, H. (2019). Inclusion in mathematics education: An ideology, A way of teaching, or both? Educational Studies in Mathematics , 100 (1), 25–41. https://doi.org/10.1007/s10649-018-9854-z

Saenz, M., Medina, A., & Urbine Holguin, B. (2020). Colombia: La prender al onda (to turn on the wave). Education continuity stories series . OECD Publishing https://oecdedutoday.com/wp-content/uploads/2020/12/Colombia-a-prender-la-onda.pdf

Schindler, M., & Bakker, A. (2020). Affective field during collaborative problem posing and problem solving: A case study. Educational Studies in Mathematics , 105 , 303–324. https://doi.org/10.1007/s10649-020-09973-0

Schoenfeld, A. H. (1999). Looking toward the 21st century: Challenges of educational theory and practice. Educational Researcher , 28 (7), 4–14. https://doi.org/10.3102/0013189x028007004

Schukajlow, S., Rakoczy, K., & Pekrun, R. (2017). Emotions and motivation in mathematics education: Theoretical considerations and empirical contributions. ZDM-Mathematics Education , 49 (3), 307–322. https://doi.org/10.1007/s11858-017-0864-6

Sfard, A. (2005). What could be more practical than good research? Educational Studies in Mathematics , 58 (3), 393–413. https://doi.org/10.1007/s10649-005-4818-5

Shimizu, Y., & Vithal, R. (Eds.). (2019). ICMI Study 24 Conference Proceedings. School mathematics curriculum reforms: Challenges, changes and opportunities . ICMI: University of Tsukuba & ICMI http://www.human.tsukuba.ac.jp/~icmi24/

Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics , 10 (3), 24–41.

Stephan, M. L., Chval, K. B., Wanko, J. J., Civil, M., Fish, M. C., Herbel-Eisenmann, B., … Wilkerson, T. L. (2015). Grand challenges and opportunities in mathematics education research. Journal for Research in Mathematics Education , 46 (2), 134–146. https://doi.org/10.5951/jresematheduc.46.2.0134

Suazo-Flores, E., Alyami, H., Walker, W. S., Aqazade, M., & Kastberg, S. E. (2021). A call for exploring mathematics education researchers’ interdisciplinary research practices. Mathematics Education Research Journal , 1–10. https://doi.org/10.1007/s13394-021-00371-0

Svensson, P., Meaney, T., & Norén, E. (2014). Immigrant students’ perceptions of their possibilities to learn mathematics: The case of homework. For the Learning of Mathematics , 34 (3), 32–37.

UNESCO. (2015). Teacher policy development guide . UNESCO, International Task Force on Teachers for Education 2030. https://teachertaskforce.org/sites/default/files/2020-09/370966eng_0_1.pdf .

Van den Heuvel-Panhuizen, M. (2005). Can scientific research answer the ‘what’ question of mathematics education? Cambridge Journal of Education , 35 (1), 35–53. https://doi.org/10.1080/0305764042000332489

Wittmann, E. C. (1995). Mathematics education as a ‘design science’. Educational Studies in Mathematics , 29 (4), 355–374.

Yoon, H., Byerley, C. O. N., Joshua, S., Moore, K., Park, M. S., Musgrave, S., Valaas, L., & Drimalla, J. (2021). United States and South Korean citizens’ interpretation and assessment of COVID-19 quantitative data. The Journal of Mathematical Behavior . https://doi.org/10.1016/j.jmathb.2021.100865 .

Download references

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.

Author information

Authors and affiliations.

Utrecht University, Utrecht, Netherlands

Arthur Bakker & Linda Zenger

University of Delaware, Newark, DE, USA

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Arthur Bakker .

Ethics declarations

In line with the guidelines of the Code of Publication Ethics (COPE), we note that the review process of this article was blinded to the authors.

Additional information

Publisher’s note.

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix 1: Explanation of Fig. 1

We have divided Fig. 1 in 12 rectangles called A1 (bottom left) up to C4 (top right) to explain the details (for image annotation go to https://www.fisme.science.uu.nl/toepassingen/28937 )

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ .

Reprints and permissions

About this article

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

Download citation

Accepted : 04 March 2021

Published : 06 April 2021

Issue Date : May 2021

DOI : https://doi.org/10.1007/s10649-021-10049-w

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Grand challenges
• Mathematics education research
• Research agenda
• Find a journal
• Publish with us
• Track your research

• Paper writing help
• Buy an Essay
• Pay for essay
• Buy Research Paper
• Write My Research Paper
• Research Paper Help
• Custom Research Paper
• Custom Dissertation
• Dissertation Help
• Buy Dissertation
• Dissertation Writer
• Write my Dissertation
• How it works

Research Paper Topics on Mathematics

Research papers in mathematics usually have a set of basic requirements that apply to all students. As a rule, they relate to the format of the research work and the requirements for it. Students need to adhere to a specific font, spacing, and writing format. It is necessary to have a title page of the research paper and the project, as well as the correct text formatting with recommendations and pagination.

Students must adhere to a clear heading hierarchy in research work and adhere to the correct abbreviations and formulas in the design of the entire project. As a rule, research work associated with the use of certain drawings, tables, diagrams, and graphs. Also, there may be charts which must also be marked following the requirements.

Students must reveal the topic given to them and bring clear arguments in favor of their statements. It should be a full-fledged mathematics research topic that fully reveals the theme and provides the reader with real evidence of opinion. It's important to choose an interesting math research topic.

The main task for each student is to relate everything they plan to write with one Topic. It will allow the teacher not to waste time looking for information and get the opportunity to see holistic work that fully meets all the established requirements. Students should also check the practical part as mathematics is an exact science that does not tolerate assumptions. So, let's take a look at the math research topics for middle schools.

Algebra & Algebraic Geometry

Algebra and algebraic geometry are the oldest branches of this science. This section arose as a result of the need to solve arithmetic problems of the same type. Now, this is a huge branch of science, which is extremely important because of its ability to make accurate calculations and definitions in various areas of our life. Choose any research topic for mathematics that suits you well.

• Formality in Deformation Theory
• The World of P-adic Numbers
• Toric Geometry and Mirror Symmetry
• Algebraic Geometry and Singularity Theory
• P-adic Dynamics and Applications of P-adic Numbers in Other Sciences
• Algebraic Geometry New Trends: The height Functions in Modern Science
• Theory of Height Functions & in Math

Algebraic Topology

This section of typology studies topological spaces using the juxtaposition of algebraic objects. At the moment, it is one of the more popular offshoots that uses homotopy groups and homomorphism. Any research topic in mathematics is a chance to create a good paper. This area of mathematics is popular thanks to several scientific studies. Here is a list of paper themes, but you can also find math education research topics by yourself.

• Main Calculus Functions and How to Use it in Real Life
• The Calculus of Functors and Applications
• How to Use the Algebraic Models for Spaces in Real Life
• New Wave of Automorphisms in Math
• The Asymptotic Properties of Polynomials in Higher Dimensions
• Moduli Spaces Geometry: Main Features
• The Lefschetz Properties and Main Parameters
• The Group Theory in Mathematics: Main Benefits
• Automorphisms of Manifolds and Related Options
• Modern Trends in Algebraic Topology and How to Use it Properly

Analysis & PDEs

This list of topics can be especially interesting for those who want to develop methods for solving equations of mathematical physics. Let's proceed to researchable topics in mathematics. In particular, students can create their work based on the approximation of the differential operator and use numerical methods to solve the tasks and open the topic of scientific research.

• Regularity Theory for Linear, Semilinear, and Fully Nonlinear Elliptic and Parabolic Equation
• New Theory for Linear, Semilinear, and Nonlinear Functions
• Qualitative Properties of Solutions to Reaction-diffusion Equations
• The Analysis of Problems in Mathematical Physics and Mathematical Modeling
• Delay Equations Formulation of Structured Population Dynamics
• Fractional differential equations and Fractals
• Mathematical Modeling of Ecological Systems and Epidemic Systems
• Delay Differential Equations is One Best Research Area in Mathematics.

According to many historians, the foundations of geometry were laid by the ancient Greeks, who took over from the Egyptians some mathematical calculations and general principles. This section received such a name in 1822. Nevertheless, all the foundations and important stages of this science branch were laid many centuries ago. Here are the applied mathematics research topics.

• The Bending Active Approach In Lightweight Design
• The Need Of Geometry In Orthodontics
• Archimedes Theory of a Circle ABCD and a Triangle K
• Applications of Toric Geometry to Geometric Representation Theory
• Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations
• Geometric Constructions of Mapping Cones in the Fukaya Category
• Euclidean and Non-Euclidean Geometries. Going Beyond the Space, We Used to know

Mathematical Logic & Foundations

This branch of mathematics may be of interest to those who want to study the nature of mathematical proof in general and various mathematical judgments as a branch of informal logic. In many respects, the foundation of this science was laid by Aristotle. Nevertheless, mathematical logic is developing in our time. Therefore, this is a good basis for scientific work.

• Deductive Parsing of Visual Languages
• Reduction of the Yablo paradox
• Gödel's: Minds are not Machines
• Ways to use the Mathematical logic in data science
• Applied Propositional Logic and digital circuit design
• Category Theory and Model Theory of Constructive Systems
• Constructive Mathematics and Point-free Methods in Topology and Analysis
• Ways to Use Martin-Löf type Theory and Univalent Foundations in Real Life

Number Theory

This is one of the popular branches of mathematics, consisting of many areas such as the Euclidean algorithm, continued fractions, Diophantine equations, and farm theory. At the moment, this industry has a lot of interesting topics to write a research paper. That's why you can find interesting mathematical topics.

• Writing Pi As the Sum of Arctangents With Recurring Linear Sequences, the Golden Mean and Lucas Numbers
• Binary Options Winning Formula: Make Consistent Wins Every Time
• A Theory of Numbers and New Operations
• The Twin Prime Conjecture in a Number Theory
• Pythagorean Triplets (Alternative Approach, Algebraic Operations, Dual of Given Triplets, and New Observations)
• A New Theory of Numbers: The Latest Discoveries in Mathematics
• Alternative Formula for the Series of Consecutive m-Squares under Alternating Signs
• A Clever Way To Factor One Less A Perfect Square
• How to use Number Theory to fight with gender stereotypes
• Goldbachs Twin Prime Conjecture

Probability & Statistics

By choosing this direction of mathematics, you can write a lot of interesting materials on modeling and data distribution and the study of two-dimensional numerical data. This is a particularly popular area of mathematics that will allow you to choose interesting topics for yourself with experiments and proofs. Let's check the research topics for mathematic paper.

• Highlighting Probability Issues in Simulated Annealing and Tabu Search
• Estimation of the Probability of Non-Response in Sampling Surveys Using Kernel Density Estimation Methods
• Importance of Statistical Tools and Methods in Data Science
• A Statistical Approach on Experimental Study for Determining Switching Frequency of Retro Reflector Sensor Using PLC
• The Statistics Analysis on Gender: Ain't ia Woman
• Quantitative Data Management, Statistical Analysis, and Graphics Using Stata
• The Data Science and Modern Statistics: the Symbiotic Connection
• Sei Shonagon's Pillow Book as an Example of Probability & Statistics Processes
• The Weibull Length Biased Exponential Distribution: Statistical Properties and Applications
• Univariate and Bivariate Transformations

Representation Theory

This branch of mathematics is especially interesting due to algebraic structures and linear transformations of vector spaces. Students can use Number Theory and Differential Geometry to describe certain nuances of Fourier theory or use topological groups to prove new theories. Here you can choose an undergraduate math research topic.

• Computing Modular Forms for the Weil Representation
• Combinatorics of the Asymmetric Simple Exclusion Process
• The Representation Theory & Asymmetric Simple Exclusion
• The Weil Representation in Tropical Mathematics
• Camel and Cactus Test in Representation Theory
• Combinatorics and Computations in Tropical Mathematics
• Quaternionic Representation and Its Use in Real Life
• The Modern Trends in Representation Theory of Diffeomorphism Groups

How to Write a Research Paper on Mathematics?

As we wrote earlier, teachers are very scrupulous about the correct design and writing of research papers. And interesting math research paper topics are a must. First of all, you need to write a clear introduction and structure of all your work. It is worth noting that you need formal and informal exposure, as well as bringing evidence of your work. One of the sections there will be special recommendations that depend on a particular educational institution.

You should understand that any such work requires a clear technical design and the ghost of the necessary graphs, tables, formulas, and calculations. The most important task of such research papers is to provide clear and evidence of your point of view to argue for each point and paragraph of your work. This is the key aspect, without which you do not get good grades.

That is why you should turn to professionals if you are not confident in your abilities or want to save a little time. Our company can help you write a research paper in mathematics and accompany you until you receive your final grade. It's important to choose good research topics in mathematics education, and we are ready to help you.

An Inspiration Sources List:

• Maths At Home
• The Holy Grail for Any Student
• The Mathunion Website
• Yahoo Mathematics Page
• Computer Algebra Group
• MacTutor History of Math
• Mathematics on the Web

Get the Reddit app

r/mathematics is a subreddit dedicated to focused questions and discussion concerning mathematics.

High school student that needs help coming up with math research topic

Hi, I’m currently in high school and have been tasked to conduct a mathematical investigation. it’s 20% of my final grade but I’m struggling to come up with some ideas that aren’t too difficult.

My teacher said to apply math to a real life situation, perhaps something we find interesting, and my mind went to astronomy (though I unfortunately suck at physics)

I was wondering if there were any research topics I could apply things like vectors and matrices to (maybe graph theory too) to something astronomy related that wouldn’t be too difficult since my knowledge of math doesn’t really go beyond first year of college level.

I have a few months to work on this and If anyone could help me come up with any ideas I would really appreciate it!! Thanks

P.S. doesn’t have to be astronomy related, but I’d really love to do something with matrices and/or vectors, but of course I’m open to other recommendations too!

share this!

April 29, 2024

This article has been reviewed according to Science X's editorial process and policies . Editors have highlighted the following attributes while ensuring the content's credibility:

fact-checked

trusted source

Intervention based on science of reading and math boosts comprehension and word problem-solving skills

by University of Kansas

New research from the University of Kansas has found that an intervention based on the science of reading and math effectively helped English learners boost their comprehension, visualize and synthesize information, and make connections that significantly improved their math performance.

The intervention , performed for 30 minutes twice a week for 10 weeks with 66 third-grade English language learners who displayed math learning difficulties, improved students' performance when compared to students who received general instruction. This indicates that emphasizing cognitive concepts involved in the science of reading and math are key to helping students improve, according to researchers.

"Word problem-solving is influenced by both the science of reading and the science of math. Key components include number sense, decoding, language comprehension and working memory. Utilizing direct and explicit teaching methods enhances understanding and enables students to effectively connect these skills to solve math problems . This integrated approach ensures that students are equipped with necessary tools to navigate both the linguistic and numerical demands of word problems," said Michael Orosco, professor of educational psychology at KU and lead author of the study.

The intervention incorporates comprehension strategy instruction in both reading and math, focusing and decoding, phonological awareness, vocabulary development, inferential thinking, contextualized learning and numeracy.

"It is proving to be one of the most effective evidence-based practices available for this growing population," Orosco said.

The study, co-written with Deborah Reed of the University of Tennessee, was published in the journal Learning Disabilities Research and Practice .

For the research, trained tutors implemented the intervention, developed by Orosco and colleagues based on cognitive and culturally responsive research conducted over a span of 20 years. One example of an intervention session tested in the study included a script in which a tutor examined a word problem explaining that a person made a quesadilla for his friend Mario and gave him one-fourth of it, then asked students to determine how much remained.

The tutor first asked students if they remembered a class session in which they made quesadillas and what shape they were, and demonstrated concepts by drawing a circle on the board, dividing it into four equal pieces, having students repeat terms like numerator and denominator. The tutor explains that when a question asks how much is left, subtraction is required. The students also collaborated with peers to practice using important vocabulary in sentences. The approach both helps students learn and understand mathematical concepts while being culturally responsive.

"Word problems are complex because they require translating words into mathematical equations, and this involves integrating the science of reading and math through language concepts and differentiated instruction," Orosco said. "We have not extensively tested these approaches with this group of children. However, we are establishing an evidence-based framework that aids them in developing background knowledge and connecting it to their cultural contexts."

Orosco, director of KU's Center for Culturally Responsive Educational Neuroscience, emphasized the critical role of language in word problems, highlighting the importance of using culturally familiar terms. For instance, substituting "pastry" for "quesadilla" could significantly affect comprehension for students from diverse backgrounds. Failure to grasp the initial scenario could impede subsequent problem-solving efforts.

The study proved effective in improving students' problem-solving abilities, despite covariates including an individual's basic calculation skills, fluid intelligence and reading comprehension scores. That finding is key, as while ideally all students would begin on equal footing and there would be few variations in a classroom, in reality, covariates exist and are commonplace.

The study had trained tutors deliver the intervention, and its effectiveness should be further tested with working teachers, the authors wrote. Orosco said professional development to help teachers gain the skills is necessary, and it is vital for teacher preparation programs to train future teachers with such skills as well. And helping students at the elementary level is necessary to help ensure success in future higher-level math classes such as algebra.

The research builds on Orosco and colleagues' work in understanding and improving math instruction for English learners. Future work will continue to examine the role of cognitive functions such as working memory and brain science, as well as potential integration of artificial intelligence in teaching math.

"Comprehension strategy instruction helps students make connections, ask questions, visualize, synthesize and monitor their thinking about word problems," Orosco and Reed wrote. "Finally, applying comprehension strategy instruction supports ELs in integrating their reading, language and math cognition…. Focusing on relevant language in word problems and providing collaborative support significantly improved students' solution accuracy."

Provided by University of Kansas

Explore further

Feedback to editors

New technique uses enzymes to create versatile nanoparticles

9 minutes ago

Chemists discover spontaneous nanoparticle formation in charged microdroplets

19 minutes ago

Thermoelectric effect between two liquid materials observed for the first time

20 minutes ago

Chinese astronomers discover a high-velocity star ejected from globular cluster Messier 15

22 minutes ago

Pseudomagic quantum states: A path to quantum supremacy

35 minutes ago

New study helps disentangle role of soil microbes in the global carbon cycle

58 minutes ago

New discovery reveals that ocean algae unexpectedly help cool the Earth

4 hours ago

Research signals major milestone in cutting harmful gases that deplete ozone and worsen global warming

Hubble Space Telescope finds surprises around a star that erupted 40 years ago

First map of outflows from nearby quasar I Zwicky 1

6 hours ago

Relevant PhysicsForums posts

How is physics taught without calculus.

8 hours ago

Is "College Algebra" really just high school "Algebra II"?

Uk school physics exam from 1967.

May 27, 2024

Physics education is 60 years out of date

May 16, 2024

Plagiarism & ChatGPT: Is Cheating with AI the New Normal?

May 13, 2024

Physics Instructor Minimum Education to Teach Community College

May 11, 2024

More from STEM Educators and Teaching

Related Stories

Study shows program improves teaching skills, students' word problem–solving

Jun 14, 2022

Study shows approach can help English learners improve at math word problems

Jun 19, 2018

Study examines role of working memory, cognitive functions in English learners learning to write

Oct 17, 2023

Cognitive study shows lack of bilingual education adversely affects English language learners' writing skills

Oct 14, 2021

How vocabulary breadth and depth influence bilingual reading comprehension

Aug 21, 2023

Study validates the simple view of reading for enhancing second and foreign language learners' experience

Aug 17, 2023

Recommended for you

Study reveals complex dynamics of philanthropic funding for US science

19 hours ago

First-generation medical students face unique challenges and need more targeted support, say researchers

Investigation reveals varied impact of preschool programs on long-term school success

May 2, 2024

Training of brain processes makes reading more efficient

Apr 18, 2024

Researchers find lower grades given to students with surnames that come later in alphabetical order

Apr 17, 2024

Earth, the sun and a bike wheel: Why your high-school textbook was wrong about the shape of Earth's orbit

Apr 8, 2024

Let us know if there is a problem with our content

Use this form if you have come across a typo, inaccuracy or would like to send an edit request for the content on this page. For general inquiries, please use our contact form . For general feedback, use the public comments section below (please adhere to guidelines ).

Please select the most appropriate category to facilitate processing of your request

Thank you for taking time to provide your feedback to the editors.

Your feedback is important to us. However, we do not guarantee individual replies due to the high volume of messages.

E-mail the story

Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. The information you enter will appear in your e-mail message and is not retained by Phys.org in any form.

Newsletter sign up

Get weekly and/or daily updates delivered to your inbox. You can unsubscribe at any time and we'll never share your details to third parties.

More information Privacy policy

Donate and enjoy an ad-free experience

We keep our content available to everyone. Consider supporting Science X's mission by getting a premium account.

E-mail newsletter

10 Hard Math Problems That Continue to Stump Even the Brightest Minds

Maybe you’ll have better luck.

For now, you can take a crack at the hardest math problems known to man, woman, and machine. For more puzzles and brainteasers, check out Puzzmo . ✅ More from Popular Mechanics :

• To Create His Geometric Artwork, M.C. Escher Had to Learn Math the Hard Way
• Fourier Transforms: The Math That Made Color TV Possible
• The Game of Trees is a Mad Math Theory That Is Impossible to Prove

The Collatz Conjecture

In September 2019, news broke regarding progress on this 82-year-old question, thanks to prolific mathematician Terence Tao. And while the story of Tao’s breakthrough is promising, the problem isn’t fully solved yet.

A refresher on the Collatz Conjecture : It’s all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. Take any natural number, apply f, then apply f again and again. You eventually land on 1, for every number we’ve ever checked. The Conjecture is that this is true for all natural numbers (positive integers from 1 through infinity).

✅ Down the Rabbit Hole: The Math That Helps the James Webb Space Telescope Sit Steady in Space

Tao’s recent work is a near-solution to the Collatz Conjecture in some subtle ways. But he most likely can’t adapt his methods to yield a complete solution to the problem, as Tao subsequently explained. So, we might be working on it for decades longer.

The Conjecture lives in the math discipline known as Dynamical Systems , or the study of situations that change over time in semi-predictable ways. It looks like a simple, innocuous question, but that’s what makes it special. Why is such a basic question so hard to answer? It serves as a benchmark for our understanding; once we solve it, then we can proceed onto much more complicated matters.

The study of dynamical systems could become more robust than anyone today could imagine. But we’ll need to solve the Collatz Conjecture for the subject to flourish.

Goldbach’s Conjecture

One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude. But we need proof for all natural numbers.

Goldbach’s Conjecture precipitated from letters in 1742 between German mathematician Christian Goldbach and legendary Swiss mathematician Leonhard Euler , considered one of the greatest in math history. As Euler put it, “I regard [it] as a completely certain theorem, although I cannot prove it.”

✅ Dive In: The Math Behind Our Current Theory of Human Color Perception Is Wrong

Euler may have sensed what makes this problem counterintuitively hard to solve. When you look at larger numbers, they have more ways of being written as sums of primes, not less. Like how 3+5 is the only way to break 8 into two primes, but 42 can broken into 5+37, 11+31, 13+29, and 19+23. So it feels like Goldbach’s Conjecture is an understatement for very large numbers.

Still, a proof of the conjecture for all numbers eludes mathematicians to this day. It stands as one of the oldest open questions in all of math.

The Twin Prime Conjecture

Together with Goldbach’s, the Twin Prime Conjecture is the most famous in Number Theory—or the study of natural numbers and their properties, frequently involving prime numbers. Since you've known these numbers since grade school, stating the conjectures is easy.

When two primes have a difference of 2, they’re called twin primes. So 11 and 13 are twin primes, as are 599 and 601. Now, it's a Day 1 Number Theory fact that there are infinitely many prime numbers. So, are there infinitely many twin primes? The Twin Prime Conjecture says yes.

Let’s go a bit deeper. The first in a pair of twin primes is, with one exception, always 1 less than a multiple of 6. And so the second twin prime is always 1 more than a multiple of 6. You can understand why, if you’re ready to follow a bit of heady Number Theory.

✅ Keep Learning: If We Draw Graphs Like This, We Can Change Computers Forever

All primes after 2 are odd. Even numbers are always 0, 2, or 4 more than a multiple of 6, while odd numbers are always 1, 3, or 5 more than a multiple of 6. Well, one of those three possibilities for odd numbers causes an issue. If a number is 3 more than a multiple of 6, then it has a factor of 3. Having a factor of 3 means a number isn’t prime (with the sole exception of 3 itself). And that's why every third odd number can't be prime.

How’s your head after that paragraph? Now imagine the headaches of everyone who has tried to solve this problem in the last 170 years.

The good news is that we’ve made some promising progress in the last decade. Mathematicians have managed to tackle closer and closer versions of the Twin Prime Conjecture. This was their idea: Trouble proving there are infinitely many primes with a difference of 2? How about proving there are infinitely many primes with a difference of 70,000,000? That was cleverly proven in 2013 by Yitang Zhang at the University of New Hampshire.

For the last six years, mathematicians have been improving that number in Zhang’s proof, from millions down to hundreds. Taking it down all the way to 2 will be the solution to the Twin Prime Conjecture. The closest we’ve come —given some subtle technical assumptions—is 6. Time will tell if the last step from 6 to 2 is right around the corner, or if that last part will challenge mathematicians for decades longer.

The Riemann Hypothesis

Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize Problems , with $1 million reward for its solution. It has implications deep into various branches of math, but it’s also simple enough that we can explain the basic idea right here.

There is a function, called the Riemann zeta function, written in the image above.

For each s, this function gives an infinite sum, which takes some basic calculus to approach for even the simplest values of s. For example, if s=2, then 𝜁(s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly 𝜋²/6. When s is a complex number—one that looks like a+b𝑖, using the imaginary number 𝑖—finding 𝜁(s) gets tricky.

So tricky, in fact, that it’s become the ultimate math question. Specifically, the Riemann Hypothesis is about when 𝜁(s)=0; the official statement is, “Every nontrivial zero of the Riemann zeta function has real part 1/2.” On the plane of complex numbers, this means the function has a certain behavior along a special vertical line. The hypothesis is that the behavior continues along that line infinitely.

✅ Stay Curious: How to Paint a Room Using Math

The Hypothesis and the zeta function come from German mathematician Bernhard Riemann, who described them in 1859. Riemann developed them while studying prime numbers and their distribution. Our understanding of prime numbers has flourished in the 160 years since, and Riemann would never have imagined the power of supercomputers. But lacking a solution to the Riemann Hypothesis is a major setback.

If the Riemann Hypothesis were solved tomorrow, it would unlock an avalanche of further progress. It would be huge news throughout the subjects of Number Theory and Analysis. Until then, the Riemann Hypothesis remains one of the largest dams to the river of math research.

The Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture is another of the six unsolved Millennium Prize Problems, and it’s the only other one we can remotely describe in plain English. This Conjecture involves the math topic known as Elliptic Curves.

When we recently wrote about the toughest math problems that have been solved , we mentioned one of the greatest achievements in 20th-century math: the solution to Fermat’s Last Theorem. Sir Andrew Wiles solved it using Elliptic Curves. So, you could call this a very powerful new branch of math.

✅ The Latest: Mathematicians Discovered Something Mind-Blowing About the Number 15

In a nutshell, an elliptic curve is a special kind of function. They take the unthreatening-looking form y²=x³+ax+b. It turns out functions like this have certain properties that cast insight into math topics like Algebra and Number Theory.

British mathematicians Bryan Birch and Peter Swinnerton-Dyer developed their conjecture in the 1960s. Its exact statement is very technical, and has evolved over the years. One of the main stewards of this evolution has been none other than Wiles. To see its current status and complexity, check out this famous update by Wells in 2006.

The Kissing Number Problem

A broad category of problems in math are called the Sphere Packing Problems. They range from pure math to practical applications, generally putting math terminology to the idea of stacking many spheres in a given space, like fruit at the grocery store. Some questions in this study have full solutions, while some simple ones leave us stumped, like the Kissing Number Problem.

When a bunch of spheres are packed in some region, each sphere has a Kissing Number, which is the number of other spheres it’s touching; if you’re touching 6 neighboring spheres, then your kissing number is 6. Nothing tricky. A packed bunch of spheres will have an average kissing number, which helps mathematically describe the situation. But a basic question about the kissing number stands unanswered.

✅ Miracles Happen: Mathematicians Finally Make a Breakthrough on the Ramsey Number

First, a note on dimensions. Dimensions have a specific meaning in math: they’re independent coordinate axes. The x-axis and y-axis show the two dimensions of a coordinate plane. When a character in a sci-fi show says they’re going to a different dimension, that doesn’t make mathematical sense. You can’t go to the x-axis.

A 1-dimensional thing is a line, and 2-dimensional thing is a plane. For these low numbers, mathematicians have proven the maximum possible kissing number for spheres of that many dimensions. It’s 2 when you’re on a 1-D line—one sphere to your left and the other to your right. There’s proof of an exact number for 3 dimensions, although that took until the 1950s.

Beyond 3 dimensions, the Kissing Problem is mostly unsolved. Mathematicians have slowly whittled the possibilities to fairly narrow ranges for up to 24 dimensions, with a few exactly known, as you can see on this chart . For larger numbers, or a general form, the problem is wide open. There are several hurdles to a full solution, including computational limitations. So expect incremental progress on this problem for years to come.

The Unknotting Problem

The simplest version of the Unknotting Problem has been solved, so there’s already some success with this story. Solving the full version of the problem will be an even bigger triumph.

You probably haven’t heard of the math subject Knot Theory . It ’s taught in virtually no high schools, and few colleges. The idea is to try and apply formal math ideas, like proofs, to knots, like … well, what you tie your shoes with.

For example, you might know how to tie a “square knot” and a “granny knot.” They have the same steps except that one twist is reversed from the square knot to the granny knot. But can you prove that those knots are different? Well, knot theorists can.

✅ Up Next: The Amazing Math Inside the Rubik’s Cube

Knot theorists’ holy grail problem was an algorithm to identify if some tangled mess is truly knotted, or if it can be disentangled to nothing. The cool news is that this has been accomplished! Several computer algorithms for this have been written in the last 20 years, and some of them even animate the process .

But the Unknotting Problem remains computational. In technical terms, it’s known that the Unknotting Problem is in NP, while we don ’ t know if it’s in P. That roughly means that we know our algorithms are capable of unknotting knots of any complexity, but that as they get more complicated, it starts to take an impossibly long time. For now.

If someone comes up with an algorithm that can unknot any knot in what’s called polynomial time, that will put the Unknotting Problem fully to rest. On the flip side, someone could prove that isn’t possible, and that the Unknotting Problem’s computational intensity is unavoidably profound. Eventually, we’ll find out.

The Large Cardinal Project

If you’ve never heard of Large Cardinals , get ready to learn. In the late 19th century, a German mathematician named Georg Cantor figured out that infinity comes in different sizes. Some infinite sets truly have more elements than others in a deep mathematical way, and Cantor proved it.

There is the first infinite size, the smallest infinity , which gets denoted ℵ₀. That’s a Hebrew letter aleph; it reads as “aleph-zero.” It’s the size of the set of natural numbers, so that gets written |ℕ|=ℵ₀.

Next, some common sets are larger than size ℵ₀. The major example Cantor proved is that the set of real numbers is bigger, written |ℝ|>ℵ₀. But the reals aren’t that big; we’re just getting started on the infinite sizes.

✅ More Mind-Blowing Stuff: Mathematicians Discovered a New 13-Sided Shape That Can Do Remarkable Things

For the really big stuff, mathematicians keep discovering larger and larger sizes, or what we call Large Cardinals. It’s a process of pure math that goes like this: Someone says, “I thought of a definition for a cardinal, and I can prove this cardinal is bigger than all the known cardinals.” Then, if their proof is good, that’s the new largest known cardinal. Until someone else comes up with a larger one.

Throughout the 20th century, the frontier of known large cardinals was steadily pushed forward. There’s now even a beautiful wiki of known large cardinals , named in honor of Cantor. So, will this ever end? The answer is broadly yes, although it gets very complicated.

In some senses, the top of the large cardinal hierarchy is in sight. Some theorems have been proven, which impose a sort of ceiling on the possibilities for large cardinals. But many open questions remain, and new cardinals have been nailed down as recently as 2019. It’s very possible we will be discovering more for decades to come. Hopefully we’ll eventually have a comprehensive list of all large cardinals.

What’s the Deal with 𝜋+e?

Given everything we know about two of math’s most famous constants, 𝜋 and e , it’s a bit surprising how lost we are when they’re added together.

This mystery is all about algebraic real numbers . The definition: A real number is algebraic if it’s the root of some polynomial with integer coefficients. For example, x²-6 is a polynomial with integer coefficients, since 1 and -6 are integers. The roots of x²-6=0 are x=√6 and x=-√6, so that means √6 and -√6 are algebraic numbers.

✅ Try It Yourself: Can You Solve This Viral Brain Teaser From TikTok?

All rational numbers, and roots of rational numbers, are algebraic. So it might feel like “most” real numbers are algebraic. Turns out, it’s actually the opposite. The antonym to algebraic is transcendental, and it turns out almost all real numbers are transcendental—for certain mathematical meanings of “almost all.” So who’s algebraic , and who’s transcendental?

The real number 𝜋 goes back to ancient math, while the number e has been around since the 17th century. You’ve probably heard of both, and you’d think we know the answer to every basic question to be asked about them, right?

Well, we do know that both 𝜋 and e are transcendental. But somehow it’s unknown whether 𝜋+e is algebraic or transcendental. Similarly, we don’t know about 𝜋e, 𝜋/e, and other simple combinations of them. So there are incredibly basic questions about numbers we’ve known for millennia that still remain mysterious.

Is 𝛾 Rational?

Here’s another problem that’s very easy to write, but hard to solve. All you need to recall is the definition of rational numbers.

Rational numbers can be written in the form p/q, where p and q are integers. So, 42 and -11/3 are rational, while 𝜋 and √2 are not. It’s a very basic property, so you’d think we can easily tell when a number is rational or not, right?

Meet the Euler-Mascheroni constant 𝛾, which is a lowercase Greek gamma. It’s a real number, approximately 0.5772, with a closed form that’s not terribly ugly; it looks like the image above.

✅ One More Thing: Teens Have Proven the Pythagorean Theorem With Trigonometry. That Should Be Impossible

The sleek way of putting words to those symbols is “gamma is the limit of the difference of the harmonic series and the natural log.” So, it’s a combination of two very well-understood mathematical objects. It has other neat closed forms, and appears in hundreds of formulas.

But somehow, we don’t even know if 𝛾 is rational. We’ve calculated it to half a trillion digits, yet nobody can prove if it’s rational or not. The popular prediction is that 𝛾 is irrational. Along with our previous example 𝜋+e, we have another question of a simple property for a well-known number, and we can’t even answer it.

Dave Linkletter is a Ph.D. candidate in Pure Mathematics at the University of Nevada, Las Vegas. His research is in Large Cardinal Set Theory. He also teaches undergrad classes, and enjoys breaking down popular math topics for wide audiences.

.css-cuqpxl:before{padding-right:0.3125rem;content:'//';display:inline;} Pop Mech Pro .css-xtujxj:before{padding-left:0.3125rem;content:'//';display:inline;}

How to Live Forever, or Die Trying

The Army & Navy Working Together Defense Missiles

How Russia Copied America’s Deadliest Missile

North Korea Sends “Poop Balloons” Into South Korea

Plane Flown by 'Ace of Aces' Pilot Finally Found

The F-35 Is Still One of the Safest Planes

The Air Force’s Upcoming “Doomsday Plane” Replacem

Meet the Marines' New Kamikaze Drone

Did Russia Launch a Weapon Near a U.S. Satellite?

A Groundbreaking Discovery For Interstellar Travel

America's Missile Defense Has Entered a New Era

Applied Mathematics Research

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

Applied Mathematics Fields

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

Applied Math Committee

Numbers, Facts and Trends Shaping Your World

Read our research on:

Full Topic List

Regions & Countries

• Publications
• Our Methods
• Short Reads
• Tools & Resources

Read Our Research On:

U.S. public, private and charter schools in 5 charts

While children in the United States are guaranteed a free education at their local public school through state constitutional law, many families weigh other educational options for their children. Even before the coronavirus pandemic upended families’ usual routines, 36% of parents with K-12 students say they considered multiple schools for their child in the 2018-19 school year.

Students’ school environments vary widely – sometimes even for children living in the same community – depending on whether they attend traditional public, private or charter schools.

Here are some key distinctions between these three types of schools, based on data from the National Center for Education Statistics (NCES). All figures reflect the most recent school year with data for all three types of schools.

Pew Research Center conducted this analysis to better understand how U.S. students’ school experiences might differ depending on whether they attend a traditional public school, a private school or a charter school.

Data comes from the Institute of Education Sciences’ National Center for Education Statistics . We used the most recent year in which data is available for all three school types.

Public, private and charter schools include those that teach students in kindergarten through 12th grade, unless otherwise specified. Racial categories used in this analysis include those who report being a single race and non-Hispanic. Hispanics are of any race. National School Lunch Program data for 2021-22 is not available for Alaska.

What’s the difference between public, private and charter schools?

Until a few decades ago, parents with kids in elementary, middle or high school could choose to send them to either a traditional public school or a private one. More recently, many states have added a third option: public charter schools.

• Traditional public schools are taxpayer funded, are tuition free and must adhere to standards set by a school district or state board of education. These are the most common schooling option in the U.S.
• Private schools are known for being selective, religiously affiliated or sometimes both, and charge tuition rather than receive public money. In addition to tuition dollars, private schools may be funded through a combination of donations, endowments or grants from other private sources. As a result, they have more autonomy when it comes to curriculum and other academic standards. During the 2021-22 school year, about three-quarters of private school K-12 students (77%) attended a religiously affiliated school. The largest share went to Catholic schools, which accounted for 35% of all private school enrollment. Another 23% of private school students attended secular institutions.
• Public charter schools are legally allowed to operate in nearly all states, plus the District of Columbia, as of 2024. Like traditional public schools, these are taxpayer funded and tuition free. They’re open to any student who wishes to enroll. But unlike their traditional counterparts, agreements – or charters – with the state or local government allow them flexibility when it comes to curriculum and other standards. They also may turn students away due to space constraints.

Differences exist in the size and locale of each type of school, NCES data from the 2021-22 school year shows.

Traditional public schools tend to be larger than the other types. For instance, 39% of public schools enroll 500 or more students, compared with 32% of charter schools and 8% of private schools. And while 31% of public schools have fewer than 300 students, 44% of charter schools and 82% of private schools do.

Public schools are relatively evenly distributed across urban, suburban and rural areas, while most charter and private school campuses are located in either cities or suburbs.

(Traditional public and charter school environment data includes prekindergarten students, who account for less than 1% of enrollment at these types of schools.)

Where is enrollment growing and shrinking?

During the 2021-22 school year, the vast majority of the country’s roughly 54.6 million public, private and charter school students in pre-K through 12th grade (83%) attended traditional public schools. Another 10% were enrolled in private schools, and 7% went to public charter schools.

Enrollment numbers have shifted over the last decade:

• Traditional public school enrollment has declined. In fall 2011, about 47.2 million students attended public elementary, middle and secondary schools, accounting for 87% of all school enrollment. By fall 2021, the number of public school students dropped to about 45.4 million, resulting in a small drop in public schools’ share of total enrollment.
• The popularity of charter schools has grown. Minnesota became the first state to pass legislation allowing charter schools in 1991. In the last 10 years alone, enrollment has risen from about 2.1 million students in fall 2011 to nearly 3.7 million in fall 2021, an increase from 4% to 7% of total enrollment.
• Private school enrollment has held relatively steady. Private school students have consistently made up about 10% of school enrollment, with numbers that have fluctuated from a 10-year low of fewer than 5.3 million in 2011 to a peak of almost 5.8 million in 2015.

How does enrollment look at the state level?

Nationwide, the vast majority of students in pre-K through 12th grade attend traditional public schools – but shares vary somewhat from state to state. In Wyoming, for example, nearly all students (97%) attend public school, while 45% do in D.C.

The states with the highest percentages of public school enrollment include some of those with the lowest population density . In addition to Wyoming, West Virginia (95%), Montana (93%), Kansas and Alaska (91% each) round out the top five states by share of public school enrollment.

In most states, students are more likely to attend a private school than a charter school. Charter school students make up a larger share of enrollment than private school students in just 12 states and D.C. (Data is unavailable for seven states because they did not have any charter schools or legislation allowing them in fall 2021.)

Among the places where students are the least likely to attend traditional public schools:

• D.C. has the highest share of charter school students, at 36%. Just 45% of K-12 students there attend traditional public schools. Another 19% attend private schools.
• D.C. and Hawaii have the largest percentage of students in private schools, at 19% each. In Hawaii, another 76% of students are enrolled in public school, and 6% are enrolled in charter schools.

How do student demographics vary by school type?

Charter schools had the most racial and ethnic diversity during the 2021-22 school year. Hispanic students make up the largest share of enrollment there (36%), followed by White (29%), Black (24%) and Asian American students (4%).

In contrast, 47% of traditional public school students and 65% of private school students are White. Smaller shares are Hispanic, Black or Asian.

Differences also exist by household income level. Nearly all public and charter schools are part of the National School Lunch Program , which provides free or reduced-price meals to students based on family income.

In general, charter school students are more likely than public school kids to qualify for the program. For instance, 31% of charter students and 21% of traditional public school students are enrolled at a school where more than three-quarters of their peers qualify for free or reduced-price lunch.

Because a relatively small share of private schools participate in this program, 2021-22 data is not available for them. However, research shows that private school enrollment rates are highest among upper-income families .

What does the teaching staff look like at each type of school?

More than 4.2 million full- and part-time teachers worked at public, private and charter schools during the 2020-21 school year, the most recent year with available data. That year, about 3.5 million teachers (83%) taught at traditional public schools. Another 466,000 (11%) worked in private schools, and 251,000 (6%) taught at public charters.

The teaching force in each environment varies based on race and ethnicity, age, experience, and educational attainment.

• Charter school teachers are the most racially and ethnically diverse: 69% of charter school teachers are White, compared with about eight-in-ten at both traditional public and private schools. Charters also employ the largest shares of Black and Hispanic teachers.
• Private school teachers skew slightly older, while charter school teachers are the youngest: About 17% of private school teachers are ages 60 and older, compared with 8% in public schools and 7% in charter schools. And in charter schools, 21% of teachers are under 30, compared with 14% each in public and private schools.
• Charters employ a larger share of teachers with fewer years of experience: For instance, 13% of both private and charter school teachers have fewer than three years of experience, compared with 7% of public school teachers. And 43% of charter school teachers have between three and nine years of experience, compared with 28% each in public and private schools.
• Public school teachers are the most likely to have a master’s degree: 52% of public school teachers have a master’s degree, compared with about 41% each in charter and private schools.

Katherine Schaeffer is a research analyst at Pew Research Center .

A quarter of U.S. teachers say AI tools do more harm than good in K-12 education

Most americans think u.s. k-12 stem education isn’t above average, but test results paint a mixed picture, about 1 in 4 u.s. teachers say their school went into a gun-related lockdown in the last school year, about half of americans say public k-12 education is going in the wrong direction, what public k-12 teachers want americans to know about teaching, most popular.

1615 L St. NW, Suite 800 Washington, DC 20036 USA (+1) 202-419-4300 | Main (+1) 202-857-8562 | Fax (+1) 202-419-4372 |  Media Inquiries

Research Topics

• Email Newsletters

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 .

© 2024 Pew Research Center