what are good topics for research in 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

260 Interesting Math Topics for Essays & Research Papers

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

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

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

🔝 Top 10 Interesting Math Topics

✅ branches of mathematics, ✨ fun math topics.

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

🔍 References

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

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

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

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

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

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

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

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

🏫 Math Essay Topics for High School

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

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

🎓 College Math Topics for a Paper

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

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

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

🤔 Advanced Topics in Math to Write a Paper on

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

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

📚 Math Research Topics for a Paper

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

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

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

✏️ Math Education Research Topics

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

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

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

🧮 Algebra Topics for a Paper

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

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

📏 Geometry Topics for a Research Paper

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

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

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

➗ Calculus Topics to Write a Paper on

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

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

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

💵 Business Math Topics to Write About

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

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

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

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

Cause and effect essays examine how an event happened and what consequences it had. Gaining weight after eating lots of fast food is an example of a cause-and-effect relationship. Possible topics cover a variety of subjects ranging from mental health to history and politics. This article gives you an outline...

An analysis essay aims to break down the subject in order to understand it. You can choose to analyze a text, a process, or an idea. This article will help you write a great essay! Selecting an interesting topic makes writing a lot easier. We’ve prepared a list of excellent...

Everybody knows that being healthy requires effort. We should exercise regularly and maintain a balanced diet. However, the reward is worth it. A healthy lifestyle prevents chronic illnesses and leads to better body performance. Besides, if you improve your physical well-being, your mental health will strengthen as well! In this...

Environment affects us all, whether we want it or not. Political leaders and students alike discuss ways to tackle environmental topics & issues. Some might argue about the role humans play in all this. The fact remains that our environment is a delicate matter. That’s why we must educate ourselves...

Our code of ethics is derived from what we think is right or wrong. On top of that, we have to agree to the moral standards established by the society we live in. Conventional norms generally label theft, murder, or harassment as bad. However, there are many influences that impact...

A definition explains the meaning of a term or a concept. In a dictionary, you’ll find a definition in a single sentence. A definition paper, however, encompasses several paragraphs. Such an essay, amongst other things, can include personal experience and examples. To write a successful definition paper, you need to...

As simple as it is, the purpose of the descriptive essay is to explain or portray its subject. It can focus on any topic or issue you want to write about. Be sure that any middle school, high school, or college student can manage this type of creative writing assignment!...

Rhetorical analysis essay focuses on assessing the method used for delivering a message. This assignment isn’t about giving an opinion on the topic. The purpose is to analyze how the author presents the argument and whether or not they succeeded. Keep reading to find out more strategies and prompts for...

A narrative essay tells a story about a series of events. At the core of this kind of essay can be a personal experience or a fictional plot. Any story can be a basis for a narrative essay! Narratives can look similar to descriptions. Still, they are different. A descriptive...

Similar to the instructions in a recipe book, process essays convey information in a step-by-step format. In this type of paper, you follow a structured chronological process. You can also call it a how-to essay. A closely related type is a process analysis essay. Here you have to carefully consider...

In a classification essay, you divide the subject into categories. To create these categories, you single out certain attributes of things. You can classify them according to their characteristics, themes, or traits. Sounds complicated? Be sure that any high school or college student can manage this type of essay!

Throughout your high school years, you are likely to write many evaluative papers. In an evaluation essay you aim is to justify your point of view through evidence.

I need a writer on algebra. I am a PhD student.Can i be helped by anybody/expert?

Please I want to do my MPhil research on algebra if you can help me

shall your university help me on research in mathematics ?

how I get the full pdf of those tittles

Print as pdf.

220 Brilliant Math Research Topics and Ideas for Students

Table of Contents

Do you have to submit a math research paper? Are you looking for the best math research topics? Well, in this blog post, we have shared a list of 220 interesting math research topics to consider for assignments and academic projects. If you are a student who is pursuing a degree in mathematics, then you can very well use the topic ideas suggested here. Also, you can check this blog post and get to know the important steps for writing a brilliant math research paper.

What is Mathematics?

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

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

Different Branches of Mathematics

Listed below are some popular branches of mathematics.

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

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

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

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

How to Write a Math Research Paper

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

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

Topic Selection

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

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

Research Paper Writing

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

Introduction

Implications.

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

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

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

Tips for Choosing a Good Math Research Topic

It might be confusing for you to identify a good topic for your math research paper, but by following these tips you can spot the appropriate topic for your study.

• Identify which area in mathematics you are interested in and then pick a topic relevant to it. For instance, you may perform math research on topics from areas such as number theory, geometry, algebra, and so on.
• To generate math research paper topics, look for open problems or conjectures in math journals, conferences, or online forums.
• Select a math research topic that is doable and has sufficient resources for the study.
• Give preference to math research ideas that have a close connection with other fields or any real-world applications.
• If the topic you have chosen is broad, narrow it down. Always focus on a specific research topic or a question with a good scope for discussion and analysis.
• Make sure the math research topic you have selected is original and can be approached from a fresh perspective
• Discuss your math research idea with professors, researchers, or classmates and then obtain their insight and feedback. This will help you refine your idea and come up with high-quality work.

List of the Best Math Research Topics

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

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

Simple and Easy Math Topics

If you want to perform math research effortlessly, then take into account any of the topics based on the fundamental concepts of mathematics rather than complex ideas. Listed below are a few simple math study topics that will be easy for you to examine.

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

Basic Math Topics for Middle School Students

Do you need math research topics for middle school students? Take a look below. In the list, we have suggested some fascinating research topics on mathematics for middle school students to get started.

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

Interesting Math Topics for High School Students

If it is challenging for you to identify a good math research topic for your high school assignment, make use of the list published here. For high school students, we have provided some interesting math research questions in the list to explore.

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

Math Research Paper Topics for Undergraduate Students

Are you an undergraduate student searching for math research ideas? Get help from the list uploaded below. To make the topic selection easier for students, in the list, we have added some exclusive research paper topics and ideas on different math disciplines.

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

Math Research Topics for College Students

In this section, we have provided a list of incredible math research topics for college students. If you require a perfect math research topic for your college assignment, from the list, choose any study topic that aligns with your needs and explore it extensively.

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

Applied Math Research Topics

Applied Mathematics is the practical application of mathematical principles to solve real-world issues in domains such as physics, engineering, and economics. If you wish to explore practical issues in your math research paper, then you may work on any of the below-listed applied math study topics.

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

Business Math Research Topics

Business Math typically focuses on the mathematical principles and methods used to analyze and solve problems in finance, accounting, and management. In case, you are passionate about finance-related things, then in your research paper, you may investigate any of these business math topics.

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

Advanced Math Research Paper Topics

In your math research paper, you may examine and write about any of the below-listed advanced math topics. Calculus, differential equations, and number theory are some examples of advanced mathematical principles used in specific domains.

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

Complex Math Research Topics

Complex Math is the study of mathematical structures including non-real numbers such as complex analysis and complex geometry. Listed below are a few complex math study topics you may investigate in your mathematics research paper.

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

Math Research Ideas on Probability and Statistics

Probability and Statistics focus on the study of chance events, data analysis, and inference for making educated judgments. Here are some amazing research questions on probability and statistics you may examine in your math thesis.

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

Algebra Research Topics

Algebra deals with the study of variables, equations, and functions used to solve systems and model relationships. If you are fascinated by numbers, then for your math research, you may choose any of the below-listed study topics on algebra.

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

Math Research Paper Topics on Geometry

Geometry is the study of object shapes, sizes, and placements, with a focus on qualities and relationships. In case, you are curious about examining shapes and sizes, then in your math research paper, you may address any of these geometry topics.

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

Calculus Research Topics

Calculus studies the rates of change and accumulation, including differential and integral calculus. When it comes to preparing your math research paper, take into consideration any of the below-suggested topics on calculus.

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

Popular Math Research Topics

To prepare your math research paper, take into account any of the below-listed popular study topics. But when you examine a topic that has been commonly chosen by others, analyze it from various dimensions and come up with optimized solutions.

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

Latest Math Research Topics

Listed below are some trending research ideas on mathematics you may consider for your studies. By preparing math research papers on the latest topics or real-world problems, you may promote innovation, advance mathematical knowledge, and drive progress in various associated fields.

• What does point zero reflect on a graph where the vertical and horizontal lines meet?
• How to recognize adjacent angles easily without any trouble?
• Compare the differential vs. analytic geometry by citing relevant examples.
• Explain how to use a graphics system for solving various types of equations.
• How to divide the feasible and non-feasible regions in linear programming?
• What are confidence intervals and how it help in statistical math?
• How to differentiate the effect of a magnetic field on a given point of the circle by using an appropriate differential formula?
• What are the different types of identities that are used in trigonometric functions?
• Why polynomials are difficult to solve as compared to monomials? Give examples.
• Explain radical expressions and their significance with examples.
• Explain how fractal geometry can be used to model natural phenomena.
• Discuss the underlying patterns in prime number distribution.
• How can machine learning be optimized using mathematical techniques?
• What new cryptographic algorithms can be developed using number theory?
• What combinatorial optimization techniques can solve complex problems?
• Explain how to efficiently solve partial differential equations
• What efficient algorithms can be developed for computational algebraic geometry?
• How do geometric groups behave in various mathematical structures?
• Explain how can complex systems be modeled and predicted using mathematical techniques
• Explore what graph theory algorithms can be applied to real-world network analysis.

The Bottom Line

The list of research topics we have suggested in this blog shows how much more is there to discover in math. Particularly, from basic questions in number theory and algebra to practical applications in machine learning and data science, the math research topics we have recommended above will allow you to explore and learn more. So, for your math research, from our list, feel free to pick any topic that is related to your area of interest and study objectives. Examining the math research topic of your passion will allow you to gain new insights, create new technologies, and solve complex problems. If it is tough for you to conduct an extensive study and prepare a research paper on math topics, then get guidance from experienced mathematicians on our team and finish your task with accurate answers.

Related Post

240 Best Dissertation Topics on Various Fields of Study

140 Intriguing Media Dissertation Topics and Ideas

List of Amazing Wedding Speech Topics and Ideas

About author.

Jacob Smith

I am an Academic Writer and have affection to share my knowledge through posts’. I do not feel tiredness while research and analyzing the things. Sometime, I write down hundred of research topics as per the students requirements. I want to share solution oriented content to the students.

Comments are closed.

• Featured Posts

140 Unique Geology Research Topics to Focus On

200+ outstanding world history topics and ideas 2023, 190 excellent ap research topics and ideas, 150+ trending group discussion topics and ideas, 170 funny speech topics to blow the minds of audience, who invented exams learn the history of examination, how to focus on reading 15 effective tips for better concentration, what is a rhetorical analysis essay and how to write it, primary school teacher in australia- eligibility, job role, career options, and salary, 4 steps to build a flawless business letter format, get help instantly.

Raise Your Grades with Assignment Help Pro

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

13 Best Tips To Write An Assignment

Whenever the new semester starts, you will get a lot of assignment writing tasks. Now you enter the new academic…

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…

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  

33k Accesses

100 Citations

18 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

Explore related subjects.

• Artificial Intelligence
• Medical Ethics

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

Mathematics Research Paper Topics

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

Academic Writing, Editing, Proofreading, And Problem Solving Services

Get 10% off with 24start discount code.

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

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

Fields of Mathematics

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

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

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

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

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

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

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

Back to Science Research Paper Topics .

ORDER HIGH QUALITY CUSTOM PAPER

• 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

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

Pure mathematics articles from across Nature Portfolio

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

Latest Research and Reviews

Various distance between generalized Diophantine fuzzy sets using multiple criteria decision making and their real life applications

• Murugan Palanikumar
• Nasreen Kausar
• Fikadu Tesgera Tolasa

AI-driven research in pure mathematics and theoretical physics

Advances in artificial-intelligence-assisted mathematical investigations suggest that human–machine collaboration will be an integral part of future theoretical research.

• Yang-Hui He

High accuracy solutions for the Pochhammer–Chree equation in elastic media

• Mostafa M. A. Khater
• Suleman H. Alfalqi

Calibration estimation of population total using multi-auxiliary information in the presence of non-response

• Anant Patel
• Menakshi Pachori

Optimizing decision-making with aggregation operators for generalized intuitionistic fuzzy sets and their applications in the tech industry

• Muhammad Wasim
• Awais Yousaf
• Hamiden Abd El-Wahed Khalifa

A new multi-attribute group decision-making method based on Einstein Bonferroni operators under interval-valued Fermatean hesitant fuzzy environment

• Jasni Mohamad Zain

News and Comment

The real value of numbers

• Mark Buchanan

Machine learning to guide mathematicians

• Fernando Chirigati

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

• Matthew B. Hastings

e is everywhere

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

• Stefanie Reichert

Imagination captured

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

• Abigail Klopper

Prime interference

• David Abergel

Quick links

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

Research Areas

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

An editorially independent publication supported by the Simons Foundation.

Get the latest news delivered to your inbox.

Type search term(s) and press enter

• Comment Comments
• Save Article Read Later Read Later

What Makes for ‘Good’ Mathematics?

February 1, 2024

Peter Greenwood for  Quanta Magazine

Introduction

We tend to think of mathematics as purely logical, but the teaching of math, its values, its usefulness and its workings are packed with nuance. So what is “good” mathematics? In 2007, the mathematician Terence Tao wrote an essay for the Bulletin of the American Mathematical Society that sought to answer this question. Today, as the recipient of a Fields Medal, a Breakthrough Prize in Mathematics and a MacArthur Fellowship, Tao is one of the most honored and prolific mathematicians alive. In this episode, he joins our host and fellow mathematician Steven Strogatz to revisit the makings of good mathematics.

Listen on  Apple Podcasts ,  Spotify ,  Google Podcasts , TuneIn  or your favorite podcasting app, or you can  stream it from  Quanta .

STEVEN STROGATZ : Back in October 2007, way back when the first-generation iPhone was still a hot commodity and the stock market was at an all-time high before the Great Recession, Terence Tao, a professor of math at UCLA, was determined to answer a question that had long been debated among mathematicians: What exactly is good mathematics?

Is it about rigor? Elegance? Real-world utility? Terry wrote a very thoughtful and generous, I would even say openhearted, essay about all the ways that math could be good. But now, more than 15 years later, do we need to rethink what good mathematics is?

I’m Steve Strogatz, and this is “The Joy of Why,” a podcast from Quanta Magazine where my co-host, Janna Levin, and I take turns exploring some of the biggest unanswered questions in math and science today.

( Theme plays )

Here today to revisit the eternal question of what makes math good is Terry Tao himself. Professor Tao has authored more than 300 research papers on an amazingly wide swath of mathematics including harmonic analysis, partial differential equations, combinatorics, number theory, data science, random matrices and much more. He’s been referred to as the “Mozart of Mathematics.” And as the winner of a Fields Medal, a Breakthrough Prize in Mathematics, a MacArthur Fellowship and many other awards, that moniker is certainly well-deserved.

Terry, welcome to “The Joy of Why.”

TERENCE TAO : Pleasure to be here.

STROGATZ : I’m very excited to be able to talk to you about this question of what it is that makes some types of mathematical research good. I can remember pretty vividly flipping through the Bulletin of the American Math Society back in 2007 and coming across your essay about this issue that you posed for us. It’s something that all mathematicians think about. But for people out there who may not be so familiar, could you tell us, how did you land on this question? How did you define good math back at the time?

TAO : Right, yes. It was actually a solicitation. So the editor of the Bulletin at the time had asked me to contribute an article. I think I had a very naive idea of what mathematics was as a student. I kind of had this idea that there was some sort of council of greybeards that would hand out problems for people to work on. And it was kind of a shock to me as a graduate student, realizing that there wasn’t actually this central authority to hand out problems, and people did self-directed research.

I kept going to talks and listening to how other mathematicians talked about what they find exciting and what makes them excited about math, and the fact that each mathematician has a different way of approaching mathematics. Like, some would pursue applications, some by sort of aesthetic beauty, some by just problem solving. They wanted to solve a problem and they would focus on sort of the most difficult, the most challenging tasks. Some would focus on technique; some would try to make things as elegant as possible.

But what struck me when sort of listening to so many of these different mathematicians talk about what they find valuable in mathematics is that, even though we all had sort of different ideals as to what good mathematics should look like, they all kind of tend to converge to the same thing.

If a piece of mathematics is really good, people who pursue beauty will eventually happen across it. People who pursue, who value, you know, technical power or applications will eventually land upon it.

Eugene Wigner had a very famous essay on the unreasonable effectiveness of mathematics in the physical sciences almost a century ago, where he just observed that there were areas of mathematics — for example, Riemannian geometry, the study of curved space — that was initially just a purely theoretical exercise to mathematicians, you know, trying to prove the parallel postulate and so forth, turning out to be precisely what Einstein and Poincaré and Hilbert needed to describe the mathematics of general relativity. And that’s just a phenomenon that occurs.

So it’s not just that mathematics, that [what] mathematicians find intellectually interesting end up being physically important. But even within mathematics, subjects that mathematicians find elegant also happen to provide deep insight.

What I feel like is that, you know, there is some platonic good mathematics out there, and all our different value systems are just different ways of accessing that objective good stuff.

STROGATZ : That’s very interesting. Being a sort of person inclined to platonic thinking myself, I’m tempted to agree. Although I’m a little surprised to hear you say that, because I would have thought where you were going initially seemed to be, like, there are so many different points of view about this. It is an interesting fact, though, kind of an empirical fact, that we do converge on agreeing about what is good or not good, even though, as you say, we come at it from so many different values.

TAO : Right. The convergence may take time. You know, so there are definitely fields, for example, where they look a lot better as measured by one metric than others. Like maybe they have a lot of applications, but their presentation is extremely disgusting, you know.

(Strogatz laughs)

Or things that are very elegant but don’t yet have many good applications in the real world. But I do feel like eventually it will converge.

STROGATZ : Well, let me ask you about this point of contact with the real world. It’s an interesting tension in math. And, you know, as little kids, let’s say, when we first learn about geometry, you might think at that point that triangles are real, or circles or straight lines are real, and that they can tell you about the rectangular shapes you see in buildings out in the world, or that surveyors need to use geometry. And after all, the word comes from the measurement of the Earth, right, “geometry.” And so, there was a time when geometry was empirical.

But what I wanted to ask you has to do with a comment that John von Neumann made. So von Neumann, for anyone not familiar, was himself a great mathematician. And he made this comment in this essay, “ The Mathematician ,” about the relationship between math and the empirical world, the real world, where he says roughly that mathematical ideas originate in empirics, but that at some point, once you get the mathematical ideas, the subject starts to take on a life of its own. And then it’s more like a creative piece of art. Aesthetic criteria become important. But he says that causes danger. That when a subject starts to become too far removed from its empirical source, like especially in its second or third generation, he says that there’s a chance the subject can suffer from too much abstract inbreeding and it’s in danger of degeneration.

Any thoughts about that? I mean, does math have to stay in contact with its empirical source?

TAO: Yes, I think it does have to be grounded. When I say that, empirically, all these different ways of doing mathematics do converge, it’s only because — this only happens when the subject is healthy. So, you know, the good news is that usually it is.

But, for example, mathematicians value short proofs over long ones, all other things being equal. But one could imagine people going overboard and, like, one subfield of mathematics being obsessed with making proofs as short as possible and having these extremely opaque two-line proofs of deep theorems. And they make it kind of this contest, and then it becomes this sort of abstruse game and then you lose all the intuition. You lose maybe deeper understanding because you’re just so obsessed with making all your proofs as short as possible. Now, this doesn’t actually happen in practice. But this is kind of a theoretical example, and I think von Neumann was making a similar point.

And in the sixties and seventies, like, there was an era of mathematics where abstraction was making huge strides in simplifying and unifying a lot of mathematics that was previously very empirical. Especially in algebra, people were realizing, you know, numbers and polynomials and many other objects that were previously treated separately, you could all think of them as members of the same algebraic class, in this case a ring.

And a lot of progress in mathematics was being made by finding the right abstraction, you know, whether it was a topological space or a vector space, whatever, and proving theorems in great generality. And this is sometimes what we call the Bourbaki era in mathematics. And it did veer a little bit too far from being grounded.

We of course had, like, the whole New Math episode in the States, where educators tried to teach math in the Bourbaki style and eventually realized that that was not the appropriate pedagogy at that level.

But now the pendulum has swung back quite a bit. We have kind of — the subject has matured quite a bit and every field of mathematics, geometry, topology, whatever, we have kind of satisfactory formalizations and we kind of know what the right abstractions are. And now the field is again focusing on interconnections and applications. It’s connecting much more to the real world now.

I mean, not just sort of physics, which is a traditional connection, but, you know, computer science, life sciences, social sciences, you know. With the rise of big data, pretty much almost any human discipline now can be mathematized to some extent.

STROGATZ : I’m very interested in the word that you just used a minute ago about “interconnections,” because that seems like a central point for us to discuss. It’s something that you mention in your essay that, along with these, what you call “local” criteria about elegance, or real-world applications, or whatever, you mention this “global” aspect of good mathematics: that good mathematics connects to other good mathematics.

That’s almost key to what makes it good, that it’s integrated with other parts. But it’s interesting because it sounds almost like circular reasoning: that good math is the math that connects to other good math. But it’s a really powerful idea, and I’m just wondering if you could expand on it a little bit more.

TAO : Yeah, so, I mean, what mathematics is about — one of the things that mathematics does is that it makes connections that are very basic and fundamental, but not obvious if you just look at it from the surface level. A very early example of this is Descartes’ invention of Cartesian coordinates that made a fundamental connection between geometry — the study of points and lines and spatial objects — and numbers, algebra.

So, for example, a circle you can think of as a geometric object, but you can also think of it as an equation: x 2 + y 2 = 1 is the equation of a circle. At the time, it was a very revolutionary connection. You know, the ancient Greeks viewed number theory and geometry as almost completely disjointed subjects.

But with Descartes, there was this fundamental connection. And now it’s internalized; you know, the way we teach mathematics. It’s not surprising anymore that if you have a geometric problem, you attack it with numbers. Or if you have a problem with numbers, you may attack it with geometry.

It’s somewhat because both geometry and numbers are aspects of the same mathematical concept. We have an entire field called algebraic geometry, which is neither algebra nor geometry, but it’s a unified subject studying objects that you can either think of as geometric shapes, like lines and circles and so forth, or as equations.

But really, it’s a holistic union of the two that we study. And as the subject has deepened, we’ve realized that that is more fundamental somehow than either algebra or geometry separately, in some ways. So, these connections are helping us discover sort of the real mathematics that initially, somehow, our empirical studies only give us a corner of the subject.

There’s this famous parable of the elephant, I forget where, that if you have… There are four blind men, and they discover an elephant. And one of them feels the leg of the elephant and they think, “Oh, this, it’s very rough. It must be like a tree or something.”

And one of them feels the trunk, and it’s only a lot later that they see that there’s a single elephant object that is explaining all their separate hypotheses. Yeah, so we’re all blind initially, you know. We’re just watching the shadows on Plato’s cave and only later realizing —

STROGATZ : Wow, you are very philosophical here. This is something. I can’t resist now: If you’re going to start talking about the elephant and the blind people, this suggests that you think mathematics is out there — that it is something like the elephant and that we are the blind… Or, you know, we’re trying to see something that exists independent of human beings. Is that really what you believe?

TAO : When you do good mathematics, like, it’s not just pushing symbols around. You do feel like there is some actual object that you’re trying to understand, and all our equations we have are just sort of approximations of that, or shadows.

You can debate the philosophical point of what is actually reality and so forth. I mean, these are things you can actually touch, and the more real things get mathematically, sometimes the less physical they seem. As you said, geometry initially, you know, was a very tangible thing about objects in physical space that you could — you know, you can actually build a circle and a square and so forth.

But in modern geometry, you know, we work in higher dimensions. We can talk about discrete geometries, all kinds of wacky topologies. And, I mean, the subject still deserves to be called geometry, even though there is no Earth being measured anymore. The ancient Greek etymology is very outdated but it’s, but there’s definitely something there. Whether — how real you want to call it. But I guess the point is that for the purpose of actually doing mathematics, it helps to believe it’s real.

STROGATZ : Yeah, isn’t that interesting? It does. It seems like that’s something that goes very deep in the history of math. I was struck by an essay by Archimedes writing to his friend, or at least colleague, Eratosthenes.

We’re talking now, like, 250 B.C. And he makes the remark, he’s discovered a way to find the area of what we would call the segment of a parabola. He’s taking a parabola, he cuts across it with a line segment that’s at an oblique angle to the axis of the parabola, and he figures out this area. He gets a very beautiful result. But he says something to Eratosthenes like, “These results were inherent in the figures all along.” You know, like, they’re there. They’re there. They’re just waiting for him to find.

It’s not like he created them. It’s not like poetry. I mean, it’s interesting, actually, isn’t it? That a lot of great artists — Michelangelo talked about releasing the statue from the stone, you know, as if it were in there to begin with. And it sounds like you and many other great mathematicians have — as you say, it’s very useful to believe this idea, that it’s there waiting for us, waiting for the right minds to discover it.

TAO : Right. Well, I think one manifestation of that is that ideas that are often very complicated to explain when they’re first discovered, they get simplified. I mean, you know, often the reason why something looks very deep or difficult at the beginning is you don’t have the right notation.

For example, we have decimal notation now to manipulate numbers, and it’s very convenient. But in the past, we had like, you know, Roman numerals and then there were even more primitive number systems that were just really, really difficult to work with if you wanted to do mathematics.

Euclid’s Elements , you know — some of the arguments in these ancient texts. Like, there’s one theorem in Euclid’s Elements I think called the Bridge of Fools or something. It’s like the statement that, I think the statement is like an isosceles triangle, the two base angles are equal. Like, this is like a two-line proof in modern geometric texts, you know, with the right axioms. But Euclid had this horrendous way of doing it. And it was where many students of geometry in the classical era just completely gave up on mathematics.

STROGATZ : True. ( laughs )

TAO : But, you know, we now have a much better way of doing that. So often the complications we see in mathematics are artifacts of our own limitations. And, so, as we mature, you know, things become simpler. And it feels more real because of that. We’re not seeing the artifacts. We’re seeing the essence.

STROGATZ : Well, so going back to your essay: When you wrote it, at the time — I mean, this was pretty early in your career, not the very beginning, but still. Why did you feel back then that it was important to try to define what good mathematics was?

TAO : I think… So by that point, I was already starting to advise graduate students, and I was noticing that, you know, there was some misconceptions about, sort of, what is good and what is not. And I was also talking to mathematicians in different fields, and what one’s field valued in mathematics seemed different from others. But yet, somehow we were all studying the same subject.

And sometimes someone would say something that sort of rubbed me the wrong way, you know, like, “This mathematics has no applications, therefore it has no value.” Or “This proof is just too complicated; therefore it has no value,” or something. Or conversely, you know, “This proof is too simple; therefore it is not worth…” You know. Like, there was some, like, sort of snobbery and so forth, sometimes I would encounter.

And in my experience, the best mathematics came when I understood a different point of view, a different way of thinking about mathematics from someone in a different field and applying it to a problem that I cared about. And so my experience of how to use mathematics properly, how to wield it, was so different from these — sort of the “one true way of doing mathematics.”

I felt like this point had to be made somehow. That there’s really a plural way of doing mathematics, but whereas mathematics is still united.

STROGATZ : That’s very revealing, because I had wondered, you know, like, in my introduction I mentioned the many different branches of math that you have explored, and I didn’t even include some. Like, I can remember just a few years ago, your work about this mystery in fluid dynamics, about whether certain equations that we think do a good job of approximating the motions of water and air. I don’t want to go into details too much, but just to say, here you are, people think of you doing number theory or harmonic analysis, and suddenly you’re working on fluid dynamics questions. I mean, I realize it’s partial differential equations. But still, your breadth of interest seems to be related to your breadth of accepting different insights, different valuable ideas from all the different ways of doing good math.

TAO : I forget who said it, but there are two types of mathematicians. There’s hedgehogs and foxes. A fox is someone who knows a little bit about everything. A hedgehog is a creature that knows one thing very, very well. And neither is better than the other. They complement each other. I mean, in mathematics, you need people who are really deep domain experts in one subfield, and they know a subject inside-out. And you need people who can see the connections between one field and another. So I definitely identify as a fox, but I work with a lot of hedgehogs. The work I’m most proud of is often a collaboration like that.

STROGATZ : Oh, yeah. Do they realize that they’re hedgehogs?

TAO : Well, okay, the roles change over time. Like, there are other collaborations where I’m the hedgehog and someone else is the fox. These are sort of not permanent — you know, these are not in your DNA.

STROGATZ : Ah, good point. We can adopt — we can wear both cloaks.

Well, what about, was there a response to the essay at the time? Did people say anything back to you?

TAO : I got a fairly positive response in general. I mean, the Bulletin of the AMS is not a hugely, widely circulated publication, I think. And also, I didn’t really say anything too controversial. Also, this kind of predated social media, so, I think maybe there’s a few math blogs that picked it up, but there was no Twitter. There was nothing to make it go viral.

Yeah, also I think, in general, mathematicians don’t spend much of their time and intellectual capital on speculation. I mean, there’s another mathematician called Minhyong Kim who had this very nice metaphor that, to mathematicians, credibility is like currency, like money. If you prove theorems and you demonstrate that you know the subject, you’re accumulating somehow this currency of credibility in the bank. And once you have enough currency, you can afford to speculate a little bit by being a bit philosophical and saying what might be true rather than what you can actually prove.

But we tend to be conservative, and we don’t want an overdraft in our bank account. You know, you don’t want most of your writing to be speculative and only like one percent to actually prove something.

STROGATZ : Fair enough. So, okay. So, lots of years have passed since then. What are we talking about? It’s more than 15 years.

TAO : Oh yeah, time flies.

STROGATZ : Has your opinion changed? Is there anything we need to revise?

TAO : Well, the culture of math is changing quite a bit. I already had a broad view of mathematics, and now I have an even broader one.

So, one very concrete example is: Computer-assisted proofs were still controversial in 2007. There was a famous conjecture called the Kepler conjecture, which concerns the most efficient way to pack unit balls in three-dimensional space. And there’s a standard packing, I think it’s called the cubic central packing or something, that Kepler conjectured to be the best possible.

This was finally resolved, but the proof was very computer-assisted . It was quite complicated, and [Thomas] Hales eventually actually created a whole computer language to formally verify this particular proof, but it was not accepted as a real proof for many years. But it illustrated how controversial the concept of a proof that you needed computer assistance to verify was.

In the years since, there’s been many, many other examples of proofs where a human can reduce a complicated problem to something which still requires a computer to verify. And then the computer goes ahead and verifies it. We’ve kind of developed practices about how to do this responsibly. You know, how to publish code and data and ways to check and new open-source things and so forth. And now, there’s widespread acceptance of computer-assisted proofs.

Now, I think, the next cultural shift will be whether AI-generated proofs will be accepted . Right now, AI tools are not at the level where they can generate proofs to really advance mathematical problems. Maybe undergraduate-level homework assignments, they can kind of manage, but research mathematics, they’re not at that level yet. But at some point, we’re going to start seeing AI-assisted papers come out and there will be a debate.

The way our culture has changed in some ways… Back in 2007, only a fraction of mathematicians made their preprints available before publishing. Authors would jealously guard their preprints until they had the notification of acceptance from the journal. And then they might share.

But now everyone puts their papers on public servers like the arXiv . There’s a lot more openness to put videos and blog posts, about where the ideas of a paper come from. Because people realize that this is what makes work more influential and more impactful. If you try to not publicize your work and be very secretive about it, it doesn’t make a splash.

Math has become much more collaborative . You know, 50 years ago, I would say that the majority of papers in mathematics were single-author. Now, definitely the majority are two or three or four authors. And we’re just beginning to see really big projects like we do in the sciences, you know, like tens, hundreds of people collaborate. That’s still difficult for mathematicians to do, but I think we’re going to get there.

Concurrently, we’re becoming much more interdisciplinary. We’re working with other sciences a lot more. We’re working between fields of mathematics. And because of the internet, we can collaborate with people across the world. So, the way we do mathematics is definitely changing.

I hope in the future, we will be able to utilize the amateur math community more. There are other fields like astronomy, where astronomers make great use of the amateur astronomy community, like, you know, a lot of comets, for example, are found by amateurs.

But mathematicians… There’s a few isolated areas of mathematics such as like, tiling, two-dimensional tiling, and maybe finding records in prime numbers. There’s some very select fields of mathematics where amateurs do contribute, and they’re welcomed. But there’s a lot of barriers. In most areas of mathematics, you need so much training and internalized or conventional wisdom that we can’t crowd source things. But this may change in the future. Maybe one impact of AI would be to allow amateur mathematicians to contribute meaningfully to mathematics.

STROGATZ : That’s very interesting.

[Break for ad insertion]

STROGATZ : So the amateurs might, with the help of AIs, either ask new questions that are good or help with good explorations of existing questions, that sort of thing?

TAO : There are many different modalities — yeah. So, for example, there are now projects to formalize proofs of big theorems in these things called formal proof assistants , which are like computer languages that can 100% verify that a theorem is true or not and — is proven or not. This actually enables large-scale collaboration in mathematics.

So in the past, if you collaborate with 10 other people to prove a theorem, and each one contributes a step, everyone has to verify everyone else’s math. Because the thing about math is that if one step has an error in it, the whole thing can fall apart.

So you need trust, and so — therefore this prevents, this really inhibits really large-scale collaborations in mathematics. But there are now, there have been successful examples of really large theorems being formalized where there’s a huge community, they don’t all know each other, they don’t all trust each other, but they communicate through uploading to some Github repository or something, like, individual proofs of individual steps in the argument. And the formal proof software verifies everything, and so you don’t need to worry about trust. So we are enabling new modes of collaboration, which we haven’t seen really in the past.

STROGATZ : It’s really interesting to hear your vision, Terry. It’s a fascinating thought. You don’t hear the phrase “citizen mathematician.” You hear of citizen science, but why not citizen math?

But I’m just wondering, are there any trends that you are worried about, for example, with computer-assisted proofs or AI-generated proofs? Will we know that certain results are true, but we won’t understand why?

TAO : So that is a problem. I mean, it’s already a problem even before the advent of AI. So, there are many fields where the papers in a subject are getting longer and longer, hundreds of pages. And I’m hopeful that AI can actually conversely help simplify and it can explain as well as prove.

So there’s already experimental software where, like, if you take a proof that has been formalized, you can actually convert it into an interactive human-readable document, where you have the proof and you see the high-level steps and if there’s a sentence you don’t understand you can double-click on it, and it will expand into smaller steps. Soon I think you can also get an AI chatbot sitting next to you while you’re going through the proof, and they can take questions and they can explain each step as if they were the author. I think we’re already very close to that.

There are concerns. We have to change the way we educate our students, particularly now that many of our traditional ways of assigning homework and so forth, we are almost at the point where these AI tools can just instantly answer many of our standard exam questions. And so, we need to teach our students new skills, like how to verify whether an AI-generated output is correct or not and how to get a second opinion.

And we may see the advent of a more experimental side to mathematics, you know. So, mathematics is almost entirely theoretical, whereas most sciences have both a theoretical and experimental component. We may eventually have results that are first only proven by computers and, as you say, we don’t understand. But then once we have the data that the AI, the computer-generated proofs provide, we may be able to run experiments.

There’s a little bit of experimental mathematics now. People do study, like, large data sets of various things, elliptic curves, say. But it could become much bigger in the future.

STROGATZ : Gee, you have a very optimistic view, it sounds like to me. It’s not like the Golden Age is in the past. If I’m hearing you right, you think that there’s a lot of very exciting stuff ahead.

TAO : Yeah, a lot of the new technological tools are very empowering. I mean, AI in general has many complex ups and downsides. And outside the sciences, there’s a lot of possible disruption to the economy, intellectual property rights and so forth. But within math, I think the ratio of good to bad is better than in many other areas.

And, you know, the internet really has transformed the way we do mathematics. I collaborate with a lot of people in a lot of different fields. I could not do this without the internet. The fact that I can go on Wikipedia or whatever and get started learning a subject, and I can email somebody, and we can collaborate online. If I had to do things old-school where I could only talk to people in my department and use physical mail for everything else, I could not do the math that I do now.

STROGATZ : Wow, all right. I just have to underline what you just said, because I never thought in a million years I was going to hear this: Terry Tao reads Wikipedia to learn math?

TAO : As a starting point. I mean, it’s not always Wikipedia, but just to get the keywords, and then I will do a more specialized search of, say, MathSciNet or some other database. But yeah.

STROGATZ : It’s not a criticism. I mean, I do the same thing. Wikipedia is actually, if there’s any criticism of the math on Wikipedia, maybe it’s that sometimes it’s a little too advanced for the readers that it’s intended for, I think. Not always. I mean, it depends. It varies a lot from article to article. But that’s just funny. I love hearing that.

TAO : I mean, these tools, you have to be able to vet the output. You know, so, I mean, the reason why I can use Wikipedia to do mathematics is because I already know enough mathematics that I can smell if a piece of Wikipedia in mathematics is suspicious or not. You know, it may get some sources and one of them is going to be a better source than the other. And I know the authors, and I have an idea of which reference is going to be better for me. If I used Wikipedia to learn about a subject that I had no experience in, then I think it would be more of a random variable.

STROGATZ : Well, so we’ve been talking quite a bit about what it is that makes good mathematics, the possible future for new kinds of good mathematics. But maybe we should address the question: Why does this even matter? Why is it important for math to be good?

TAO : Well, so, first of all, I mean, why do we have mathematicians at all? Why does society value mathematicians and give us the resources to do what we do? You know, it’s because we do provide some value. We can have applications to the real world. There’s intellectual interest, and some of the theories we develop eventually end up providing insight into other phenomena.

And not all mathematics is of equal value. I mean, you could compute more and more digits of pi, but at some point, you don’t learn anything. Any subject needs some sort of value judgment because you have to allocate resources. There’s so much mathematics out there. What advances do you want to highlight and publicize and let other people know about, and which ones maybe should just be sitting quietly on a journal somewhere?

Even if you think of a subject as being completely objective and, you know, there’s only true or false, we still have to make choices. You know, just because time is a limited resource. Attention is a limited resource. Money is a limited resource. So, these are always important questions.

STROGATZ : Well, interesting that you mention about publicizing, because it is something that I think is a distinctive feature of your work, that you’ve also put in a lot of effort to make math publicly accessible through your blog, through various articles you’ve written. I remember discussing one that you wrote in American Scientist about universality and that idea. Why is it important to make math publicly accessible and understandable? I mean, what is it that you’re trying to do?

TAO : It kind of happened organically. Early in my career, the World Wide Web was still very new, and mathematicians started having webpages with various content, but there wasn’t much of a central directory. Before Google and so forth, it was actually hard to find individual resources.

So, I started sort of making little directories on my webpage . And I would also make webpages for my own papers, and I’d make some commentary. Initially, it was more for my own benefit, just as an organizational tool, just to help me find things. As a byproduct, it was available to the public, but I was kind of the primary consumer, or at least so I thought, of my own webpages.

But I remember very distinctly, there was one time when I wrote a paper and I put it on my webpage, and I had a little subpage called “What’s New?” And I just said, “Here’s a paper. There’s a question in it that I still couldn’t answer, and I don’t know how to solve it.” And I just made this comment. And then like two days later, I got an email saying, “Oh, I was just checking your homepage. I know the answer to this. There’s a paper which will solve your problem.”

And it made me realize, first of all, that people were actually visiting my webpage, which I didn’t really know. But that interaction with the community could really — well, it could help me directly solve my questions.

There’s this law called Metcalfe’s law in networking that, you know, if you have n people, and they all talk to each other, there’s about n 2 connections between them. And so, the larger the audience and the larger the forum where everyone can talk to everybody else, the more potential connections you can make and the more good things can happen.

I mean, in my career, so much of the discoveries I’ve made, or the connections I’ve made is because of an unexpected connection. My whole career experience has been sort of the more connections equals just better stuff happening.

STROGATZ : I think a beautiful example of what you’re just referring to, but I’d love to hear you talk about it, is the connections that you made with people in data science who are interested in questions having to do with medical resonance imaging, MRI. Could you tell us a little about that story?

TAO : So, this was about 2006, 2005, I think. So, there was an interdisciplinary program here on campus at UCLA on, I think, multiscale geometric analysis, or something like that, where they were bringing together pure mathematicians who were interested in sort of multiscale type geometry in its own right, and then, you know, people who had very concrete data type problems.

And I had just started working on some problems in random matrix theory, so I was sort of known as someone who could manipulate matrices. And I met someone who I already knew, Emmanuel Candès , because at the time he worked right next door in Caltech. And he and another collaborator, Justin Romberg , they had discovered this unusual phenomena.

So they were looking at MRI images, but they’re very slow. To collect enough really high-resolution image of a human body, or enough to maybe catch a tumor, or whatever medically important feature you want to find, it often takes several minutes because they have to scan all these different angles and then synthesize the data. And this was a problem, actually, because little kids, for example, just to sit still for three minutes in the MRI machine was quite problematic.

So they were experimenting with a different way, using some linear algebra. They were hoping to get a 10%, 20% better performance improvement. You know, a slightly sharper image by tweaking the standard algorithm a little bit.

So the standard algorithm was called least squares approximation, and they were doing something else, called total variation minimization. But then when they ran the computer software, they got like almost perfect reconstruction of their test image. Massive, massive improvement. And they couldn’t explain this.

But Emmanuel was at this program, and we were chatting at tea or something. And he just mentioned this and, actually, my first thought was that you must have made a mistake in your calculation, that what you’re saying is not actually possible. And I remember going back home that night and trying to write down an actual proof that what they were seeing could not actually happen. And then halfway through, I realized I had made an assumption which wasn’t true. And then I realized that actually it could work. And then I figured out what might be the explanation. And then we worked together, and we actually found a good explanation and we published that.

And once we did that, people realized that there were many other situations where you had to take a measurement which normally required lots and lots of data, and in some cases you can take a much smaller amount of data and still get a really high-resolution measurement.

So now, modern MRI machines, for example — a scan that used to take three minutes can now take 30 seconds because this software, this algorithm is hardwired, hard-coded into the machines now.

STROGATZ : It’s a beautiful story, it’s such a great story. I mean, talk about important mathematics that is changing lives, literally, in this context of medical imaging. I love the serendipity of it and your open-mindedness, you know, to hear this idea and then think, well, “this is impossible, I can prove it.” And then realizing, no, actually. Fantastic to see math making such an impact.

Well, okay, I think I better let you go, Terry. It’s been a real pleasure discussing the essence of good mathematics with you. Thanks so much for joining us today.

TAO : Yeah, no, it’s been a pleasure.  

STROGATZ : “The Joy of Why” is a podcast from Quanta Magazine , an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests or other editorial decisions in this podcast or in Quanta Magazine .

“The Joy of Why” is produced by PRX Productions . The production team is Caitlin Faulds, Livia Brock, Genevieve Sponsler and Merritt Jacob. The executive producer of PRX Productions is Jocelyn Gonzales. Morgan Church and Edwin Ochoa provided additional assistance. From Quanta Magazine , John Rennie and Thomas Lin provided editorial guidance, with support from Matt Carlstrom, Samuel Velasco, Nona Griffin, Arleen Santana and Madison Goldberg.

Our theme music is from APM Music. Julian Lin came up with the podcast name. The episode art is by Peter Greenwood and our logo is by Jaki King and Kristina Armitage. Special thanks to the Columbia Journalism School and Burt Odom-Reed at the Cornell Broadcast Studios.

I’m your host, Steve Strogatz. If you have any questions or comments for us, please email us at [email protected] . Thanks for listening.

Get highlights of the most important news delivered to your email inbox

Also in Mathematics

Mathematicians Discover New Shapes to Solve Decades-Old Geometry Problem

‘Groups’ Underpin Modern Math. Here’s How They Work.

Perplexing the Web, One Probability Puzzle at a Time

Comment on this article.

Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation. Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected. Moderators are staffed during regular business hours (New York time) and can only accept comments written in English. 

Next article

Use your social network.

Forgot your password ?

We’ll email you instructions to reset your password

Enter your new password

• Welcome from the Chair
• Michalik Distinguished Lecture Series
• Pittsburgh Mathematical Horizons 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.

Guide to Graduate Studies

The PhD Program The Ph.D. program of the Harvard Department of Mathematics is designed to help motivated students develop their understanding and enjoyment of mathematics. Enjoyment and understanding of the subject, as well as enthusiasm in teaching it, are greater when one is actively thinking about mathematics in one’s own way. For this reason, a Ph.D. dissertation involving some original research is a fundamental part of the program. The stages in this program may be described as follows:

• Acquiring a broad basic knowledge of mathematics on which to build a future mathematical culture and more detailed knowledge of a field of specialization.
• Choosing a field of specialization within mathematics and obtaining enough knowledge of this specialized field to arrive at the point of current thinking.
• Making a first original contribution to mathematics within this chosen special area.

Students are expected to take the initiative in pacing themselves through the Ph.D. program. In theory, a future research mathematician should be able to go through all three stages with the help of only a good library. In practice, many of the more subtle aspects of mathematics, such as a sense of taste or relative importance and feeling for a particular subject, are primarily communicated by personal contact. In addition, it is not at all trivial to find one’s way through the ever-burgeoning literature of mathematics, and one can go through the stages outlined above with much less lost motion if one has some access to a group of older and more experienced mathematicians who can guide one’s reading, supplement it with seminars and courses, and evaluate one’s first attempts at research. The presence of other graduate students of comparable ability and level of enthusiasm is also very helpful.

University Requirements

The University requires a minimum of two years of academic residence (16 half-courses) for the Ph.D. degree. On the other hand, five years in residence is the maximum usually allowed by the department. Most students complete the Ph.D. in four or five years. Please review the program requirements timeline .

There is no prescribed set of course requirements, but students are required to register and enroll in four courses each term to maintain full-time status with the Harvard Kenneth C. Griffin Graduate School of Arts and Sciences.

Qualifying Exam

The department gives the qualifying examination at the beginning of the fall and spring terms. The qualifying examination covers algebra, algebraic geometry, algebraic topology, complex analysis, differential geometry, and real analysis. Students are required to take the exam at the beginning of the first term. More details about the qualifying exams can be found here .

Students are expected to pass the qualifying exam before the end of their second year. After passing the qualifying exam students are expected to find a Ph.D. dissertation advisor.

Minor Thesis

The minor thesis is complementary to the qualifying exam. In the course of mathematical research, students will inevitably encounter areas in which they have gaps in knowledge. The minor thesis is an exercise in confronting those gaps to learn what is necessary to understand a specific area of math. Students choose a topic outside their area of expertise and, working independently, learns it well and produces a written exposition of the subject.

The topic is selected in consultation with a faculty member, other than the student’s Ph.D. dissertation advisor, chosen by the student. The topic should not be in the area of the student’s Ph.D. dissertation. For example, students working in number theory might do a minor thesis in analysis or geometry. At the end of three weeks time (four if teaching), students submit to the faculty member a written account of the subject and are prepared to answer questions on the topic.

The minor thesis must be completed before the start of the third year in residence.

Language Exam

Mathematics is an international subject in which the principal languages are English, French, German, and Russian. Almost all important work is published in one of these four languages. Accordingly, students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination. Students are asked to translate one page of mathematics into English with the help of a dictionary. Students may request to substitute the Italian language exam if it is relevant to their area of mathematics. The language requirement should be fulfilled by the end of the second year. For more information on the graduate program requirements, a timeline can be viewed at here .

Non-native English speakers who have received a Bachelor’s degree in mathematics from an institution where classes are taught in a language other than English may request to waive the language requirement.

Upon completion of the language exam and eight upper-level math courses, students can apply for a continuing Master’s Degree.

Teaching Requirement

Most research mathematicians are also university teachers. In preparation for this role, all students are required to participate in the department’s teaching apprenticeship program and to complete two semesters of classroom teaching experience, usually as a teaching fellow. During the teaching apprenticeship, students are paired with a member of the department’s teaching staff. Students attend some of the advisor’s classes and then prepare (with help) and present their own class, which will be videotaped. Apprentices will receive feedback both from the advisor and from members of the class.

Teaching fellows are responsible for teaching calculus to a class of about 25 undergraduates. They meet with their class three hours a week. They have a course assistant (an advanced undergraduate) to grade homework and to take a weekly problem session. Usually, there are several classes following the same syllabus and with common exams. A course head (a member of the department teaching staff) coordinates the various classes following the same syllabus and is available to advise teaching fellows. Other teaching options are available: graduate course assistantships for advanced math courses and tutorials for advanced undergraduate math concentrators.

Final Stages

How students proceed through the second and third stages of the program varies considerably among individuals. While preparing for the qualifying examination or immediately after, students should begin taking more advanced courses to help with choosing a field of specialization. Unless prepared to work independently, students should choose a field that falls within the interests of a member of the faculty who is willing to serve as dissertation advisor. Members of the faculty vary in the way that they go about dissertation supervision; some faculty members expect more initiative and independence than others and some variation in how busy they are with current advisees. Students should consider their own advising needs as well as the faculty member’s field when choosing an advisor. Students must take the initiative to ask a professor if she or he will act as a dissertation advisor. Students having difficulty deciding under whom to work, may want to spend a term reading under the direction of two or more faculty members simultaneously. The sooner students choose an advisor, the sooner they can begin research. Students should have a provisional advisor by the second year.

It is important to keep in mind that there is no technique for teaching students to have ideas. All that faculty can do is to provide an ambiance in which one’s nascent abilities and insights can blossom. Ph.D. dissertations vary enormously in quality, from hard exercises to highly original advances. Many good research mathematicians begin very slowly, and their dissertations and first few papers could be of minor interest. The ideal attitude is: (1) a love of the subject for its own sake, accompanied by inquisitiveness about things which aren’t known; and (2) a somewhat fatalistic attitude concerning “creative ability” and recognition that hard work is, in the end, much more important.

50 IB Maths IA Topic Ideas

IB Maths is a struggle for most people going through their diploma. To make matters worse, on top of just doing the dreaded maths exam, we’re also expected to write a Maths IA exploration into a topic of our choice! Where do you even begin such a task? How do you even choose a topic? To make things easier, we have plenty of free Maths resources! Firstly though, we’ve compiled 50 common Maths IA topics that may spark some creative juices and set you on your way to conquering one of the hardest assignments of the diploma!

Once you have chosen your topic, you may want to check out our other posts on how to structure and format your Maths IA or how to write your IA .

NOTE: These topics are purely meant as inspiration and are not to be chosen blindly. Even though many of these topics led to high scores for some of our graduates in the past, it is important that you listen to the advice of your subject teacher before choosing any topic!

• Pascal’s triangle : Discovering patterns within this famous array of numbers
• Pythagorean triples : Can you find patterns in what numbers form a pythagorean triple?
• Monty Hall problem : How does Bayesian probability work in this real-life example, and can you add a layer of complexity to it?
• The Chinese Remainder Theorem : An insight into the mathematics of number theory
• Sum of all positive integers is -1/12? Explore this fascinating physics phenomenon through the world of sequences and series
• Birthday paradox : Why is it that in a room of people probability dictates that people are very likely to share a birthday? 
• Harmonic series : Explore why certain notes/chords in music sound dissonant, and others consonant, by looking at the ratios of frequencies between the notes.
• Optimizing areas : Optimizing the area of a rectangle is easy, but can you find a way to do it for any polygon?
• Optimizing volumes : Explore the mathematics of finding a maximum volume of a cuboid subject to some constraint
• Flow of traffic : How does mathematics feed into our traffic jams that we endure every morning?
• Football statistics : Does spending a lot of cash during the transfer window translate to more points the following year? Or is there a better predictor of a team’s success like wages, historic performance, or player valuation?
• Football statistics #2 : How does a manager sacking affect results? 
• Gini coefficient : Can you use integration to derive the gini coefficient for a few countries, allowing you to accurately compare their levels of economic inequality?
• Linear regressions : Run linear regressions using OLS to predict and estimate the effect of one variable on another.
• The Prisoner’s Dilemma : Use game theory in order to deduce the optimal strategy in this famous situation
• Tic Tac Toe : What is the optimal strategy in this legendary game? Will my probability of winning drastically increase by some move that I can make?
• Monopoly : Is there a strategy that dominates all others? Which properties should I be most excited to land on?
• Rock Paper Scissors : If I played and won with rock already, should I make sure to change what I play this time? Or is it better to switch? 
• The Toast problem : If there is a room of some number of people, how many toasts are necessary for everyone to have toasted with everyone?
• Cracking a Password : How long would it take to be able to correctly guess a password? How much safer does a password get by adding symbols or numbers?
• Stacking Balls : Suppose you want to place balls in a cardboard box, what is the optimal way to do this to use your space most effectively?
• The Wobbly Table : Many tables are wobbly because of uneven ground, but is there a way to orient the tables to make sure they are always stable?
• The Stable Marriage Problem : Is there a matching algorithm that ensures each person in society ends up with their one true love? What is the next best alternative if this is not viable?
• Mathematical Card Tricks : Look at the probabilities at play in the famous 3 card monte scam. 
• Modelling the Spread of a Virus : How long would it take for us all to be wiped out if a deadly influenza spreads throughout the population?
• The Tragedy of the Common s : Our population of fish is dwindling, but how much do we need to reduce our production by in order to ensure the fish can replenish faster than we kill?
• The Risk of Insurance : An investigation into asymmetric information and how being unsure about the future state of the world may lead us to be risk-averse
• Gabriel’s Horn : This figure has an infinite surface area but a finite volume, can you p rove this?
• Modelling the Shape of an Egg : Although it may sound easy, finding the surface area or volume of this common shape requires some in-depth mathematical investigation
• Voting Systems : What voting system ensures that the largest amount of people get the official that they would prefer? With 2 candidates this is logical, but what if they have more than 2?
• Probability : Are Oxford and Cambridge biased against state-school applicants?
• Statistics : With Tokyo 2020 around the corner, how aboutmodelling change in record performances for a particular discipline?
• Analysing Data : In the 200 meter dash, is there an advantage to a particular lane in track? 
• Coverage : Calculation of rate of deforestation, and afforestation. How long will our forests last?
• Friendly numbers, Solitary numbers, perfect numbers : Investigate what changes the condition of numbers
• Force : Calculating the intensity of a climber’s fall based upon their distance above where they last clamped in
• Königsberg bridge problem : Using networks to solve problems. 
• Handshake problem : How many handshakes are required so that everyone shakes hands with all the other people in the room? 
• The mathematics of deceit : How con artists use pyramid schemes to get rich quick!
• Modelling radioactive decay : The maths of Chernobyl – when will it be safe to live there?
• Mathematics and photography : Exploring the relationship between the aperture of a camera and a geometric sequence
• Normal Distribution : Using distributions to examine the 2008 financial crisis
• Mechanics : Body Proportions for Track and Field events
• Modelling : How does a cup of Tea cool?
• Relationships : Do BMI ratings and country wealth share a significant relationship?
• Modelling : Can we mathematically model musical chords and concepts like dissonance?
• Evaluating limits : Exploring L’Hôpital’s rule
• Chinese postman problem : How do we calculate shortest possible routes?
• Maths and Time : Exploring ideas regarding time dilation
• Plotting Planets : Using log functions to track planets!

So there we have it: 50 IB Maths IA topic ideas to give you a head-start for attacking this piece of IB coursework ! We also have similar ideas for Biology , Chemistry , Economics , History , Physics , TOK … and many many more tips and tricks on securing those top marks on our free resources page – just click the ‘Maths resources’ button!

Still feeling confused, or want some personalised help? We offer online private tuition from experienced IB graduates who got top marks in their Maths IA. 

Share article links

Related Articles

• Most Popular

30 IB Biology IA Topic Ideas!

Are you struggling with choosing your topic for your IB Biology IA? Don’t worry, we’ve all been there. Finding a topic is one of the – if not THE – most important part of writing your IA, so we want to make sure that you get it right! Luckily, there are so many great topics […]

20 IB Physics IA Topic Ideas!

Choosing where to start with an IA can be the hardest part, and this is definitely true for the Physics IA. We know that our topic has to be somewhat related to the syllabus, but where should we focus? Thankfully, we’ve asked some of our favourite IB graduates for some of the ideas they pursued! […]

25 History IA Topic Ideas!

Are you about to start your History internal assessment? We know the struggle. One of the most difficult parts about the task is finding a good History IA topic because it feels like you can just write about anything. The IB breaks it down into 7 main different types of topics that you can choose, […]

• Future Students
• Current Students
• Faculty/Staff

News and Media

• News & Media Home
• Research Stories
• School’s In
• In the Media

You are here

Research shows the best ways to learn math.

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

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

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

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

Number sense is critical

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

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

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

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

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

Role of the brain

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

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

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

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

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

Math treated differently

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

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

Strategies, activities

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

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

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

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

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

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

More Stories

⟵ Go to all Research Stories

Get the Educator

Subscribe to our monthly newsletter.

Stanford Graduate School of Education

482 Galvez Mall Stanford, CA 94305-3096 Tel: (650) 723-2109

• Contact Admissions
• GSE Leadership
• Site Feedback
• Web Accessibility
• Career Resources
• Faculty Open Positions
• Explore Courses
• Academic Calendar
• Office of the Registrar
• Cubberley Library
• StanfordWho
• StanfordYou

Improving lives through learning

• Stanford Home
• Maps & Directions
• Search Stanford
• Emergency Info
• Terms of Use
• Non-Discrimination
• Accessibility

© Stanford University , Stanford , California 94305 .

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

• Publications
• Account settings

The PMC website is updating on October 15, 2024. Learn More or Try it out now .

• Advanced Search
• Journal List
• Springer Nature - PMC COVID-19 Collection

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

Arthur bakker.

1 Utrecht University, Utrecht, Netherlands

2 University of Delaware, Newark, DE USA

Linda Zenger

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.

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 ).

Methodological approach

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.

Numbers of responses per continent (2019)

Note : When a respondent filled in two countries on two continents, we attributed half to one and the other half to the other continent

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 ​ Table2), 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).

Percentages of responses mentioned in each theme (2019)

Note. Percentages do not add up to 100, because many respondents mentioned multiple themes

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 ​ Table2 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. ​ Fig.1 1 to capture themes in their relationships.

Artistic impression of the future themes

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.

The most frequently mentioned theme was what we labeled approaches to teaching (64% of the respondents, see Table ​ Table2). 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.”

Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

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.

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.

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.”

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.

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.

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).

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.

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 ).

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.

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 ).

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.

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.

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 )

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.

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. ​ Fig.1. 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.

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 ).

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

Appendix 1: Explanation of Fig. ​ Fig.1 1

We have divided Fig. ​ Fig.1 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 )

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.

Publisher’s note

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

• Akkerman SF, Bakker A. Boundary crossing and boundary objects. Review of Educational Research. 2011; 81 (2):132–169. doi: 10.3102/0034654311404435. [ CrossRef ] [ Google Scholar ]
• Arendt, H. (1958/1998). The human condition (2nd ed.). University of Chicago Press.
• Backes B, Cowan J. Is the pen mightier than the keyboard? The effect of online testing on measured student achievement. Economics of Education Review. 2019; 68 :89–103. doi: 10.1016/j.econedurev.2018.12.007. [ CrossRef ] [ Google Scholar ]
• 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. 10.1016/j.learninstruc.2009.09.001
• Bakker A. What is worth publishing? A response to Niss. For the Learning of Mathematics. 2019; 39 (3):43–45. [ Google Scholar ]
• Bakker A, Gravemeijer KP. An historical phenomenology of mean and median. Educational Studies in Mathematics. 2006; 62 (2):149–168. doi: 10.1007/s10649-006-7099-8. [ CrossRef ] [ Google Scholar ]
• Bakx A, Bakker A, Koopman M, Beijaard D. Boundary crossing by science teacher researchers in a PhD program. Teaching and Teacher Education. 2016; 60 :76–87. doi: 10.1016/j.tate.2016.08.003. [ CrossRef ] [ Google Scholar ]
• 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. 10.1016/j.caeo.2020.100016
• Beckers D, Beckers A. ‘Newton was heel exact wetenschappelijk – ook in zijn chemische werk’. Nederlandse wetenschapsgeschiedenis in niet-wetenschapshistorische tijdschriften, 1977–2017. Studium. 2019; 12 (4):185–197. doi: 10.18352/studium.10203. [ CrossRef ] [ Google Scholar ]
• 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. 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. 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. Constructing and employing theoretical frameworks in (mathematics) education research. For the Learning of Mathematics. 2019; 39 (3):44–47. [ Google Scholar ]
• Cai J, Jiang C. An analysis of problem-posing tasks in Chinese and U.S. elementary mathematics textbooks. International Journal of Science and Mathematics Education. 2017; 15 (8):1521–1540. doi: 10.1007/s10763-016-9758-2. [ CrossRef ] [ Google Scholar ]
• Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics , 105 , 287–301. 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. 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. 10.1207/s15327833mtl0803_6
• Cobb P, Gresalfi M, Hodge LL. An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education. 2009; 40 (1):40–68. [ Google Scholar ]
• Darragh L. Identity research in mathematics education. Educational Studies in Mathematics. 2016; 93 (1):19–33. doi: 10.1007/s10649-016-9696-5. [ CrossRef ] [ Google Scholar ]
• 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. 10.1007/s11858-018-1007-4
• Deng, Z. (2018). Contemporary curriculum theorizing: Crisis and resolution. Journal of Curriculum Studies , 50 (6), 691–710. 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 . 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. 10.1007/s11858-020-01185-3 [ PMC free article ] [ PubMed ]
• 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. 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. 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. 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. Proportional reasoning in nursing practice. Journal for Research in Mathematics Education. 2001; 32 (1):4–27. doi: 10.2307/749619. [ CrossRef ] [ Google Scholar ]
• 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. Approaching participation in school-based mathematics as a cross-setting phenomenon. The Journal of the Learning Sciences. 2011; 20 (1):111–150. doi: 10.1080/10508406.2011.528319. [ CrossRef ] [ Google Scholar ]
• 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. 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 . 10.1007/s10649-021-10029-0 [ PMC free article ] [ PubMed ]
• Lefebvre, H. (2004). Rhythmanalysis: Space, time and everyday life (Original 1992; Translation by S. Elden & G. Moore) . Bloomsbury Academic. 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. 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. The very multi-faceted nature of mathematics education research. For the Learning of Mathematics. 2019; 39 (2):2–7. [ Google Scholar ]
• OECD. (2020). Back to the Future of Education: Four OECD Scenarios for Schooling. Educational Research and Innovation . OECD Publishing. 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. 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. 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. 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. 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. 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. 10.1007/s11858-017-0864-6 [ PMC free article ] [ PubMed ]
• Sfard A. What could be more practical than good research? Educational Studies in Mathematics. 2005; 58 (3):393–413. doi: 10.1007/s10649-005-4818-5. [ CrossRef ] [ Google Scholar ]
• 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. Some remarks on understanding in mathematics. For the Learning of Mathematics. 1990; 10 (3):24–41. [ Google Scholar ]
• 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. 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. 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. Can scientific research answer the ‘what’ question of mathematics education? Cambridge Journal of Education. 2005; 35 (1):35–53. doi: 10.1080/0305764042000332489. [ CrossRef ] [ Google Scholar ]
• Wittmann EC. Mathematics education as a ‘design science’ Educational Studies in Mathematics. 1995; 29 (4):355–374. doi: 10.1007/BF01273911. [ CrossRef ] [ Google Scholar ]
• 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 . 10.1016/j.jmathb.2021.100865.

IB Maths Resources from Intermathematics

IB Maths Resources: 300 IB Maths Exploration ideas, video tutorials and Exploration Guides

Maths IA – 300 Maths Exploration Topics

Maths ia – 300 maths exploration topics:.

Scroll down this page to find over  300 examples of maths IA exploration topics and ideas for IB mathematics students doing their internal assessment (IA) coursework.  Topics include Algebra and Number (proof), Geometry, Calculus, Statistics and Probability, Physics, and links with other subjects.  Suitable for Applications and Interpretations students (SL and HL) and also Analysis and Approaches students (SL and HL).

New online Maths IA Course!

I have just made a comprehensive online course:  Getting a 7 in IB Maths Coursework . 

Gain the inside track on what makes a good coursework piece from an IB Maths Examiner as you learn all the skills necessary to produce something outstanding.  This course is written for current IB Mathematics students.  There is more than 240 minutes of video tutorial content as well as a number of multiple choice quizzes to aid understanding.  There are also a number of pdf downloads to support the lesson content.  I think this will be really useful – check it out!

Modelling allows us to predict real world events using mathematical functions.  

Why is this topic a good idea?

This topic is a nice combination of graphical skills, regression and potentially calculus.  It easily links to the real world and so is easy to find engaging ideas.

Some suggested ideas:

Calculus and Physics

Calculus allows us to understand rates of change and therefore motion over time.  It’s one of the most powerful tools ever invented.

This topic allows a nice demonstration of calculus skills, often links with graphical ideas and is easy to create real world links.

Data and Probability

The modern currency of the internet is data – and data collection and data interpretation skills are essential.

This topic if done well can bring in ideas of probability, statistics and other branches of mathematics.

Statistics and data analysis are important skills in business and science.  There are many different tests which help us understand the significance of results

This topic if done well can allow students to do some experiments and investigations

Geometry connects us with mathematics done for over 2000 years by the likes of Euclid done with compasses and rulers.

This topic is a nice combination of graphical skills and the ability to apply to new situations.

Pure Mathematics

Pure mathematics allows us to experience ideas of proof and gets us closer to what “real” mathematicians do.  

This topic is a nice chance to explore ideas in proof, number theory and complex numbers.

Matrices and computing

Here are some more interesting topic ideas spanning a variety of mathematical fields – linking to matrices and computational ideas

Using matrices to make fractals

Google page rank – billion dollar maths!

Further ideas

If the ideas above aren’t enough I’ve also added even more ideas and links below.  Please explore!

1) Modular arithmetic – This technique is used throughout Number Theory. For example, Mod 3 means the remainder when dividing by 3.

2) Goldbach’s conjecture: “Every even number greater than 2 can be expressed as the sum of two primes.” One of the great unsolved problems in mathematics.

3) Probabilistic number theory

4) Applications of complex numbers : The stunning graphics of Mandelbrot and Julia Sets are generated by complex numbers.

5) Diophantine equations : These are polynomials which have integer solutions. Fermat’s Last Theorem is one of the most famous such equations.

6) Continued fractions : These are fractions which continue to infinity. The great Indian mathematician Ramanujan discovered some amazing examples of these.

7) Patterns in Pascal’s triangle : There are a large number of patterns to discover – including the Fibonacci sequence.

8) Finding prime numbers : The search for prime numbers and the twin prime conjecture are some of the most important problems in mathematics. There is a $1 million prize for solving the Riemann Hypothesis and $250,000 available for anyone who discovers a new, really big prime number.

9) Random numbers

10) Pythagorean triples : A great introduction into number theory – investigating the solutions of Pythagoras’ Theorem which are integers (eg. 3,4,5 triangle).

11) Mersenne primes : These are primes that can be written as 2^n -1.

12) Magic squares and cubes : Investigate magic tricks that use mathematics. Why do magic squares work?

13) Loci and complex numbers

14) Egyptian fractions : Egyptian fractions can only have a numerator of 1 – which leads to some interesting patterns. 2/3 could be written as 1/6 + 1/2. Can all fractions with a numerator of 2 be written as 2 Egyptian fractions?

15) Complex numbers and transformations

16) Euler’s identity: An equation that has been voted the most beautiful equation of all time, Euler’s identity links together 5 of the most important numbers in mathematics.

17) Chinese remainder theorem . This is a puzzle that was posed over 1500 years ago by a Chinese mathematician. It involves understanding the modulo operation.

18) Fermat’s last theorem : A problem that puzzled mathematicians for centuries – and one that has only recently been solved.

19) Natural logarithms of complex numbers

20) Twin primes problem : The question as to whether there are patterns in the primes has fascinated mathematicians for centuries. The twin prime conjecture states that there are infinitely many consecutive primes ( eg. 5 and 7 are consecutive primes). There has been a recent breakthrough in this problem.

21) Hypercomplex numbers

22) Diophantine application: Cole numbers

23) Perfect Numbers: Perfect numbers are the sum of their factors (apart from the last factor). ie 6 is a perfect number because 1 + 2 + 3 = 6.

24) Euclidean algorithm for GCF

25) Palindrome numbers: Palindrome numbers are the same backwards as forwards.

26) Fermat’s little theorem : If p is a prime number then a^p – a is a multiple of p.

27) Prime number sieves

28) Recurrence expressions for phi (golden ratio): Phi appears with remarkable consistency in nature and appears to shape our understanding of beauty and symmetry.

29) The Riemann Hypothesis – one of the greatest unsolved problems in mathematics – worth $1million to anyone who solves it (not for the faint hearted!)

30) Time travel to the future : Investigate how traveling close to the speed of light allows people to travel “forward” in time relative to someone on Earth. Why does the twin paradox work?

31) Graham’s Number – a number so big that thinking about it could literally collapse your brain into a black hole.

32) RSA code – the most important code in the world? How all our digital communications are kept safe through the properties of primes.

33) The Chinese Remainder Theorem : This is a method developed by a Chinese mathematician Sun Zi over 1500 years ago to solve a numerical puzzle. An interesting insight into the mathematical field of Number Theory.

34) Cesaro Summation: Does 1 – 1 + 1 – 1 … = 1/2? . A post which looks at the maths behind this particularly troublesome series.

35) Fermat’s Theorem on the sum of 2 squares – An example of how to use mathematical proof to solve problems in number theory.

36) Can we prove that 1 + 2 + 3 + 4 …. = -1/12 ? How strange things happen when we start to manipulate divergent series.

37) Mathematical proof and paradox – a good opportunity to explore some methods of proof and to show how logical errors occur.

38) Friendly numbers, Solitary numbers, perfect numbers. Investigate what makes a number happy or sad, or sociable! Can you find the loop of infinite sadness?

39) Zeno’s Paradox – Achilles and the Tortoise – A look at the classic paradox from ancient Greece – the philosopher “proved” a runner could never catch a tortoise – no matter how fast he ran.

40) Stellar Numbers – This is an excellent example of a pattern  sequence investigation. Choose your own pattern investigation for the exploration.

41) Arithmetic number puzzle – It could be interesting to do an exploration where you solve number problems – like this one.

42)  Normal Numbers – and random number generators  – what is a normal number – and how are they connected to random number generators?

43)  Narcissistic Numbers  – what makes a number narcissistic – and how can we find them all?

44)  Modelling Chaos  – how we can use grahical software to understand the behavior of sequences

45) The Mordell Equation .  What is the Mordell equation and how does it help us solve mathematical problems in number theory?

46) Ramanujan’s Taxi Cab and the Sum of 2 Cubes .  Explore this famous number theory puzzle.

47) Hollow cubes and hypercubes investigation.  Explore number theory in higher dimensions!

48) When do 2 squares equal 2 cubes?  A classic problem in number theory which can be solved through computational power.

49) Rational approximations to irrational numbers.  How accurately can be approximate irrationals?

50) Square triangular numbers.  When do we have a square number which is also a triangular number?

51) Complex numbers as matrices – Euler’s identity.  We can use a matrix representation of complex numbers to test whether Euler’s identity still holds.

52) Have you got a Super Brain?  How many different ways can we use to solve a number theory problem?

1a)  Non-Euclidean geometries:  This allows us to “break” the rules of conventional geometry – for example, angles in a triangle no longer add up to 180 degrees. In some geometries triangles add up to more than 180 degrees, in others less than 180 degrees.

1b)  The shape of the universe  – non-Euclidean Geometry is at the heart of Einstein’s theories on General Relativity and essential to understanding the shape and behavior of the universe.

2)  Hexaflexagons:  These are origami style shapes that through folding can reveal extra faces.

3)  Minimal surfaces and soap bubbles : Soap bubbles assume the minimum possible surface area to contain a given volume.

4)  Tesseract – a 4D cube : How we can use maths to imagine higher dimensions.

5)  Stacking cannon balls:  An investigation into the patterns formed from stacking canon balls in different ways.

6)  Mandelbrot set and fractal shapes : Explore the world of infinitely generated pictures and fractional dimensions.

7)  Sierpinksi triangle : a fractal design that continues forever.

8)  Squaring the circle : This is a puzzle from ancient times – which was to find out whether a square could be created that had the same area as a given circle. It is now used as a saying to represent something impossible.

9)  Polyominoes : These are shapes made from squares. The challenge is to see how many different shapes can be made with a given number of squares – and how can they fit together?

10)  Tangrams:  Investigate how many different ways different size shapes can be fitted together.

11)  Understanding the fourth dimension:  How we can use mathematics to imagine (and test for) extra dimensions.

12)  The Riemann Sphere  – an exploration of some non-Euclidean geometry. Straight lines are not straight, parallel lines meet and angles in a triangle don’t add up to 180 degrees.

13)  Graphically understanding complex roots  – have you ever wondered what the complex root of a quadratic actually means graphically? Find out!

14)  Circular inversion  – what does it mean to reflect in a circle? A great introduction to some of the ideas behind non-euclidean geometry.

15)  Julia Sets and Mandelbrot Sets  – We can use complex numbers to create beautiful patterns of infinitely repeating fractals. Find out how!

16)  Graphing polygons investigation.   Can we find a function that plots a square?  Are there functions which plot any polygons?  Use computer graphing to investigate.

17)  Graphing Stewie from Family Guy.  How to use graphic software to make art from equations.

18)  Hyperbolic geometry  – how we can map the infinite hyperbolic plane onto the unit circle, and how this inspired the art of Escher.

19)  Elliptical Curves – how this class of curves have importance in solving Fermat’s Last Theorem and in cryptography.

20)  The Coastline Paradox  – how we can measure the lengths of coastlines, and uses the idea of fractals to arrive at fractional dimensions.

21)  Projective geometry  –  the development of geometric proofs based on points at infinity.

22)  The Folium of Descartes . This is a nice way to link some maths history with studying an interesting function.

23)  Measuring the Distance to the Stars . Maths is closely connected with astronomy – see how we can work out the distance to the stars.

24)  A geometric proof for the arithmetic and geometric mean . Proof doesn’t always have to be algebraic. Here is a geometric proof.

25)  Euler’s 9 Point Circle . This is a lovely construction using just compasses and a ruler.

26)  Plotting the Mandelbrot Set  – using Geogebra to graphically generate the Mandelbrot Set.

27)  Volume optimization of a cuboid   – how to use calculus and graphical solutions to optimize the volume of a cuboid.

28)  Ford Circles – how to generate Ford circles and their links with fractions.

29)  Classical Geometry Puzzle: Finding the Radius . This is a nice geometry puzzle solved using a variety of methods.

30)  Can you solve Oxford University’s Interview Question? .  Try to plot the locus of a sliding ladder.

31)  The Shoelace Algorithm to find areas of polygons .  How can we find the area of any polygon?

32)  Soap Bubbles, Wormholes and Catenoids . What is the geometric shape of soap bubbles?

33)  Can you solve an Oxford entrance question?   This problem asks you to explore a sliding ladder.

34)  The Tusi circle  – how to create a circle rolling inside another circle using parametric equations.

35)  Sphere packing  – how to fit spheres into a package to minimize waste.

36)  Sierpinski triangle  – an infinitely repeating fractal pattern generated by code.

37)  Generating e through probability and hypercubes .  This amazing result can generate e through considering hyper-dimensional shapes.

38)  Find the average distance between 2 points on a square .  If any points are chosen at random in a square what is the expected distance between them?

39)  Finding the average distance between 2 points on a hypercube .  Can we extend our investigation above to a multi-dimensional cube?

40)  Finding focus with Archimedes.   The Greeks used a very different approach to understanding quadratics – and as a result had a deeper understanding of their physical properties linked to light and reflection.

41)  Chaos and strange Attractors: Henon’s map .  Gain a deeper understanding of chaos theory with this investigation.

Calculus/analysis and functions

1)  The harmonic series:  Investigate the relationship between fractions and music, or investigate whether this series converges.

2)  Torus – solid of revolution : A torus is a donut shape which introduces some interesting topological ideas.

3)  Projectile motion:  Studying the motion of projectiles like cannon balls is an essential part of the mathematics of war. You can also model everything from Angry Birds to stunt bike jumping. A good use of your calculus skills.

4)  Why e is base of natural logarithm function:  A chance to investigate the amazing number e.

5)  Fourier Transforms – the most important tool in mathematics?  Fourier transforms have an essential part to play in modern life – and are one of the keys to understanding the world around us. This mathematical equation has been described as the most important in all of physics. Find out more! (This topic is only suitable for IB HL students).

6)  Batman and Superman maths  – how to use Wolfram Alpha to plot graphs of the Batman and Superman logo

7)  Explore the Si(x) function  – a special function in calculus that can’t be integrated into an elementary function.

8)  The Remarkable Dirac Delta Function . This is a function which is used in Quantum mechanics – it describes a peak of zero width but with area 1.

9)  Optimization of area – an investigation . This is an nice example of how you can investigation optimization of the area of different polygons.

10)  Envelope of projectile motion .  This investigates a generalized version of projectile motion – discover what shape is created.

11)  Projectile Motion Investigation II . This takes the usual projectile motion ideas and generalises them to investigate equations of ellipses formed.

12)  Projectile Motion III: Varying gravity .  What would projectile motion look like on different planets?

13)  The Tusi couple – A circle rolling inside a circle . This is a lovely result which uses parametric functions to create a beautiful example of mathematical art.

14)  Galileo’s Inclined Planes .  How did Galileo achieve his breakthrough understanding of gravity?  Follow in the footsteps of a genius!

Statistics and modelling 1 [topics could be studied in-depth]

1) Traffic flow : How maths can model traffic on the roads.

2) Logistic function and constrained growth

3)  Benford’s Law  – using statistics to catch criminals by making use of a surprising distribution.

4)  Bad maths in court  – how a misuse of statistics in the courtroom can lead to devastating miscarriages of justice.

5)  The mathematics of cons  – how con artists use pyramid schemes to get rich quick.

6)  Impact Earth  – what would happen if an asteroid or meteorite hit the Earth?

7)  Black Swan events  – how usefully can mathematics predict small probability high impact events?

8)  Modelling happiness  – how understanding utility value can make you happier.

9)  Does finger length predict mathematical ability?  Investigate the surprising correlation between finger ratios and all sorts of abilities and traits.

10) Modelling epidemics/spread of a virus

11)  The Monty Hall problem  – this video will show why statistics often lead you to unintuitive results.

12) Monte Carlo simulations

13) Lotteries

14)  Bayes’ theorem : How understanding probability is essential to our legal system.

15)  Birthday paradox:  The birthday paradox shows how intuitive ideas on probability can often be wrong. How many people need to be in a room for it to be at least 50% likely that two people will share the same birthday? Find out!

16)  Are we living in a computer simulation?  Look at the Bayesian logic behind the argument that we are living in a computer simulation.

17)  Does sacking a football manager affect results ? A chance to look at some statistics with surprising results.

18)  Which times tables do students find most difficult?  A good example of how to conduct a statistical investigation in mathematics.

19)  Introduction to Modelling.  This is a fantastic 70 page booklet explaining different modelling methods from Moody’s Mega Maths Challenge.

20)  Modelling infectious diseases  – how we can use mathematics to predict how diseases like measles will spread through a population

21)  Using Chi Squared to crack codes  – Chi squared can be used to crack Vigenere codes which for hundreds of years were thought to be unbreakable. Unleash your inner spy!

22)  Modelling Zombies  – How do zombies spread? What is your best way of surviving the zombie apocalypse? Surprisingly maths can help!

23)  Modelling music with sine waves  – how we can understand different notes by sine waves of different frequencies. Listen to the sounds that different sine waves make.

24)  Are you psychic?  Use the binomial distribution to test your ESP abilities.

25)  Reaction times  – are you above or below average? Model your data using a normal distribution.

26)  Modelling volcanoes  – look at how the Poisson distribution can predict volcanic eruptions, and perhaps explore some more advanced statistical tests.

27)  Could Trump win the next election ? How the normal distribution is used to predict elections.

28)  How to avoid a Troll  – an example of a problem solving based investigation

29)  The Gini Coefficient  – How to model economic inequality

30)  Maths of Global Warming – Modeling Climate Change  – Using Desmos to model the change in atmospheric Carbon Dioxide.

31)  Modelling radioactive decay   – the mathematics behind radioactivity decay, used extensively in science.

32)  Circular Motion: Modelling a Ferris wheel . Use Tracker software to create a Sine wave.

33)  Spotting Asset Bubbles . How to use modeling to predict booms and busts.

34)  The Rise of Bitcoin . Is Bitcoin going to keep rising or crash?

35)  Fun with Functions! . Some nice examples of using polar coordinates to create interesting designs.

36)  Predicting the UK election using linear regression . The use of regression in polling predictions.

37)  Modelling a Nuclear War . What would happen to the climate in the event of a nuclear war?

38)  Modelling a football season .  We can use a Poisson model and some Excel expertise to predict the outcome of sports matches – a technique used by gambling firms.

39) Modeling hours of daylight  – using Desmos to plot the changing hours of daylight in different countries.

40)  Modelling the spread of Coronavirus (COVID-19) . Using the SIR model to understand epidemics.

41)  Finding the volume of a rugby ball (or American football) .  Use modeling and volume of revolutions.

42)  The Martingale system paradox.   Explore a curious betting system still used in currency trading today.

Statistics and modelling 2 [more simplistic topics: correlation, normal, Chi squared]

1)  Is there a correlation between hours of sleep and exam grades? Studies have shown that a good night’s sleep raises academic attainment.

2)  Is there a correlation between height and weight?  (pdf).  The NHS use a chart to decide what someone should weigh depending on their height. Does this mean that height is a good indicator of weight?

3)  Is there a correlation between arm span and foot height?   This is also a potential opportunity to discuss the  Golden Ratio  in nature.

4) Is there a correlation between smoking and lung capacity?

5)  Is there a correlation between GDP and life expectancy?  Run the Gapminder graph to show the changing relationship between GDP and life expectancy over the past few decades.

7)  Is there a correlation between numbers of yellow cards a game and league position? Use the Guardian Stats data to find out if teams which commit the most fouls also do the best in the league.

8)  Is there a correlation between Olympic 100m sprint times and Olympic 15000m times? Use the Olympic database to find out if the 1500m times have got faster in the same way the 100m times have got quicker over the past few decades.

9) Is there a correlation between time taken getting to school and the distance a student lives from school?

10) Does eating breakfast affect your grades?

11) Is there a correlation between  stock prices of different companies?  Use Google Finance to collect data on company share prices.

12) Is there a correlation between  blood alcohol laws and traffic accidents ?

13) Is there a correlation between  height and basketball ability?   Look at some stats for NBA players to find out.

14) Is there a correlation between  stress and blood pressure ?

15) Is there a correlation between  Premier League wages and league positions ?

16) Are a sample of student heights  normally distributed?  We know that adult population heights are normally distributed – what about student heights?

17) Are a sample of flower heights normally distributed?

18) Are a sample of student weights normally distributed?

19)  Are the IB maths test scores normally distributed?  (pdf). IB test scores are designed to fit a bell curve. Investigate how the scores from different IB subjects compare.

20) Are the weights of “1kg” bags of sugar normally distributed?

21)  Does gender affect hours playing sport?  A UK study showed that primary school girls play much less sport than boys.

22) Investigation into the distribution of  word lengths in different languages . The English language has an average word length of 5.1 words.  How does that compare with other languages?

23)  Do bilingual students have a greater memory recall than non-bilingual students? Studies have shown that bilingual students have better “working memory” – does this include memory recall?

Games and game theory

1) The prisoner’s dilemma : The use of game theory in psychology and economics.

3)  Gambler’s fallacy:  A good chance to investigate misconceptions in probability and probabilities in gambling. Why does the house always win?

4)  Bluffing in Poker:  How probability and game theory can be used to explore the the best strategies for bluffing in poker.

5)  Knight’s tour in chess:  This chess puzzle asks how many moves a knight must make to visit all squares on a chess board.

6) Billiards and snooker

7) Zero sum games

8)  How to “Solve” Noughts and Crossess  (Tic Tac Toe) – using game theory. This topics provides a fascinating introduction to both combinatorial Game Theory and Group Theory.

9)  Maths and football  – Do managerial sackings really lead to an improvement in results? We can analyse the data to find out. Also look at the  finances  behind Premier league teams

10) Is there a correlation between  Premier League wages and league position?  Also look at how the  Championship  compares to the Premier League.

11)  The One Time Pad  – an uncrackable code? Explore the maths behind code making and breaking.

12)  How to win at Rock Paper Scissors . Look at some of the maths (and psychology behind winning this game.

13)  The Watson Selection Task  – a puzzle which tests logical reasoning.  Are maths students better than history students?

Topology and networks

2) Steiner problem

3)  Chinese postman problem  – This is a problem from graph theory – how can a postman deliver letters to every house on his streets in the shortest time possible?

4) Travelling salesman problem

5)  Königsberg bridge problem : The use of networks to solve problems. This particular problem was solved by Euler.

6)  Handshake problem : With n people in a room, how many handshakes are required so that everyone shakes hands with everyone else?

7)  Möbius strip : An amazing shape which is a loop with only 1 side and 1 edge.

8) Klein bottle

9) Logic and sets

10)  Codes and ciphers : ISBN codes and credit card codes are just some examples of how codes are essential to modern life. Maths can be used to both make these codes and break them.

11)  Zeno’s paradox of Achilles and the tortoise : How can a running Achilles ever catch the tortoise if in the time taken to halve the distance, the tortoise has moved yet further away?

12)  Four colour map theorem  – a puzzle that requires that a map can be coloured in so that every neighbouring country is in a different colour. What is the minimum number of colours needed for any map?

13)  Telephone Numbers  – these are numbers with special properties which grow very large very quickly. This topic links to graph theory.

14) The Poincare Conjecture and Grigori Perelman  – Learn about the reclusive Russian mathematician who turned down $1 million for solving one of the world’s most difficult maths problems.

Mathematics and Physics

1)  The Monkey and the Hunter – How to Shoot a Monkey  – Using Newtonian mathematics to  decide where to aim when shooting a monkey in a tree.

2)  How to Design a Parachute  – looking at the physics behind parachute design to ensure a safe landing!

3)  Galileo: Throwing cannonballs off The Leaning Tower of Pisa  – Recreating Galileo’s classic experiment, and using maths to understand the surprising result.

4)  Rocket Science and Lagrange Points  – how clever mathematics is used to keep satellites in just the right place.

5)  Fourier Transforms – the most important tool in mathematics?  – An essential component of JPEG, DNA analysis, WIFI signals, MRI scans, guitar amps – find out about the maths behind these essential technologies.

6)  Bullet projectile motion experiment  – using Tracker software to model the motion of a bullet.

7)  Quantum Mechanics – a statistical universe?   Look at the inherent probabilistic nature of the universe with some quantum mechanics.

8)  Log Graphs to Plot Planetary Patterns . The planets follow a surprising pattern when measuring their distances.

9)  Modeling with springs and weights . Some classic physics – which generates some nice mathematical graphs.

10)  Is Intergalactic space travel possible?  Using the physics of travel near the speed of light to see how we could travel to other stars.

Maths and computing

1)  The Van Eck Sequence  – The Van Eck Sequence is a sequence that we still don’t fully understand – we can use programing to help!

2)  Solving maths problems using computers  – computers are really useful in solving mathematical problems.  Here are some examples solved using Python.

3)  Stacking cannonballs – solving maths with code  – how to stack cannonballs in different configurations.

4)  What’s so special about 277777788888899?  – Playing around with multiplicative persistence – can you break the world record?

5)  Project Euler: Coding to Solve Maths Problems . A nice starting point for students good at coding – who want to put these skills to the test mathematically.

6)  Square Triangular Numbers .  Can we use a mixture of pure maths and computing to solve this problem?

7)  When do 2 squares equal 2 cubes?  Can we use a mixture of pure maths and computing to solve this problem?

8)  Hollow Cubes and Hypercubes investigation .  More computing led investigations

9)  Coding Hailstone Numbers .  How can we use computers to gain a deeper understanding of sequences?

Further ideas:

1)  Radiocarbon dating  – understanding radioactive decay allows scientists and historians to accurately work out something’s age – whether it be from thousands or even millions of years ago.

2)  Gravity, orbits and escape velocity  – Escape velocity is the speed required to break free from a body’s gravitational pull. Essential knowledge for future astronauts.

3)  Mathematical methods in economics  – maths is essential in both business and economics – explore some economics based maths problems.

4)  Genetics  – Look at the mathematics behind genetic inheritance and natural selection.

5)  Elliptical orbits  – Planets and comets have elliptical orbits as they are influenced by the gravitational pull of other bodies in space. Investigate some rocket science!

6)  Logarithmic scales  – Decibel, Richter, etc. are examples of log scales – investigate how these scales are used and what they mean.

7)  Fibonacci sequence and spirals in nature  – There are lots of examples of the Fibonacci sequence in real life – from pine cones to petals to modelling populations and the stock market.

8)  Change in a person’s BMI over time  – There are lots of examples of BMI stats investigations online – see if you can think of an interesting twist.

9)  Designing bridges  – Mathematics is essential for engineers such as bridge builders – investigate how to design structures that carry weight without collapse.

10)  Mathematical card tricks  – investigate some maths magic.

11)  Flatland by Edwin Abbott  – This famous book helps understand how to imagine extra dimension. You can watch a short video on it  here

12)  Towers of Hanoi puzzle  – This famous puzzle requires logic and patience. Can you find the pattern behind it?

13)  Different number systems  – Learn how to add, subtract, multiply and divide in Binary. Investigate how binary is used – link to codes and computing.

14)  Methods for solving differential equations  – Differential equations are amazingly powerful at modelling real life – from population growth to to pendulum motion. Investigate how to solve them.

15)  Modelling epidemics/spread of a virus  – what is the mathematics behind understanding how epidemics occur? Look at  how infectious Ebola really is .

16)  Hyperbolic functions  – These are linked to the normal trigonometric functions but with notable differences. They are useful for modelling more complex shapes.

17)  Medical data mining  – Explore the use and misuse of statistics in medicine and science.

18) Waging war with maths: Hollow squares .  How mathematical formations were used to fight wars.

19)  The Barnsley Fern: Mathematical Art  – how can we use iterative processes to create mathematical art?

Share this:

39 thoughts on “ maths ia – 300 maths exploration topics ”.

this website is a real reference for me, thanks!

Very interesting topics for the students who whish to go beyond the curriculum

My name is Eden and this is so good resource. Much learn of math over up in here!! $$$ E.B.D.

Dis resources halp meh al0t

Very very good thanks

I need some good quality feedback on my exploration paper! Please contact me if you can help me out!

orayt rock n roll

Hi my name is Layal and I really like this Hong does too

it really good topics in math and i really like it and its new in math.

Awesome, thx. <3

amazing thank you

It help so much when i see math topic for studie and i use experiment to expand ideas among mathmatical world for further education. I was feared zombies so i did equation to predicatate

Thank you! Good resources for ideas

Hi would investigating the formula 1 racing strategy using game theory and monte carlo simulation be a good enough topic for hl?

Thanx for great ideas

very very useful 🙂

i tried to copy the name “i tried to copy the name but it was too long” but it was too long.

very good really useful.

+ first comment in 2 years

Powered by WordPress.com .

Frontiers | Science News

• Science News

Research Topics

Navigating the trust gap: three research topics on tech, ai, and the future of public confidence in science.

Technological advancements have changed how we consume and interact with information. Social media, search engines, and artificial intelligence (AI)-powered algorithms make knowledge accessible at an unprecedented scale. However, these developments have also created new opportunities for misinformation and manipulation, ultimately undermining public trust in institutions, including scientific ones.

In the information age, fostering trust and confidence among the public –particularly in the science community—prompted a community of scientists to establish the Research Topic The Erosion of Trust in the 21st Century: Origins, Implications, and Solutions . Inspired by the work of this group of researchers, we’ve curated three Research Topics that investigate the complex interplay between technology, health, and society's trust in scientific research.

These topics explore reflections on how to develop technology that benefits society, the implications of using AI in public health data, and public risk perception and its impacts on policy, decision-making, and innovation.

All articles are openly available to view and download.

1 | Technology For the Greater Good? The Influence of (Ir)responsible Systems on Human Emotions, Thinking and Behavior

56.500 views | 11 articles

This Research Topic underlines the (un)intended effects technology may have on individuals and societies, stressing dilemmas for stakeholders and pointing out good and bad practices in designing, creating, and using technology.

In recent years, technological advancements, particularly in AI, have significantly shaped the perceptions, cognitions, emotions, and behavior of a large population.

This underscores the need for deep considerations in technology, as it can work in both beneficial and detrimental ways. Notable examples include:

the use of social media for crowdfunding

chatbots for public opinion influence

the challenges of fallible biometrics

the impact of recommender systems on decision-making

the use of companion robots.

View Research Topic

2 | Extracting Insights from Digital Public Health Data using Artificial Intelligence, Volume II

31.700 views | 12 articles

This Research Topic platforms the current state-of-the-art artificial intelligence techniques on digital public health data. It contains critical insights to inform researchers, health practitioners, policymakers, and governments' decision-making.

Recent advancements in AI techniques and graphics processing unit computing capabilities have made it possible to process large volumes of data and extract valuable insights within short periods.

Although harnessing the power of AI can lead to exciting and groundbreaking digital public health research, it should be accompanied by effective health communication and mechanisms for detecting misinformation.

3 | Public risk perception in public health policies

21.000 views | 23 articles

This Research Topic investigates the complexities of public risk perception and its impacts on policy, decision-making, innovation, and public health. It addresses how public risk perception affects technology adoption and public health interventions, including cases where reluctance hinders beneficial innovations despite the evidence.

Public risk perception influences policymakers' responses to potential risks, impacting innovation and people's choices. The efforts of these scientists enhance our understanding of these drivers to improve risk communication and management strategies.

Post related info

September 24, 2024

Frontiers Science Communications

Post categories, related subjects, research topics, related content.

Three Research Topics exploring dementia diagnosis and treatment

Youth + technology: three Research Topics on empowering the next generation

Frontiers' Research Topic publishing program: pioneering the future of scientific publishing

Latest posts.

Rising Waters, Waning Forests: How scientists are using tree rings to study how rising sea levels affect coastal forests

New indoor vertical farming research could help future-proof food demand for a changing planet

Oceanic life found to be thriving thanks to Saharan dust blown from thousands of kilometers away

Your Zoom background could influence how tired you feel after a video call

Get the latest research updates, subscribe to our newsletter

Pure Mathematics Research

Pure mathematics fields.

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

Pure Math Committee

Masks Strongly Recommended but Not Required in Maryland

Respiratory viruses continue to circulate in Maryland, so masking remains strongly recommended when you visit Johns Hopkins Medicine clinical locations in Maryland. To protect your loved one, please do not visit if you are sick or have a COVID-19 positive test result. Get more resources on masking and COVID-19 precautions .

• Vaccines  
• Masking Guidelines
• Visitor Guidelines  

Specially Designed Video Games May Benefit Mental Health of Children and Teenagers

Can video games be good for children’s health? An @HopkinsKids study shows that video games designed for children’s mental well-being may have some benefit for certain conditions. ›

In a review of previous studies, a Johns Hopkins Children’s Center team concludes that some video games created as mental health interventions can be helpful – if modest – tools in improving the mental well-being of children and teens with anxiety, depression and attention-deficit/hyperactivity disorder (ADHD).

A report on the review of studies from peer-reviewed journals between 2011 and March 20, 2024, was published Sept. 23, 2024, in JAMA Pediatrics .

An estimated 20% of children and teenagers between the ages of three and 17 in the U.S. have a mental, emotional, developmental or behavioral disorder. Suicidal behaviors among high school students also increased by more than 40% in the 10 years prior to 2019, according to a report by the Agency for Healthcare Research and Quality . Other studies provide evidence that the COVID-19 pandemic’s disruptions worsened these trends, and while research suggests parents and other care givers are seeking out mental health care for children, wait times for appointments have increased.

“We found literature that suggests that even doubling the number of pediatric mental health providers still wouldn’t meet the need,” says Barry Bryant, M.D., a resident in the Department of Psychiatry and Behavioral Sciences in the Johns Hopkins University School of Medicine and first author of the new study.

In a bid to determine if so-called “gamified digital mental health interventions,” or video games designed to treat mental health conditions, benefited those with anxiety, depression and ADHD, the research team analyzed their use in randomized clinical trials for children and adolescents.

Bryant and child and adolescent psychologist Joseph McGuire, Ph.D. , identified 27 such trials from the U.S. and around the world. The studies overall included 2,911 participants with about half being boys and half being girls, between the ages of six and 17 years old.

The digital mental health interventions varied in content, but were all created with the intent of treating ADHD, depression and anxiety. For example, for ADHD, some of the games involved racing or splitting attention, which required the user to pay attention to more than one activity to be successful in gameplay. For depression and anxiety, some of the interventions taught psychotherapy-oriented concepts in a game format. All games were conducted on technology platforms, such as computers, tablets, video game consoles and smartphones. The video games are available to users in a variety of ways — some are available online, while others required access through specific research teams involved in the studies.

Outcome measurements varied depending on the study. However, the Johns Hopkins research team was able to standardize effect sizes using a random-effects model so that a positive result indicated when interventions performed better than control conditions. Hedges g , a statistic used to measure effect size, was used to quantify treatment effects overall in the studies reviewed.

The research team’s analysis found that video games designed for patients with ADHD and depression provided a modest reduction (both with an effect size of .28) in symptoms related to ADHD and depression, such as improved ability to sustain attention and decreased sadness, based on participant and family feedback from the studies. (An effect size of .28 is consistent with a smaller effect size, where as in-person interventions often produce moderate — .50 — to large — .80 — effects.)  By contrast, video games designed for anxiety did not show meaningful benefits (effect size of .07) for reducing anxiety symptoms for participants, based on participant and family feedback.

Researchers also examined factors that led to improved benefit from digital mental health interventions. Specific factors related to video game delivery (i.e., interventions on computers and those with preset time limits) and participants (i.e., studies that involved more boys) were found to positively influence therapeutic effects. Researchers say these findings suggest potential ways to improve upon the current modest symptom benefit.

“While the benefits are still modest, our research shows that we have some novel tools to help improve children’s mental health — particularly for ADHD and depression — that can be relatively accessible to families,” says Joseph McGuire, Ph.D., an author of the study and an associate professor of psychiatry and behavioral sciences in the school of medicine. “So if you are a pediatrician and you’re having trouble getting your pediatric patient into individual mental health care, there could be some gamified mental health interventions that could be nice first steps for children while waiting to start individual therapy.”

The team cautioned that their review did not indicate why certain video game interventions performed better than others. They also note that some of the trials included in the study used parent- or child-reported outcome measures, rather than standardized clinician ratings, and the studies did not uniformly examine the same factors or characteristics, such as participants’ engagement and social activities, which could have influenced the effects of the treatment. They also found that some of the video games included in the studies are not easily accessible, since they are not available online or are behind pay walls.

The researchers also noted that while video game addiction and the amount of screen time can be concerns, those children who played the games studied in a structured, time-limited format tended to do best. “If a child has a video game problem, they are often playing it for several hours a day as opposed to a gamified digital mental health intervention that might be 20-45 minutes, three times a week,” Bryant says.

“I think having many tools in the toolbox can be helpful to confront the increasing demand for child mental health care,” McGuire says.

Morgan Sisk from University of Alabama at Birmingham was also a study author.

The study was funded by generous donors and Johns Hopkins Medicine. The authors affiliated with The Johns Hopkins University did not declare any conflicts of interest under Johns Hopkins University policies.

Johns Hopkins Children’s Center Johns Hopkins Children’s Center Division of Child and Adolescent Psychiatry