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 realworld 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 timebound (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 welldefined, relevant, and poised to make a meaningful contribution to the field of mathematics.
“Exploring the Dynamics of Chaos: A Study of Fractal Patterns and Nonlinear Systems” 
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 SelfSimilarity 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: NonEuclidean 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: BlackScholes 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 DecisionMaking 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 GameTheoretic 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
 HighPerformance 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 HighDimensional 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 MeasurePreserving 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 ErrorCorrecting Codes
 Information Theory: Study of Information and Communication
 Coding Theory: Study of ErrorCorrecting 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 HigherDegree Equations
 Modular Forms: Analytic Functions with Certain Transformation Properties
 Lfunctions: 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
 Lfunctions: 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: Onedimensional 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 CalabiYau 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
 GromovWitten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
 Mirror Symmetry: Duality Between Symplectic and Complex Geometry
 CalabiYau Manifolds: RicciFlat Complex Manifolds
 Moduli Spaces: Spaces Parameterizing Geometric Objects
 DonaldsonThomas Invariants: Invariants Counting Sheaves on CalabiYau Manifolds
 Algebraic KTheory: 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 realworld 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 topnotch 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 reallife 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
 Realworld 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)
 Indepth 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
 Indepth analysis of the squarefree polynomial
 Discuss the SternBrocot 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 indepth 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 zeroone law
 What is a discrete random variable?
 Analyze the Hewitt–Savage zeroone law
 What is a transferable belief model?
 Discuss 3 major mathematical theorems
 Describe and analyze the DempsterShafer theory
 An indepth analysis of a continuous stochastic process
 Identify and analyze GaussMarkov 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 realworld 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 indepth analysis of Morley’s categoricity theorem
 How does the Backandforth method work?
 Discuss the Ehrenfeucht–Fraïssé game technique
 Discuss Aleph numbers (Alephnull and Alephone)
 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 ArtinWedderburn 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 MyhillNerode 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 twoelement 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 indepth 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 indepth analysis of fractals
 Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
 Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
 An indepth 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 indepth 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 MoivreLaplace 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 bestrated online writing company on the Internet because we always deliver highquality academic content at the most competitive prices. Give us a try today!
Leave a Reply Cancel reply
Research Areas
Analysis & pde, applied math, combinatorics, financial math, number theory, probability, representation theory, symplectic geometry & topology.
 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
 PartTime Faculty
 Emeritus Faculty
 PostDoctoral Associates
 Graduate Students
 Stay in Touch
 Newsletter Archive
 Upcoming Events
 Past Events
 Prospective Students
Department members engage in cuttingedge 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 CarnotCaratheodory 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 hightech 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 Mtheory.
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 everburgeoning 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 halfcourses) 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 fulltime 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 twohour, 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 .
Nonnative 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 upperlevel 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.
Applied Mathematics
Faculty and students interested in the applications of mathematics are an integral part of the department of mathematics; there is no formal separation between pure and applied mathematics, and the department takes pride in the many ways in which they enrich each other. we also benefit tremendously from close collaborations with faculty and students in other departments at uc berkeley as well as scientists at lawrence berkeley national laboratory and visitors to the mathematical sciences research institute.
The Department regularly offers courses in ordinary and partial differential equations and their numerical solution, discrete applied mathematics, the methods of mathematical physics, mathematical biology, the mathematical aspects of fluid and solid mechanics, approximation theory, scientific computing, numerical linear algebra, and mathematical aspects of computer science. Courses in probability theory, stochastic processes, data analysis and bioinformatics are offered by the Department of Statistics, while courses in combinatorial and convex optimization are offered by the Department of Industrial Engineering and Operations Research. Our students are encouraged to take courses of mathematical interest in these and many other departments. Topics explored intensively by our faculty and students in recent years include scientific computation and the mathematical aspects of quantum theory, computational genomics, image processing and medical imaging, inverse problems, combinatorial optimization, control, robotics, shape optimization, turbulence, hurricanes, microchip failure, MEMS, biodemography, population genetics, phylogenetics, and computational approaches to historical linguistics. Within the department we also have a Laboratory for Mathematical and Computational Biology .
Chair: PerOlof Persson
Applied Mathematics Faculty, Courses, Dissertations
Senate faculty, graduate students, visiting faculty, meet our faculty, mina aganagic, david aldous, robert m. anderson, sunčica čanić, jennifer chayes, alexandre j. chorin, paul concus, james w. demmel, l. craig evans, steven n. evans, f. alberto grünbaum, venkatesan guruswami, ole h. hald, william m. kahan, richard karp, michael j. klass, hendrik w. lenstra, jr., lin lin (林霖), michael j. lindsey, c. keith miller, john c. neu, beresford n. parlett.
 1 of 2 Grid: People (Current page)
 2 of 2 Grid: People
 next › Grid: People
 last » Grid: People
MSc by Research in Mathematics
 Entry requirements
 Funding and costs
College preference
 How to apply
About the course
The MSc by Research is an advanced research degree which provides the opportunity to investigate a project in depth and write a thesis which makes a significant contribution in the field. The research project is however designed to take less time than a Doctorate degree (normally two years, though it is possible to complete the requirements in a single year). It is not intended as a first step towards a DPhil, but rather as an alternative to a DPhil. Few students opt to apply for the MSc by Research unless there are limitations on the time or funding available.
There are no associated lectures, classes or written examinations. Your project can be in any of the subject areas for which supervision is available. The department’s research covers the entire spectrum of mathematics, with subject areas including:
 Algebra (primarily group theory and representation theory)
 Mathematical biology and ecology
 Industrial and applied mathematics (including fluid and solid mechanics, geosciences, mathematical physiology, methodologies, and networks)
 Numerical analysis.
You will be asked to outline your research interests when you apply by listing at least one but no more than three of the fields of research above on your application form. More information about the Research Groups in the Mathematical Institute can be found on the department's website. Full instructions for completing this section of the application form can be found in the How to apply section of this page.
You will be expected to acquire transferable skills as part of your training, which will require you to attend courses, lectures, workshops and colloquia. You will have the opportunity to develop other valuable skills and to contribute to the teaching work of the department, both by marking students’ work and later by leading classes of around eight to twelve students.
Supervision
You will be invited to suggest a specific supervisor or supervisors in your application form, and your preferences will be taken into account in allocating you a supervisor (which will be done before your arrival). The allocation of graduate supervision for this course is the responsibility of the Mathematical Institute and it is not always possible to accommodate the preferences of incoming graduate students to work with a particular member of staff. Under exceptional circumstances a cosupervisor may be found outside the Mathematical Institute.
Students are expected to meet with their supervisors at least four times per term. A more typical pattern is weekly, at least until you reach the stage of writing up your thesis.
Initially, you will be admitted as a Probationer Research Student in the same way as those intending to do a DPhil.
Within four terms of admission as a PRS student you will be expected to apply for transfer of status from Probationer Research Student to MSc by Research status.
A successful transfer of status from PRS to MSc by Research status will require satisfactory attendance and a successful performance in an oral examination in front of two appointed assessors.
You will be expected to submit a substantial original thesis after two or, at most, three years from the date of admission. To be successfully awarded a MSc by Research in Mathematics you will need to defend your thesis orally ( viva voce ) in front of two appointed examiners.
Graduate destinations
The department, working alongside the University’s Careers Service, supports graduate students as they move from the MSc by Research to the next stage of their career. Many graduate students stay in academia, by taking up a postdoctoral position, and many move into employment in a range of industries and sectors where the expertise and skills developed during the degree are highly valued.
Changes to this course and your supervision
The University will seek to deliver this course in accordance with the description set out in this course page. However, there may be situations in which it is desirable or necessary for the University to make changes in course provision, either before or after registration. The safety of students, staff and visitors is paramount and major changes to delivery or services may have to be made in circumstances of a pandemic, epidemic or local health emergency. In addition, in certain circumstances, for example due to visa difficulties or because the health needs of students cannot be met, it may be necessary to make adjustments to course requirements for international study.
Where possible your academic supervisor will not change for the duration of your course. However, it may be necessary to assign a new academic supervisor during the course of study or before registration for reasons which might include illness, sabbatical leave, parental leave or change in employment.
For further information please see our page on changes to courses and the provisions of the student contract regarding changes to courses.
Entry requirements for entry in 202425
Proven and potential academic excellence.
The requirements described below are specific to this course and apply only in the year of entry that is shown. You can use our interactive tool to help you evaluate whether your application is likely to be competitive .
Please be aware that any studentships that are linked to this course may have different or additional requirements and you should read any studentship information carefully before applying.
Degreelevel qualifications
As a minimum, applicants should hold or be predicted to achieve the following UK qualifications or their equivalent:
 a firstclass undergraduate degree with honours in mathematics or a related discipline.
A previous master's degree is not required, though the requirement for a firstclass undergraduate degree with honours can be alternatively demonstrated by strong performance in a master's degree.
For applicants with a degree from the USA, the minimum GPA sought is 3.7 out of 4.0.
If your degree is not from the UK or another country specified above, visit our International Qualifications page for guidance on the qualifications and grades that would usually be considered to meet the University’s minimum entry requirements.
GRE General Test scores
No Graduate Record Examination (GRE) or GMAT scores are sought.
Other qualifications, evidence of excellence and relevant experience
 Research or working experience in the proposed research may be an advantage.
 Publications are not expected.
English language proficiency
This course requires proficiency in English at the University's standard level . If your first language is not English, you may need to provide evidence that you meet this requirement. The minimum scores required to meet the University's standard level are detailed in the table below.
Test  Minimum overall score  Minimum score per component 

IELTS Academic (Institution code: 0713)  7.0  6.5 
TOEFL iBT, including the 'Home Edition' (Institution code: 0490)  100  Listening: 22 Reading: 24 Speaking: 25 Writing: 24 
C1 Advanced*  185  176 
C2 Proficiency  185  176 
*Previously known as the Cambridge Certificate of Advanced English or Cambridge English: Advanced (CAE) † Previously known as the Cambridge Certificate of Proficiency in English or Cambridge English: Proficiency (CPE)
Your test must have been taken no more than two years before the start date of your course. Our Application Guide provides further information about the English language test requirement .
Declaring extenuating circumstances
If your ability to meet the entry requirements has been affected by the COVID19 pandemic (eg you were awarded an unclassified/ungraded degree) or any other exceptional personal circumstance (eg other illness or bereavement), please refer to the guidance on extenuating circumstances in the Application Guide for information about how to declare this so that your application can be considered appropriately.
You will need to register three referees who can give an informed view of your academic ability and suitability for the course. The How to apply section of this page provides details of the types of reference that are required in support of your application for this course and how these will be assessed.
Supporting documents
You will be required to supply supporting documents with your application. The How to apply section of this page provides details of the supporting documents that are required as part of your application for this course and how these will be assessed.
Performance at interview
Technical interviews are normally held as part of the admissions process.
If invited you can expect to be interviewed by at least two people and for the interview to last around 30 minutes. The interview could take place facetoface or remotely.
It is expected that interviews will take place around three to five weeks after an application deadline.
How your application is assessed
Your application will be assessed purely on your proven and potential academic excellence and other entry requirements described under that heading.
References and supporting documents submitted as part of your application, and your performance at interview (if interviews are held) will be considered as part of the assessment process. Whether or not you have secured funding will not be taken into consideration when your application is assessed.
An overview of the shortlisting and selection process is provided below. Our ' After you apply ' pages provide more information about how applications are assessed .
Shortlisting and selection
Students are considered for shortlisting and selected for admission without regard to age, disability, gender reassignment, marital or civil partnership status, pregnancy and maternity, race (including colour, nationality and ethnic or national origins), religion or belief (including lack of belief), sex, sexual orientation, as well as other relevant circumstances including parental or caring responsibilities or social background. However, please note the following:
 socioeconomic information may be taken into account in the selection of applicants and award of scholarships for courses that are part of the University’s pilot selection procedure and for scholarships aimed at underrepresented groups ;
 country of ordinary residence may be taken into account in the awarding of certain scholarships; and
 protected characteristics may be taken into account during shortlisting for interview or the award of scholarships where the University has approved a positive action case under the Equality Act 2010.
Initiatives to improve access to graduate study
This course is taking part in a continuing pilot programme to improve the selection procedure for graduate applications, in order to ensure that all candidates are evaluated fairly.
For this course, socioeconomic data (where it has been provided in the application form) will be used to contextualise applications at the different stages of the selection process. Further information about how we use your socioeconomic data can be found in our page about initiatives to improve access to graduate study.
If you wish, you may submit an additional contextual statement (using the instructions in the How to apply section of this page) to provide further information on your socioeconomic background or personal circumstances in support of your application. Further information about how your contextual statement will be used can be found in our page about initiatives to improve access to graduate study.
Processing your data for shortlisting and selection
Information about processing special category data for the purposes of positive action and using your data to assess your eligibility for funding , can be found in our Postgraduate Applicant Privacy Policy.
Admissions panels and assessors
All recommendations to admit a student involve the judgement of at least two members of the academic staff with relevant experience and expertise, and must also be approved by the Director of Graduate Studies or Admissions Committee (or equivalent within the department).
Admissions panels or committees will always include at least one member of academic staff who has undertaken appropriate training.
Other factors governing whether places can be offered
The following factors will also govern whether candidates can be offered places:
 the ability of the University to provide the appropriate supervision for your studies, as outlined under the 'Supervision' heading in the About section of this page;
 the ability of the University to provide appropriate support for your studies (eg through the provision of facilities, resources, teaching and/or research opportunities); and
 minimum and maximum limits to the numbers of students who may be admitted to the University's taught and research programmes.
Offer conditions for successful applications
If you receive an offer of a place at Oxford, your offer will outline any conditions that you need to satisfy and any actions you need to take, together with any associated deadlines. These may include academic conditions, such as achieving a specific final grade in your current degree course. These conditions will usually depend on your individual academic circumstances and may vary between applicants. Our ' After you apply ' pages provide more information about offers and conditions .
In addition to any academic conditions which are set, you will also be required to meet the following requirements:
Financial Declaration
If you are offered a place, you will be required to complete a Financial Declaration in order to meet your financial condition of admission.
Disclosure of criminal convictions
In accordance with the University’s obligations towards students and staff, we will ask you to declare any relevant, unspent criminal convictions before you can take up a place at Oxford.
Academic Technology Approval Scheme (ATAS)
Some postgraduate research students in science, engineering and technology subjects will need an Academic Technology Approval Scheme (ATAS) certificate prior to applying for a Student visa (under the Student Route) . For some courses, the requirement to apply for an ATAS certificate may depend on your research area.
The Mathematical Institute's home is the purposebuilt Andrew Wiles Building, opened in 2013. This provides ample teaching facilities for lectures, classes and seminars. Each research student is allocated an office in the Andrew Wiles Building that they will share with 3 or 4 other students: each student has their own desk, with a computer. The Mathematical Institute provides IT support, and students can use the department's Whitehead Library, with an extensive range of books and journals.
In addition to the common room, where graduate students regularly gather for coffee and other social occasions, there is also a café in the Andrew Wiles Building.
The department offers extensive support to students, from regular skills training and career development sessions to a variety of social events in a welcoming and inclusive atmosphere. You will have the opportunity to interact with fellow students and other members of your research groups, and more widely across the department. The department is committed to offering you the best supervision and to providing a stimulating research environment.
Mathematics
Mathematics has been studied in Oxford since the University was first established in the 12th century. The Mathematical Institute aims to preserve and expand mathematical culture through excellence in teaching and research.
The Mathematical Institute offers a wide range of graduate courses, including both taught master’s courses and research degrees. Research and teaching covers the spectrum of pure and applied mathematics with researchers working in fields including:
 number theory
 combinatorics
 mathematical physics
 mathematical finance
 mathematical modelling
 mathematical biology
 numerical analysis.
Graduate students are an integral part of the department, interacting with each other and with academic staff as part of a vibrant community that strives to further mathematical study. As a graduate student at Oxford you will benefit from excellent resources, extensive training opportunities and supportive guidance from your supervisor or course director.
The Mathematical Institute has strong ties with other University departments including Computer Science, Statistics and Physics, teaching several courses jointly. Strong links with industrial and other partners are also central to the department.
View all courses View taught courses View research courses
The University expects to be able to offer over 1,000 full or partial graduate scholarships across the collegiate University in 202425. You will be automatically considered for the majority of Oxford scholarships , if you fulfil the eligibility criteria and submit your graduate application by the relevant December or January deadline. Most scholarships are awarded on the basis of academic merit and/or potential.
For further details about searching for funding as a graduate student visit our dedicated Funding pages, which contain information about how to apply for Oxford scholarships requiring an additional application, details of external funding, loan schemes and other funding sources.
Please ensure that you visit individual college websites for details of any collegespecific funding opportunities using the links provided on our college pages or below:
Please note that not all the colleges listed above may accept students on this course. For details of those which do, please refer to the College preference section of this page.
Annual fees for entry in 202425
Home  £9,500 
Overseas  £26,290 
Further details about fee status eligibility can be found on the fee status webpage.
Information about course fees
Course fees are payable each year, for the duration of your fee liability (your fee liability is the length of time for which you are required to pay course fees). For courses lasting longer than one year, please be aware that fees will usually increase annually. For details, please see our guidance on changes to fees and charges .
Course fees cover your teaching as well as other academic services and facilities provided to support your studies. Unless specified in the additional information section below, course fees do not cover your accommodation, residential costs or other living costs. They also don’t cover any additional costs and charges that are outlined in the additional information below.
Continuation charges
Following the period of fee liability , you may also be required to pay a University continuation charge and a college continuation charge. The University and college continuation charges are shown on the Continuation charges page.
Where can I find further information about fees?
The Fees and Funding section of this website provides further information about course fees , including information about fee status and eligibility and your length of fee liability .
Additional information
There are no compulsory elements of this course that entail additional costs beyond fees (or, after fee liability ends, continuation charges) and living costs. However, please note that, depending on your choice of research topic and the research required to complete it, you may incur additional expenses, such as travel expenses, research expenses, and field trips. You will need to meet these additional costs, although you may be able to apply for small grants from your department and/or college to help you cover some of these expenses.
Living costs
In addition to your course fees, you will need to ensure that you have adequate funds to support your living costs for the duration of your course.
For the 202425 academic year, the range of likely living costs for fulltime study is between c. £1,345 and £1,955 for each month spent in Oxford. Full information, including a breakdown of likely living costs in Oxford for items such as food, accommodation and study costs, is available on our living costs page. The current economic climate and high national rate of inflation make it very hard to estimate potential changes to the cost of living over the next few years. When planning your finances for any future years of study in Oxford beyond 202425, it is suggested that you allow for potential increases in living expenses of around 5% each year – although this rate may vary depending on the national economic situation. UK inflationary increases will be kept under review and this page updated.
Students enrolled on this course will belong to both a department/faculty and a college. Please note that ‘college’ and ‘colleges’ refers to all 43 of the University’s colleges, including those designated as societies and permanent private halls (PPHs).
If you apply for a place on this course you will have the option to express a preference for one of the colleges listed below, or you can ask us to find a college for you. Before deciding, we suggest that you read our brief introduction to the college system at Oxford and our advice about expressing a college preference . For some courses, the department may have provided some additional advice below to help you decide.
The following colleges accept students on the MSc by Research in Mathematics:
 Balliol College
 Brasenose College
 Christ Church
 Corpus Christi College
 Exeter College
 Green Templeton College
 Hertford College
 Jesus College
 Keble College
 Kellogg College
 Lady Margaret Hall
 Linacre College
 Lincoln College
 Magdalen College
 Mansfield College
 Merton College
 Oriel College
 Pembroke College
 The Queen's College
 Reuben College
 St Anne's College
 St Catherine's College
 St Cross College
 St Edmund Hall
 St Hilda's College
 St Hugh's College
 St Peter's College
 Somerville College
 University College
 Wadham College
 Wolfson College
 Worcester College
 Wycliffe Hall
Before you apply
Our guide to getting started provides general advice on how to prepare for and start your application. You can use our interactive tool to help you evaluate whether your application is likely to be competitive .
If it's important for you to have your application considered under a particular deadline – eg under a December or January deadline in order to be considered for Oxford scholarships – we recommend that you aim to complete and submit your application at least two weeks in advance . Check the deadlines on this page and the information about deadlines and when to apply in our Application Guide.
Application fee waivers
An application fee of £75 is payable per course application. Application fee waivers are available for the following applicants who meet the eligibility criteria:
 applicants from lowincome countries;
 refugees and displaced persons;
 UK applicants from lowincome backgrounds; and
 applicants who applied for our Graduate Access Programmes in the past two years and met the eligibility criteria.
You are encouraged to check whether you're eligible for an application fee waiver before you apply.
Readmission for current Oxford graduate taught students
If you're currently studying for an Oxford graduate taught course and apply to this course with no break in your studies, you may be eligible to apply to this course as a readmission applicant. The application fee will be waived for an eligible application of this type. Check whether you're eligible to apply for readmission .
Do I need to contact anyone before I apply?
You do not need to make contact with the department before you apply but you are encouraged to visit the relevant departmental webpages to read any further information about your chosen course.
However, if you would like to speak to an academic member of staff involved in your preferred area of research, you may get in touch with them directly or via the course administrator using the contact details provided on this page.
Completing your application
You should refer to the information below when completing the application form, paying attention to the specific requirements for the supporting documents .
For this course, the application form will include questions that collect information that would usually be included in a CV/résumé. You should not upload a separate document. If a separate CV/résumé is uploaded, it will be removed from your application .
If any document does not meet the specification, including the stipulated word count, your application may be considered incomplete and not assessed by the academic department. Expand each section to show further details.
Proposed field and title of research project
Under the section titled 'Field and title of research project', you are strongly encouraged to name at least one but no more than three research groups that you would like your application to be seen by. More information about the Research Groups in the Mathematical Institute can be found on the department's website.
You should not use this field to type out a full research proposal. You will be able to upload your research supporting materials separately if they are required (as described below).
Proposed supervisor
If known, under 'Proposed supervisor name' enter the name of the academic(s) whom you would like to supervise your research. Otherwise, leave this field blank.
You can enter up to four names and you should list them in order of preference or indicate equal preference.
Referees: Three overall, academic preferred
Whilst you must register three referees, the department may start the assessment of your application if two of the three references are submitted by the course deadline and your application is otherwise complete. Please note that you may still be required to ensure your third referee supplies a reference for consideration.
Your references should generally be academic, though up to one professional reference will be accepted.
Your references should describe your intellectual ability, academic achievement, motivation, and aptitude for advanced research.
Official transcript(s)
Your transcripts should give detailed information of the individual grades received in your universitylevel qualifications to date. You should only upload official documents issued by your institution and any transcript not in English should be accompanied by a certified translation.
More information about the transcript requirement is available in the Application Guide.
Contextual statement
If you wish to provide a contextual statement with your application, you may also submit an additional statement to provide contextual information on your socioeconomic background or personal circumstances in support of your application.
Submit a contextual statement
It is not necessary to anonymise this document, as we recognise that it may be necessary for you to disclose certain information in your statement. This statement will not be used as part of the initial academic assessment of applications at shortlisting, but may be used in combination with socioeconomic data to provide contextual information during decisionmaking processes.
Please note, this statement is in addition to completing the 'Extenuating circumstances’ section of the standard application form .
You can find more information about the contextual statement on our page that provides details of the continuing pilot programme to improve the assessment procedure for graduate applications.
Statement of purpose/personal statement: A maximum of 1,000 words
Your statement should be written in English and explain your reasons for applying for the course at Oxford, your relevant experience and education, and the specific areas that interest you and/or you intend to specialise in. This will be assessed for evidence of motivation for and understanding of the proposed area of study and whether a suitable supervisor can be provided.
If possible, please ensure that the word count is clearly displayed on the document.
Start or continue your application
You can start or return to an application using the relevant link below. As you complete the form, please refer to the requirements above and consult our Application Guide for advice . You'll find the answers to most common queries in our FAQs.
Application Guide Apply
ADMISSION STATUS
Closed to applications for entry in 202425
Register to be notified via email when the next application cycle opens (for entry in 202526)
12:00 midday UK time on:
Friday 5 January 2024 Latest deadline for most Oxford scholarships Final application deadline for entry in 202425
Full Time Only  

Course code  RM_MS1 
Expected length  2 to 3 years 
Places in 202425^  See note 
Applications/year*  9 
Expected start  
English language 
^Included in 2024/25 places for the DPhil in Mathematics *Threeyear average (applications for entry in 202122 to 202324)
Further information and enquiries
This course is offered by the Mathematical Institute
 Course page on the Mathematical Institute's website
 Academic and research staff
 Departmental research
 Mathematical, Physical and Life Sciences
 Residence requirements for fulltime courses
 Postgraduate applicant privacy policy
Courserelated enquiries
Advice about contacting the department can be found in the How to apply section of this page
✉ [email protected] ☎ +44 (0)1865 615208
Applicationprocess enquiries
See the application guide
Quick links
 Directories
Research Areas
It is possible to apply mathematics to almost any field of human endeavor. Here are some of the fields we’re working on now.
Scientific Computing and Numerical Analysis
Researchers : Loyce Adams , Bernard Deconinck , Randy LeVeque , Ioana Dumitriu , Anne Greenbaum , James Riley
Many practical problems in science and engineering cannot be solved completely by analytical means. Research in the area of numerical analysis and scientific computation is concerned with the development and analysis of numerical algorithms, the implementation of these algorithms on modern computer architectures, and the use of numerical methods in conjunction with mathematical modeling to solve largescale practical problems. Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.Current research topics in CFD include:
 high resolution methods for solving nonlinear conservation laws with shock wave solutions
 numerical methods for atmospheric flows, particularly cloud formation
 Cartesian grid methods for solving multidimensional problems in complicated geometries on uniform grids
 spectral methods for fluid stability problems
 front tracking methods for fluid flow problems with free surfaces or immersed interfaces in the context of porous media flow (ground water or oil reservoir simulation) and in physiological flows with elastic membranes.
 nonequilibrium flows in combustion and astrophysical simulation
 immersed interface methods for solidification or melting problems and seismic wave equations with discontinuous coefficients that arise in modeling the geological structure of the earth.
Another research focus is the development of methods for largescale scientific computations that are suited to implementation on parallel computer architectures. Current interests include:
 preconditioners for the iterative solution of large linear or nonlinear systems
 methods for the symmetric and nonsymmetric eigenvalue problems
 methods for general interface problems in complicated domains.
The actual implementation and testing of methods on parallel architectures is possible through collaboration with the Department of Computer Science, the Boeing Company, and the Pacific Northwest Labs.
Nonlinear Waves and Coherent Structures
Researchers : Bernard Deconinck , Nathan Kutz , Randy LeVeque
Most problems in applied mathematics are inherently nonlinear. The effects due to nonlinearities may become important under the right circumstances. The area of nonlinear waves and coherent structures considers how nonlinear effects influence problems involving wave propagation. Sometimes these effects are desirable and lead to new applications (modelocked lasers, optical solitons and nonlinear optics). Other times one has no choice but to consider their impact (water waves). The area of nonlinear waves encompasses a large collection of phenomena, such as the formation and propagation of shocks and solitary waves. The area received renewed interest starting in the 1960s with the development of soliton theory, which examines completely integrable systems and classes of their special solutions.
Mathematical Biology
Researchers : Mark Kot , Hong Qian , Eric SheaBrown , Elizabeth Halloran , Suresh Moolgavkar , Eli Shlizerman , Ivana Bozic
Mathematical biology is an increasingly large and wellestablished branch of applied mathematics. This growth reflects both the increasing importance of the biological and biomedical sciences and an appreciation for the mathematical subtleties and challenges that arise in the modelling of complex biological systems. Our interest, as a group, lies in understanding the spatial and temporal patterns that arise in dynamic biological systems. Our mathematical activities range from reactiondiffusion equations, to nonlinear and chaotic dynamics, to optimization. We employ a variety of tools and models to study problems that arise in development, epidemiology, ecology, neuroscience, resource management, and biomechanics; and we maintain active collaborations with a large number and variety of biologists and biomedical departments both in the University and elsewhere. For more information, please see the Mathematical Biology page .
Atmospheric Sciences and Climate Modeling
Researchers : Chris Bretherton , KaKit Tung , Dale Durran
Mathematical models play a crucial role in our understanding of the fluid dynamics of the atmosphere and oceans. Our interests include mathematical methods for studying the hydrodynamical instability of shear flows, transition from laminar flow to turbulence, applications of fractals to turbulence, twodimensional and quasigeostrophic turbulence theory and computation, and largescale nonlinear wave mechanics.We also develop and apply realistic coupled radiative chemicaldynamical models for studying stratospheric chemistry, and coupled radiativemicrophysicaldynamical models for studying the interaction of atmospheric turbulence and cloud systems These two topics are salient for understanding how man is changing the earth’s climate.Our work involves a strong interaction of computer modelling and classical applied analysis. This research group actively collaborates with scientists in the Atmospheric Science, Oceanography, and Geophysics department, and trains students in the emerging interdisciplinary area of earth system modeling, in addition to providing a traditional education in classical fluid dynamics.
Mathematical Methods
Researchers : Bernard Deconinck , Robert O'Malley , Jim Burke , Archis Ghate , John Sylvester , Gunther Uhlmann
The department maintains active research in fundamental methods of applied mathematics. These methods can be broadly applied to a vast number of problems in the engineering, physical and biological sciences. The particular strengths of the department of applied mathematics are in asymptotic and perturbation methods, applied analysis, optimization and control, and inverse problems.
Mathematical Finance
Researchers : Tim Leung , Matt Lorig , Doug Martin
The department’s growing financial math group is active in the areas of derivative pricing & hedging, algorithmic trading, portfolio optimization, insurance, risk measures, credit risk, and systemic risk. Research includes collaboration with students as well as partners from both academia and industry.
 Mailing List
 YouTube
 News Feed
 Student intranet /
 Staff intranet
Department of Mathematics
Research projects
Find a postgraduate research project in your area of interest by exploring the research projects we offer in the Department of Mathematics.
Programme directors
If you are not sure which supervisors are the best match for your interests, contact the postgraduate programme directors:
 Sean Holman (applied mathematics and numerical analysis)
 Olatunji Johnson (probability, statistics and financial mathematics)
 Marcus Tressl (pure mathematics)
You can also get in touch with the postgraduate research leads through our research themes page.
Opportunities within the department are advertised by supervisors as either:
 Specific, welldefined individual projects : which you can apply for directly after contacting the named supervisor
 Research fields with suggestions for possible projects : where you can discuss a range of potential projects available in a specific area with the supervisor.
Choosing the right PhD project depends on matching your interests to those of your supervisor.
Our research themes page gives an overview of the research taking place in the Department and contacts for each area. Potential supervisors can also be contacted directly through the academic staff list . They will be able to tell you more about the type of projects they offer and/or you can suggest a research project yourself.
Please note that all PhD projects are eligible for funding via a variety of scholarships from the Department, the Faculty of Science and Engineering and/or the University; see our funding page for further details. All scholarships are awarded competitively by the relevant postgraduate funding committees.
Academics regularly apply for research grants and may therefore be able to offer funding for specific projects without requiring approval from these committees. Some specific funded projects are listed below, but many of our students instead arrive at a project through discussion with potential supervisors.
Specific, individual projects
Browse all of our specific, individual projects listed on FindAPhD:
Research field projects
In addition to individual projects listed on FindAPhD, we are also looking for postgraduate researchers for potential projects within a number of other research fields.
Browse these fields below and get in contact with the named supervisor to find out more.
Continuum mechanics
Maths: continuum mechanics, maths: mathematics in the life sciences, numerical analytics and scientific computing, statistics, inverse problems, uncertainty quantification and data science, algebra, logic and number theory, maths: algebra, logic and number theory, mathematics in the life sciences, maths: statistics, inverse problems, uncertainty quantification and data science, probability, financial mathematics and actuarial science, applied mathematics and numerical analysis, complex deformations of biological soft tissues.
Supervisor: [email protected]  [email protected]
The answers to many open questions in medicine depend on understanding the mechanical behaviour of biological soft tissues. For example, which tendon is most appropriate to replace the anterior cruciate ligament in reconstruction surgery? what causes the onset of aneurysms in the aorta? and how does the mechanics of the bladder wall affect afferent nerve firing? Current work at The University of Manchester seeks to understand how the microstructure of a biological soft tissue affects its macroscale mechanical properties. We have previously focused on developing nonlinear elastic models of tendons and are now seeking to incorporate more complex physics such as viscoelasticity, and to consider other biological soft tissues, using our “in house” finite element software oomphlib. The work will require development and implementation of novel constitutive equations as well as formulation of nonstandard problems in solid mechanics. The project is likely to appeal to students with an interest in continuum mechanics, computational mathematics and interdisciplinary science.
Fluid flow, interfaces, bifurcations, continuation and control
Supervisor: [email protected]
My research interests are in fluid dynamical systems with deformable interfaces, for example bubbles in very viscous fluid, or inkjet printed droplets. The deformability of the interface can lead to complex nonlinear behaviour, and often occurs in configurations where full numerical simulation of the threedimensional system is computationally impossible. This computational difficulty leaves an important role for mathematical modelling, in using asymptotic or physical arguments to devise simpler models which can help us understand underlying physical mechanisms, make testable predictions, and to directly access control problems for active (feedback) or passive control mechanisms. Most recently I am interested in how different modelling methodologies affect whether models are robust in the forward or control problems. I am also interested in how controlbased continuation methods can be used in continuum mechanics to directly observe unstable dynamical behaviour in experiments, even without access to a physical model. This research combines fluid dynamics, mathematical modelling, computational methods (e.g. with the finiteelement library oomphlib), experiments conducted in the Manchester Centre for Nonlinear Dynamics, control theory and nonlinear dynamics. I would not expect any student to have experience in all these areas and there is scope to shape any project to your interests.
Granular materials in industry and nature
Supervisor: [email protected]
The field of granular materials encompasses a vast range of materials and processes, from the formation of sand dunes on a beach and snow avalanches in the mountains, to the roasting of coffee beans and the manufacture of pharmaceutical tablets. The science of granular materials is still in its relative infancy, and many aspects of flowing grains cannot yet be predicted with a continuum rheology. Insights into granular material behaviour come from a range of methods, and my research therefore combines mathematical modelling, computation, and laboratory experiments, undertaken at the Manchester Centre for Nonlinear Dynamics laboratories. Some example areas of work suitable for a PhD project include:  Debris flows and their deposits Debris flows are rapid avalanches of rock and water, which are triggered on mountainsides when erodible sediment is destabilised by heavy rainfall or snowmelt. These flows cause loss of life and infrastructure across the world, but many of the physical mechanisms underlying their motion remain poorly understood. Because it is difficult to predict where and when a debris flow will occur, scientific observations are rarely made on an active flow. More often, all we have to work from is the deposit left behind, and some detective work is required to infer properties of the flow (such as its speed and composition) from this deposit. This project focuses on developing theoretical models for debris flows that predict both a debris flow and its deposit  in particular the way in which grains of different sizes are distributed throughout the deposit. The aim is then to invert such models, allowing observations of a deposit, when combined with model simulations, to constrain what must have happened during the flow.  Modelling polydispersity Much of the current theory of granular materials has been formulated with the assumption of a single type of grain. When grains vary in size, shape or density, it opens up the possibility that such grains with different properties separate from one another, a process called segregation. A fundamental question in this area is predicting the rate of segregation from a description of a granular material, such as the distribution of particle sizes. Thanks to some recent developments, we are approaching a point where this can be done for very simple granular materials (in particular those containing only two, similar, sizes sizes of grain), but many practical granular materials are much more complex. For example, it is common for mixtures of grains used in industry to vary in diameter by a factor of more than 100, and the complex segregation that can occur in these mixtures is poorly understood. This project will make measurements of the segregation behaviour of such mixtures and use these to put together a theoretical framework for describing segregation in complex granular materials.
Mathematical modelling of nanoreinforced foams
Supervisor: [email protected]
Complex materials are important in almost every aspect of our lives, whether that is using a cell phone, insulating a house, ensuring that transport is environmentally friendly or that packaging is sustainable. An important facet of this is to ensure that materials are optimal in some sense. This could be an optimal stiffness for a given weight or an optimal conductivity for a given stiffness. Foams are an important class of material that are lightweight but also have the potential for unprecedented mechanical properties by adding nanoreinforcements (graphene flakes or carbon nanotubes) into the background or matrix material from which the foam is fabricated. When coupled with experimentation such as imaging and mechanical testing, mathematical models allow us to understand how to improve the design and properties of such foams. A number of projects are available in this broad area and interested parties can discuss these by making contact with the supervisor.
Wave manipulation using metamaterials
The ability to control electromagnetic waves, sound, vibration has been of practical interest for decades. Over the last century a number of materials have been designed to assist with the attenuation of unwanted noise and vibration. However, recently there has been an explosion of interest in the topic of metamaterials and metasurfaces. Such media have special microstructures, designed to provide overall (dynamic) material properties that natural materials can never hope to attain and lead to the potential of negative refraction, wave redirection and the holy grail of cloaking. Many of the mechanisms to create these artificial materials rely on low frequency resonance. Frequently we are interested in the notion of homogenisation of these microstructures and this requires a mathematical framework. A number of projects are available in this broad area and interested parties can discuss these by making contact with the supervisor.
Multiscale modelling of structurefunction relationships in biological tissues
Supervisor: [email protected]
Biological tissues have an intrinsically multiscale structure. They contain components that range in size from individual molecules to the scale of whole organs. The organisation of individual components of a tissue, which often has a stochastic component, is intimately connected to biological function. Examples include exchange organs such as the lung and placenta, and developing multicellular tissues where mechanical forces play an crucial role in growth. To describe such materials mathematically, new multiscale approaches are needed that retain essential elements of tissue organisation at small scales, while providing tractable descriptions of function at larger scales. Projects are available in these areas that offer opportunities to collaborate with life scientists while developing original mathematical models relating tissue structure to its biological function.
Adaptive finite element approximation strategies
Supervisor: [email protected]
I would be happy to supervise projects in the general area of efficient solution of elliptic and parabolic partial differential equations using finite elements. PhD projects would involve a mix of theoretical analysis and the development of proofofconcept software written in MATLAB or Python. The design of robust and efficient error estimators is an open problem in computational fluid dynamics. Recent papers on this topic include Alex Bespalov, Leonardo Rocchi and David Silvester, TIFISS: a toolbox for adaptive FEM computation, Computers and Mathematics with Applications, 81: 373390, 2021. https://doi.org/10.1016/j.camwa.2020.03.005 Arbaz Khan, Catherine Powell and David Silvester, Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity, International Journal for Numerical Methods in Engineering, 119: 120, 2019. https://doi.org/10.1002/nme.6040 John Pearson, Jen Pestana and David Silvester, Refined saddlepoint preconditioners for discretized Stokes problems, Numerische Mathematik, 138: 331363, 2018. https://doi.org/10.1007/s0021101709084
Efficient solution for PDEs with random data
I would be happy to supervise projects in the general area of efficient solution of elliptic and parabolic partial differential equations with random data. PhD projects would involve a mix of theoretical analysis and the development of proofofconcept software written in MATLAB or Python. The design of robust and efficient error estimators for stochastic collocation approximation methods is an active area of research within the uncertainty quantification community. Recent papers on this topic include Alex Bespalov, David Silvester and Feng Xu. Error estimation and adaptivity for stochastic collocation finite elements Part I: singlelevel approximation, SIAM J. Scientific Computing, 44: A3393A3412, 2022. {\tt https://doi.org/10.1137/21M1446745} Arbaz Khan, Alex Bespalov, Catherine Powell and David Silvester, Robust a posteriori error estimators for stochastic Galerkin formulations of parameterdependent linear elasticity equations, Mathematics of Computation, 90: 613636, 2021. https://doi.org/10.1090/mcom/3572 Jens Lang, Rob Scheichl and David Silvester, A fully adaptive multilevel collocation strategy for solving elliptic PDEs with random data, J. Computational Physics, 419, 109692, 2020. https://doi.org/10.1016/j.jcp.2020.109692
Bayesian and machine learning methods for statistical inverse problems
Supervisor: [email protected]
A range of projects are available on the topic of statistical inverse problems, in particular with application to problems in applied mathematics. Our aim is to construct new methods for the solution of statistical inverse problems, and to apply them to real problems from science, biology, engineering, etc. These may be more traditional Markov chain Monte Carlo (MCMC) methods, Piecewisedeterministic Markov processes (PDMPs), gradient flows (e.g. Stein gradient descent), or entirely new families of methods. Where possible the methods will be flexible and widely applicable, which will enable us to also apply them to real problems and datasets. Some recent applications involve cell matching in biology, and characterisation of physical properties of materials, for example the thermal properties of a manmade material, or the Young's modulus of a tendon or artery. The project will require the candidate to be proficient in a modern programming language (e.g. Python).
Machine learning with partial differential equations
Supervisor: [email protected]
Machine learning and artificial intelligence play a major part in our everyday life. Selfdriving cars, automatic diagnoses from medical images, face recognition, or fraud detection, all profit especially from the universal applicability of deep neural networks. Their use in safety critical applications, however, is problematic: no interpretability, missing mathematical guarantees for network or learning process, and no quantification of the uncertainties in the neural network output. Recently, models that are based on partial differential equations (PDEs) have gained popularity in machine learning. In a classification problem, for instance, a PDE is constructed whose solution correctly classifies the training data and gives a suitable model to classify unlabelled feature vectors. In practice, feature vectors tend to be high dimensional and the natural space on which they live tends to have a complicated geometry. Therefore, partial differential equations on graphs are particularly suitable and popular. The resulting models are interpretable, mathematically wellunderstood, and uncertainty quantification is possible. In addition, they can be employed in a semisupervised fashion, making them highly applicable in small data settings. I am interested in various mathematical, statistical, and computational aspects of PDEbased machine learning. Many of those aspects translate easily into PhD projects; examples are  Efficient algorithms for pLaplacianbased regression and clustering  Bayesian identification of graphs from flow data  PDEs on random graphs  Deeply learned PDEs in data science Depending on the project, applicants should be familiar with at least one of: (a) numerical analysis and numerical linear algebra; (b) probability theory and statistics; (c) machine learning and deep learning.
Pure Mathematics and Logic
Algebraic differential equations and model theory.
Supervisor: [email protected]
Differential rings and algebraic differential equations have been a crucial source of examples for model theory (more specifically, geometric stability theory), and have had numerous application in number theory, algebraic geometry, and combinatorics (to name a few). In this project we propose to establish and analyse deep structural results on the model theory of (partial) differential fields. In particular, in the setup of differentially large fields. There are interesting questions around inverse problems in differential Galois theory that can be address as part of this project. On the other hand, there are (still open) questions related to the different notions of rank in differentially closed fields; for instance: are there infinite dimensional types that are also strongly minimal? This is somewhat related to the understanding of regular types, which interestingly are quite far from being fully classified. A weak version of Zilber's dichotomy have been established for such types, but is the full dichotomy true?
Algebraic invariants of abelian varieties
Supervisor: [email protected]
Project: Algebraic invariants of abelian varieties Abelian varieties are higherdimensional generalisations of elliptic curves, objects of algebraic geometry which are of great interest to number theorists. There are various open questions about how properties of abelian varieties vary across a families of abelian varieties. The aim of this project is to study the variation of algebraic objects attached to abelian varieties, such as endomorphism algebras, MumfordTate groups or isogenies. These algebraic objects control much of the behaviour of the abelian variety. We aim to bound their complexity in terms of the equations defining the abelian variety. Potential specific projects include: (1) Constructing "relations between periods" from the MumfordTate group. This involves concrete calculations of polynomials, similar in style to classical invariant theory of reductive groups. (2) Understanding the interactions between isogenies and polarisations of abelian varieties. This involves calculations with fundamental sets for arithmetic group actions, generalising reduction theory for quadratic forms. A key tool is the theory of reductive groups and their finitedimensional representations (roots and weights).
Algebraic Model Theory of Fields with Operators
Model theory is a branch of Mathematical Logic that has had several remarkable applications with other areas of mathematics, such as Combinatorics, Algebraic Geometry, Number Theory, Arithmetic Geometry, Complex and Real Analysis, Functional Analysis, and Algebra (to name a few). Some of these applications have come from the study of modeltheoretic properties of fields equipped with a family of operators. For instance, this includes differential/difference fields. In this project, we look at the model theory of fields equipped with general classes of operators and also within certain natural classes of arithmetic fields (such as large fields). Several foundational questions remain open around what is called "modelcompanion", "elimination of imaginaries", and the "trichotomy", this is a small sample of the problems that can be tackled.
Homeomorphism groups from a geometric viewpoint
Supervisor: [email protected]
A powerful technique for studying groups is to use their actions by isometries on metric spaces. Properties of the action can be translated into algebraic properties of the group, and vice versa. This is called geometric group theory, and has played a key role in different fields of mathematics e.g. random groups, mapping class groups of surfaces, fundamental groups of 3manifolds, the Cremona group. In this project we will study the homeomorphisms of a surface by using geometric group theoretic techniques recently introduced by Bowden, Hensel, and myself. This is a new research initiative at the frontier between dynamics, topology, and geometric group theory, and there are many questions waiting to be explored using these tools. These range from new questions on the relationship between the topology/dynamics of homeomorphisms and their action on metric spaces, to older questions regarding the algebraic structure of the homeomorphism group.
The Existential Closedness problem for exponential and automorphic functions
Supervisor: [email protected]
The Existential Closedness problem asks when systems of equations involving field operations and certain classical functions of a complex variable, such as exponential and modular functions, have solutions in the complex numbers. There are conjectures predicting when such systems should have solutions. The general philosophy is that when a system is not "overdetermined" (e.g. more equations than variables) then it should have a solution. The notion of an overdetermined system of equations is related to Schanuel's conjecture and its analogues and is captured by some purely algebraic conditions. The aim of this project is to make progress towards the Existential Closedness conjectures (EC for short) for exponential and automorphic functions (and the derivatives of automorphic functions). These include the usual complex exponential function, as well as the exponential functions of semiabelian varieties, and modular functions such as the jinvariant. Significant progress has been made towards EC in recent years, but the full conjectures are open. There are many special cases which are within reach and could be tackled as part of a PhD project. Methods used to approach EC come from complex analysis and geometry, differential algebra, model theory (including ominimality), tropical geometry. Potential specific projects are: (1) proving EC in low dimensions (e.g. for 2 or 3 variables), (2) proving EC for certain relations defined in terms of the function under consideration, e.g. establishing new EC results for "blurred" exponential and/or modular functions, (3) proving EC under additional geometric assumptions on the system of equations, (4) using EC to study the modeltheoretic properties of exponential and automorphic functions.
Statistics and Probability
Mathematical epidemiology.
Supervisor: [email protected]
Understanding patterns of disease at the population level  Epidemiology  is inherently a quantitative problem, and increasingly involves sophisticated researchlevel mathematics and statistics in both infectious and chronic diseases. The details of which diseases and mathematics offer the best PhD directions are likely to vary over time, but this broad area is available for PhD research.
Spatial and temporal modelling for crime
Supervisor: [email protected]  [email protected]
A range of projects are available on the topic of statistical spatial and temporal modelling for crime. These projects will focus on developing novel methods for modelling crime related events in space and time, and applying these to real world datasets, mostly within the UK, but with the possibility to use international datasets. Some examples of recent applications include spatiotemporal modelling of drug overdoses and related crime. These projects will aim to use statistical spatiotemporalpoint processes methods, Bayesian methods, and machine learning methods. The project will require the candidate to be proficient in a modern programming language (e.g., R or Python). Applicants should have achieved a firstclass degree in Statistics or Mathematics, with a significant component of Statistics, and be proficient in a statistical programming language (e.g., R, Python, Stata, S). We strongly recommend that you contact the supervisor(s) for this project before you apply. Please send your CV and a brief cover letter to [email protected] before you apply. At Manchester we offer a range of scholarships, studentships and awards at university, faculty and department level, to support both UK and overseas postgraduate researchers. For more information, visit our funding page or search our funding database for specific scholarships, studentships and awards you may be eligible for.
Distributional approximation by Stein's method Theme
Supervisor: [email protected]
Stein's method is a powerful (and elegant) technique for deriving bounds on the distance between two probability distributions with respect to a probability metric. Such bounds are of interest, for example, in statistical inference when samples sizes are small; indeed, obtaining bounds on the rate of convergence of the central limit theorem was one of the most important problems in probability theory in the first half of the 20th century. The method is based on differential or difference equations that in a sense characterise the limit distribution and coupling techniques that allow one to derive approximations whilst retaining the probabilistic intuition. There is an active area of research concerning the development of Stein's method as a probabilistic tool and its application in areas as diverse as random graph theory, statistical mechanics and queuing theory. There is an excellent survey of Stein's method (see below) and, given a strong background in probability, the basic method can be learnt quite quickly, so it would be possible for the interested student to make progress on new problems relatively shortly into their PhD. Possible directions for research (although not limited) include: extend Stein's method to new limit distributions; generalisations of the central limit theorem; investigate `faster than would be expected' convergence rates and establish necessary and sufficient conditions under which they occur; applications of Stein's method to problems from, for example, statistical inference. Literature: Ross, N. Fundamentals of Stein's method. Probability Surveys 8 (2011), pp. 210293.
Longterm behaviour of Markov Chains
Supervisor: [email protected]
Several projects are available, studying idealised Markovian models of epidemic, population and network processes. The emphasis will mostly be on theoretical aspects of the models, involving advanced probability theory. For instance, there are a number of stochastic models of epidemics where the course of the epidemic is known to follow the solution of a differential equation over short time intervals, but where little or nothing has been proved about the longterm behaviour of the stochastic process. Techniques have been developed for studying such problems, and a project might involve adapting these methods to new settings. Depending on the preference of the candidate, a project might involve a substantial computational component, gaining insights into the behaviour of a model, via simulations, ahead of proving rigorous theoretical results.
Secondary Menu
 Math Intranet
 Computational Mathematics
Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations, developing computational tools to better understand data and structure, etc. Computational mathematics is a field closely connected with a variety of other mathematical branches, as for often times a better mathematical understanding of the problem leads to innovative numerical techniques.
Duke's Mathematics Department has a large group of mathematicians whose research involves scientific computing, numerical analysis, machine learning, computational topology, and algorithmic algebraic geometry. The computational mathematics research of our faculty has applications in data analysis and signal processing, fluid and solid mechanics, electronic structure theory, biological networks, and many other topics.
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/0000000296043448 1 ,
 Jinfa Cai ORCID: orcid.org/0000000205013826 2 &
 Linda Zenger 1
32k Accesses
98 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 futureoriented 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 COVID19 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 followup 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 1question 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 Geoscience of Utrecht University (Bèta L19247). 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 doublecheck 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, selfreflection (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 COVID19 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, antiracism, 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, problembased learning, crosscurricular 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 challengebased 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 highquality 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 datadriven 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 antiexpertise and postfact 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 K12 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 HeuvelPanhuizen, 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 pandemicspecific 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 crossingrelated 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 selfregulation. 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 (sociopolitical 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, preservice 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, outoffield 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 subjectmatter knowledge and particularly noted that many teachers seem illprepared 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 postsecondary 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 COVID19 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 crosscutting 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 highquality 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 highquality mathematics education, and unfair distribution of resources. A related future research theme is how the socalled 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 psychologicalsocial phenomena such as emotion, love, belief, attitudes, interest, curiosity, fun, engagement, joy, involvement, motivation, selfesteem, 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 socalled eassessment, 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. BiknerAhsbahs & 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 socalled 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 multifaceted 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 crosssectional research, in particular in the context of teacher professional development:
It is not enough to investigate what happens in preservice 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 crosssectional 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 selfcriticism 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 reflectioninviting 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 socalled didactical, topicspecific research is not less important today but perhaps less fashionable for funding schemes that promote innovative, groundbreaking research. On the other hand, over time it has become clear that mathematics education is a multifaceted sociocultural 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 [contentspecific vs sociopolitical] 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 longerterm 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 teacherresearchers (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 nonresearchers. 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. SuazoFlores 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 nonresearchers (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 makinginequitiesexplicit, 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 listwise 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 (paperfolded 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 inbetween 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 inbetween 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 mathematicsfocused 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’ problemposing skills. Students should be provided opportunities to formulate their own problems based on situations. However, there are few problemposing 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 nonresearchers (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 HeuvelPanhuizen, 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 researchinformed 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 K12 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 “unthought”: 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 lowtech 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 futureoriented 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 lowtech 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 WeChatbased minilessons has been developed and delivered through the WeChat video function during the pandemic. Even when the pandemic is under control, minilessons are still developed and circulated through WeChat. We therefore think it is important to study the use and influence of lowtech 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 testtaking 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 paperandpencil 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 highquality work related to mathematics education in fields such as educational psychology and the cognitive and learning sciences. Encouraging the reporting of highquality 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/s1064900670998
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 SARSCOV2: 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 nietwetenschapshistorische 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 .
BiknerAhsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education . Springer.
Bini, G., Robutti, O., & BiknerAhsbahs, 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.archivesouvertes.fr/hal01849607
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 problemposing 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/s1076301697582
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/s1064902010008x
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/articlep518.xml
Chronaki, A. (2019). Affective bodying of mathematics, children and difference: Choreographing ‘sad affects’ as affirmative politics in early mathematics teacher education. ZDMMathematics Education , 51 (2), 319–330. https://doi.org/10.1007/s11858019010459
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/articlep40.xml
Darragh, L. (2016). Identity research in mathematics education. Educational Studies in Mathematics , 93 (1), 19–33. https://doi.org/10.1007/s1064901696965
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. ZDMMathematics Education , 51 (2), 305–318. https://doi.org/10.1007/s1185801810074
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/s10649020100198
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? ZDMMathematics Education , 52 (5), 821–824. https://doi.org/10.1007/s11858020011853
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/10.1007/9783030624088
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/s1076301798146
Hannula, M. S. (2019). Young learners’ mathematicsrelated affect: A commentary on concepts, methods, and developmental trends. Educational Studies in Mathematics , 100 (3), 309–316. https://doi.org/10.1007/s1064901898659
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 COVID19: 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 schoolbased mathematics as a crosssetting phenomenon. The Journal of the Learning Sciences , 20 (1), 111–150. https://doi.org/10.1080/10508406.2011.528319
Jansen, A., HerbelEisenmann, 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). Challengebased 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 unthoughtof 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 HeuvelPanhuizen, M., & Bakker, A. (2009). Nonroutine 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 COVID19 news: A mathematics audit of newspapers in Korea. Educational Studies in Mathematics . https://doi.org/10.1007/s10649021100290
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. BiknerAhsbahs & M. Peters (Eds.), Unterrichtsentwicklung macht Schule (pp. 11–26) . Springer. https://doi.org/10.1007/9783658204877_2
Meeter, M., Bele, T., den Hartogh, C., Bakker, T., de Vries, R. E., & Plak, S. (2020). College students’ motivation and study results after COVID19 stayathome 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 multifaceted 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 reformoriented mathematics curriculum: Contradictions and boundaries across teaching, research, and policy. Educational Studies in Mathematics , 102 (3), 417–434. https://doi.org/10.1007/s1064901898343
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/s10649019099137
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/s106490189854z
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/wpcontent/uploads/2020/12/Colombiaaprenderlaonda.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/s10649020099730
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. ZDMMathematics Education , 49 (3), 307–322. https://doi.org/10.1007/s1185801708646
Sfard, A. (2005). What could be more practical than good research? Educational Studies in Mathematics , 58 (3), 393–413. https://doi.org/10.1007/s1064900548185
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., HerbelEisenmann, 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
SuazoFlores, 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/s13394021003710
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/202009/370966eng_0_1.pdf .
Van den HeuvelPanhuizen, 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 COVID19 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 )
4   Dark clouds: Negative affect  Parabola mountain  Rainbow: equity, diversity, inclusion Ships in the distance Bell curve volcano  Sun: positive affect, energy source 
3   Pyramids, one with Pascal’s triangle  Elliptic lake with triangle  Shinto temple resembling Pi  Platonic solids  Climbers: ambition, curiosity   Gherkin (London)  NEMO science museum (Amsterdam)  Cube houses (Rotterdam)  Hundertwasser waste incineration (Vienna)  Los Manantiales restaurant (Mexico City)  The sign post “this way” pointing two ways signifies the challenge for students to find their way in society  Series of prime numbers. 43*47 = 2021, the year in which Lizzy Angerer made this drawing  Students in the crow’s nest: interest, attention, anticipation, technology use  The picnic scene refers to the video (Eames & Eames, )   Bridge with graduates happy with their diplomas  Vienna University building representing higher education 
2   Fractal tree  Pythagoras’ theorem at the house wall   Lady with camera and man measuring, recording, and discussing: research and assessment  The drawing hand represents design (inspired by M. C. Escher’s 1948 drawing hands lithograph) 
1  Home setting:  Rodin’s thinker sitting on hyperboloid stool, pondering how to save the earth  Boy drawing the fractal tree; mother providing support with tablet showing fractal  Paperfolded boat  Möbius strips as scaffolds for the tree  Football (sphere)  Ripples on the water connecting the home scene with the teaching boat  School setting:  Child’s small toy boat in the river  Larger boat with students and a teacher  Technology: compass, laptop (distance education)  Magnifying glass represents research into online and offline learning  Students in a circle throwing dice (learning about probability)  Teacher with book: professional selfdevelopment  Sunflowers hinting at Fibonacci sequence and Fermat’s spiral, and culture/art (e.g., Van Gogh) 
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/s1064902110049w
Download citation
Accepted : 04 March 2021
Published : 06 April 2021
Issue Date : May 2021
DOI : https://doi.org/10.1007/s1064902110049w
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 contentsharing initiative
 Grand challenges
 Mathematics education research
 Research agenda
 Find a journal
 Publish with us
 Track your research
Recent Master's Theses  Applied Mathematics
2007  2008  2009  2010  2011  2012  2013  2014  2015  2016  2017  2018  2019  2022  2023  2024
Master's Theses 2024
Author  Title 

Sonja Cotra  
Nat KendalFreedman  
Sierra Legare  
Joeseph Poulin  
Omer Ege Kara  
Haocheng Chang  
Koi McArthur  
Mackenzie Cameron  
Ben Simpson  
Sounak Majumder 
Master's Theses 2023
Author  Title 

Michael Willette 

Adam Winick  
Akihiro Takigawa 

Kaiwen Jiang  
Amelia Kunze 

Shervin Hakimi  
Martin F. Diaz Robles 

Zhuqing Li 

Delaney Smith  
Thanin Quartz  
Brian Mao  
Dorsa Sadat Hosseini Khajouei 

Master's Theses 2022
Author  Title  

Funmilayo Adeku  
Darian McLaren  
Oluyemi Momoiyioluwa  
Kevin Siu  
Andrew Bernakevitch 
 
Maria Rosa Preciado Rivas 
 
Dylan Ruth  
Aiden Huffman  
Daniel Hogg  
Jonathan Tessier 
 
Melissa Maria Stadt 
 
Duy Nguyen 
 
Nicholas Joseph Emile Richardson  
Nicolas CastroFolker 
 
Aaron BaierReinio 
Master's Theses 2019
Author  Title  

Cameron Meaney  
Jennie Newman  
Jesse Legaspi  
Kate Clements  
Maliha Ahmed  
Rishiraj Chakraborty  
Stanislav Zonov  
Petar Simidzija  
Fabian Germ 
Master's Theses 2018
Author  Title  

Jonathan Horrocks  
Luc Larocque  
Ding Jia  
Sarah Hyatt  
Andrew Grace  
Adam Morgan  
Heming Wang  
Khalida Parveen 

Master's Theses 2017
Author  Title  

Robert GoodingTownsend 
 
Hyung Jin Kim 
 
Christian Ogbonna 
 
Paul Tiede 
 
Anthony Caterini 
 
Matthew Ambacher 
 
Shawn Corvec 
 
Lorena CidMontiel 
 
An Zhou 
 
Guillaume VerdonAkzam 

Master's Theses 2016
Author  Title  

Erik Maki 
 
Athanasios (Demetri) Pananos 
 
Joanna Boneng 
 
Justin Shaw 
 
Brian Fernandes 
 
David Layden 
 
Hanzhe Chen 
 
Aaron Coutino 
 
Michael Hynes 
 
Cesar Ramirez Ibanez 
 
Minxin Zhang 
 
Sandhya Harnanan 
 
Alexander Ashbourne 

Master's Theses 2015
Author  Title  

Tian Qiao 
 
Sumit Sijher 
 
Jason Pye 
 
Tawsif Khan 
 
Shaun Sawyer  
Wenzhe Jiang 
 
Marie Barnhill 
 
Seyed Ali Madani Tonekaboni  
Naijing Kang 
 
Chengzhu Xu  
Brandon May  
James Sandham 
Master's Theses 2014
Author  Title  

Anson Maitland  
Oleg Kabernik  
Abdulhamed Alsisi  
Eric Bembenek  
Jason Boisselle (QI)  
Keenan Lyon  
Robert Leslie Irwin  
Junyu Lai  
Arman Tavakoli  
John Ladan  
Yangxin He  
Daniel Puzzuoli (QI)  
Krishan Rajaratnam 
Master's Theses 2013
Author  Title  

Luke Bovard 
 
Mikhail Panine  
Remziye Karabekmez  
Rastko Anicic  
Aidan ChatwinDavies  
Victor Veitch  
Shyamila Wickramage  
Jason Olsthoorn  
Martin Fuhry  
Janelle Resch  
Eric Webster (QI) 
Master's Theses 2012
Author  Title  

Boglarka Soos  
Jonathan Murley  
Jared Penney  
Zhaoxin Wan  
Dale Connor  
Zhao Jin  
Nazgol Shahbandi 
Master's Theses 2011
Author  Title  

Anton Baglaenko  
Christopher Morley  
Eduardo Dos Santos Lobo Brandao 
 
Drew Lloyd  
Sonia Markes  
William Ko  
Lisa Nagy  
Devin Glew  
Tyler Holden  
Todd Murray Kemp  
Kelly Anne Ogden 
Master's Theses 2010
Author  Title  

Joshua Fletcher  
Benjamin Turnbull  
Mathieu Cliche  
Mahmoudreza Ghaznavi  
Nitin Upadhyaya  
Antonia Sanchez  
Adley Au 
Master's Theses 2009
Author  Title  

Colin Phipps  
Herbert Tang  
Ryan Morris  
Michael Dumphy  
Robert Huneault  
Colin Turner  
Chad Wells  
Scott Rostrup  
Edward Dupont  
Katie Ferguson  
Peter Stechlinski  
Derek Steinmoeller  
Alen Shun  
Subasha Wickramarachichi  
Wentao Liu 
Master's Theses 2008
Author  Title  

Edward Platt  
Ranmal Perera  
Ilya Kobelevskiy  
Christopher Scott Ferrie  
Cameron Christou  
Easwar Magesan  
William Donnelly  
Angus Prain  
Jeff Timko 
Master's Theses 2007
Author  Title  

Elham Monifi  
Yasunori Aoki  
James Gordon  
Youna Hu  
Lei Tang  
Paul Ullrich  
Yijia Li  
Scott Sitar  
Anthony Chak Tong Chan  
Tyler Wilson  
Roger Chor Chun Chau  
Eduardo Barrenechea 
 Departments
 Giving to SSE
Information for
Sse math dept., account login.
 GIBSON ONLINE
Graduate Studies in Mathematics
Our department offers Masters degrees in Mathematics, Applied Mathematics, and Statistics as well as a Ph.D. Degree in Mathematics, which can have an emphasis in any of the three areas mentioned. The Masters degrees emphasize different aspects of theory and applications in order to prepare the students for either continuing studies at the Ph.D. Level or employment. The Ph.D. Program prepares the students for a career in research at a university, a government agency or in industry. Our faculty members are actively engaged in research and mentoring so that doctoral students can choose a faculty advisor according to the students’ interests.
Tulane is a privately endowed university located in New Orleans, Louisiana. At present it has an enrollment of about 10,000 students from almost every state and from 60 foreign countries.
The Mathematics program has, if anything, been strengthened by the reorganization of Tulane University in the aftermath of Hurricane Katrina.
Tulane's mathematical tradition can be traced back to the late nineteenth century, when Florian Cajori, later an expert in the history of mathematical notation, and the most famous translator of Isaac Newton's Principia, was the recipient of the first doctorate in mathematics from Tulane University (1894). Four undergraduates from the period up to the end of World War II (as well as Cajori) later became presidents of the Mathematical Association of America (Duren, McShane, Moise, Young); one (McShane) was a member of the National Academy of Sciences. In the 1950's Tulane became a major center in mathematical research. From 1970 to 2021, 207 Ph.D.'s were awarded.
The mathematics life at Tulane is enlivened by the distinguished mathematicians who visit each year for long or short periods, and by the international symposia which are held here from time to time. The department holds the annual Clifford Lectures, a weeklong series of talks by a distinguished mathematician. A miniconference supported by the National Science Foundation is held in conjunction with each of the Clifford lecture series. The first Clifford lecturer, in 1984, was Fields Medal recipient Charles Fefferman of Princeton University. In 1985 and 1986, the Clifford lecturers were Fields Medal winners, S. T. Yau of UC San Diego and William Thurston of Princeton University. The Clifford lecturers from 1987 through 1990 were Saharon Shelah of Hebrew University in Jerusalem, Clifford Taubes of Harvard University, Charles Peskin of Courant Institute and Haim Brezis of Université de Paris and Rutgers. From 1991 to 1996 they were Sylvain Cappell of Courant Institute of Mathematical Sciences, Nigel Hitchin of the University of Warwick and Persi Diaconis of Harvard University, Peter Sarnak of Princeton University and Dan Voiculescu of UC Berkeley. In 1994, a special conference on semigroups was held in honor of Alfred H. Clifford. In 1997 the Clifford lecturers were Paul Fife of University of Utah (Spring) and Peter Kronheimer of Harvard University (Fall). The speakers from 1998 to the present were Peter Bickel and Alexander Chorin of UC Berkeley, Robert Friedman of Columbia University, Sergei N. Artemov of City University of New York, T. J. Pedley of Cambridge University and Yakov Eliashberg of Stanford University.
The Mathematics Department at Tulane University offers a Ph.D. degree in Mathematics as well as Master of Science degrees in Mathematics, Applied Mathematics and in Statistics. These programs are described below. Undergraduate students majoring in mathematics or other sciences (like engineering, physics or computer science) with a strong interest in mathematics are encouraged to apply for admission to any one of the graduate programs. People who already hold undergraduate degrees in mathematics or other sciences are also encouraged to apply.
Requirements for admission into the Tulane Graduate School include:
 The GRE Test is not required as part of the graduate application for any program.
The way to apply is to fill out and submit a Webbased application form.
Webbased application form
If you have any problems receiving the application, you may inquire at:
Director of Graduate Studies
Mathematics Department Tulane University 6823 St. Charles Ave. New Orleans, LA 70118 phone: (504) 8655727 fax: (504) 8655063 [email protected]
Most graduate students receive tuition waivers and teaching assistantships , which carry a stipend of $26,500 approximately. Teaching Assistants typically teach two laboratories (each meets weekly), although more advanced students may teach one section of an undergraduate course. All Ph.D. students are required to teach an undergraduate course, or to serve as teaching assistants in problem sessions attached to undergraduate courses, for at least two semesters during their residence.
Ph.D. students (U.S. Citizens or Legally Permanent Residents) are strongly recommended to apply the external funding resources such as NSF GRFP (Graduate Research Fellowship Program). More details on here https://www.nsfgrfp.org . For nonU.S. Citizens, there is a bountiful funding resources from several fields. Please find the information on https://immigrantsrising.org .
The Tulane Mathematics Department is known for its friendly atmosphere and its practice of fostering close contact and cordial relations between faculty and graduate students. To us, this is a very important aspect of life here, and we strive to maintain it. The ratio of graduate students to faculty members is kept between 1.0 and 1.5. This is important to us because it allows all new graduate students to soon become familiar with everyone and feel at home. It also allows the faculty to get to know the students during their first semester.
The incoming graduate students are advised by the Director of Graduate Studies of the Mathematics Department. The Director, in consultation with the students, determines appropriate firstyear courses for each student, according to their preparation and interests. Throughout the program, the Director of Graduate Studies continues to help the students plan their studies and realize their mathematical interests.
All graduate (both Master and Ph.D.) students are given a cubicle in either 6 rooms at Gibson or Stanley Thomas 4th floor equipped with desks, desktop, bookcases and chalkboards. Students also have unlimited access to the lounge and the Mathematics Library, all located within the 4th floor at Gibson Hall. The lounge, or "commons room", is a place where people gather after seminars and colloquia for refreshments and discussion.
Ph.D. in Mathematics
Completing this degree takes about 5 years, depending on the student's preparation and progress satisfying the requirements. For advanced incoming students, limited transfer credit is possible. The Ph.D. prepares the students for a research career in mathematics in industry or academia.
Mathematicians with a Ph.D. from Tulane University have been successful getting jobs in a variety of colleges, research universities, government research laboratories and industries, including biotechnology, ecommerce and financial institutions.
More details on Ph.D. Requirements
Master of Science in Mathematics
This program is designed to provide students with the opportunity to broaden and deepen their knowledge of core areas of mathematics. The course work is designed to provide both breadth of knowledge and depth in an area of interest to the student. This experience will prepare the student for further studies leading to a Ph.D. degree in mathematics.
More details on MS in Mathematics
Master of Science in Applied Mathematics
This program is designed to provide students with the opportunity to broaden and deepen their knowledge of mathematics with an emphasis on those areas that have been most important in science and engineering. Students will also examine, through seminars and case studies, examples of significant applications of mathematics to other areas. This expanded base of knowledge, together with extensive experience in problem solving, is excellent preparation for further studies leading to the Ph.D. degree or for immediate employment in many areas of industry and government.
The program is open to students who have a Bachelor's degree in mathematics or a related field, and have completed undergraduate courses equivalent to Linear Algebra, Numerical Methods, and Analytical Methods. Proficiency in a programming language is essential. Students who have not completed all of these courses may be admitted and are required to take them during the first year.
More details on MS in Applied Mathematics
Master of Science in Statistics
The Master of Science degree in Statistics combines theory and application. Students in statistics will be trained in data collection, the editing and presentation of large data sets, the analyses of these sets and the mathematical foundations upon which all of these areas are based. The training has the twofold purpose of preparing the student to enter commercial, governmental and other work areas which extensively rely on statistical information and to prepare the student to continue in pursuit of a more advanced degree. Students with appropriate background (three semesters of Calculus and some knowledge of elementary statistics) usually complete the program in one or two academic years.
More details on MS in Statistics
Master of Data Science
The M.S. in Data Science (MSDS) program is a professional, nonthesis degree that is jointly offered by the Mathematics and Computer Science departments. The burst of data in the modern world has fundamentally changed many fields of human activity, including healthcare, energy, manufacturing and scientific research. It has also generated an everincreasing demand for a new type of professional: the data scientist. The MSDS program aims at providing the next generation of practitioners with cuttingedge datadriven problemsolving skills. These are based on rigorous mathematical foundations, and include data management, advanced statistical modeling, as well as the practical implementation and use of stateoftheart algorithms.
More details on M.S. in Data Science
Math 6030: Stochastic Processes
Math 60506060: Real Analysis I & II
Math 6070: Introduction to Probability
Math 6080: Introduction to Statistical Inference
Math 6090: Linear Algebra
Math 61106120: Abstract Algebra I & II
Math 6210: Differential Geometry
Math 6240: Ordinary Differential Equations
Math 6250: Mathematical Foundation of Computer Security
Math 6280: Information Theory
Math 6300: Complex Analysis
Math 6310: Scientific Computing
Math 6350: Numerical Optimization
Math 6370: Time Series Analysis
Math 6470: Analytic Methods of Applied Mathematics
Math 70107020: Topology I & II
Math7150: Probability Theory I
Math 71107120: Algebra I & II
Math 72107220: Analysis I & II
Math 7240: Mathematical Statistics
Math 7260: Linear Models
Math 72917292: Algebraic Geometry I & II
Math 73107320: Applied Math I & II
Math 7360: Data Analysis
Math 75107520: Differential Geometry I& II
Math 75307540: Partial Differential Equations I & II
Math 7550: Probability Theory II
Math 7560: Stochastic Processes II
Math 75707580: Scientific Computation II & III
Math 77107790: Special Topics Courses
More information on the courses
The Mathematics Department consists of 24 regular faculty members, several postdoctoral researchers and frequent visiting faculty in many areas of mathematics.
Its faculty enjoys national and international recognition in Algebra, Analysis, Differential Geometry, Mathematical Physics, Probability and Statistics, Scientific Computation, Theoretical Computer Science, and Topology. The researchers in Scientific Computation and in Statistics, and an increasing number of faculty in other areas, collaborate actively with colleagues in other units of the university such as the Schools of Engineering, Liberal Arts and Sciences, Medicine, and Public Health.
During the past five years our regular faculty have published over 100 research articles and several books. The regular faculty direct theses in very diverse areas which range through all of Pure Mathematics, Applied Mathematics, and Statistics. Detailed information can be found on the faculty page.
The Mathematics Department is housed in the upper floors of Gibson Hall, a stone structure built in 1894. Here are located faculty, graduate students, and staff offices, as well as classrooms, seminar rooms and computers linked to Tulane's main computing system. The department also contains the A. H. Clifford Mathematics Research Library, housing some 28,000 bound volumes and subscribing to 243 journals devoted to all areas of mathematics.
The department has a Microsoft Network with Windows and Mac workstations. The Math Department is connected by a cloud based network and the ability to store your files on Box (a cloud base storage). Graduate students are provided with adequate computing resources, Ethernet connections, Matlab, Mathematica, Microsoft office and other essential programs that are aimed to help aid in your success.
Tulane University is located in America's most exciting and most visited city. Our department is on St. Charles Avenue, across from Audubon Park, in a quiet residential area full of majestic oak trees and fine old antebellum homes. Oftenphotographed streetcars provide an easy ride to the picturesque French Quarter. New Orleans has a rich cultural life, with a symphony orchestra, operas, ballets, plays, a noted art museum, many art galleries, excellent jazz, a major jazz festival and many other events. During Mardi Gras (40 days before Easter) the town fills with parades and revelry. New Orleans is also famous for its cuisine; it boasts a number of great restaurants, and many more with good inexpensive meals.
Remember Me
 Forgot your password?
 Forgot your username?
 Create an account
 Math & Stat Office Automation
 Class and Seminar Room Booking
 New Core Lab (Classroom Booking)
 Office Automation Portal
 OARS Port 6060
 OARS Port 4040
 OARS Port 2020
 Departmental Committees
 Annual Reports
 Plan Your Visit
 Faculty Opening
 Post Doc Opening
Intranet  Webmail  Forms  IITK Facility
 Regular Faculty
 Visiting Faculty
 Inspire faculty
 Former Faculty
 Former Heads
 Ph.D. Students
 BS (Stat. & Data Sc.)
 BSMS (Stat. & Data Sc.)
 Double Major (Stat. & Data Sc.)
 BS (Math. & Sc. Comp.)
 BSMS (Math. & Sc. Comp.)
 Double Major (Math. & Sc. Comp.)
 BSH (Math & Sc. Comp.)
 BSM (Math & Sc. Comp.)
 Minor in Stat. & Data Sc.
 M.Sc. Mathematics (2 Year)
 M.Sc. Statistics (2 Year)
 Mathematics & Scientific Computing
 Statistics and Data Science
 Project Guidelines
 BS(SDS) Internship Courses Guidelines
 Courses in the Summer Term
 Analysis, Topology, Geometry
 Algebra, Number Theory and Mathematical Logic
 Numerical Analysis & Scientific Computing, ODE, PDE, Fluid Mechanics
 Probability , Statistics
 Research Areas in Mathematics and Statistics
 Publication
 UG/ PG Admission
 PhD Admission
 Financial Aid
 International Students
 Research scholars day
 National Mathematics Day
 Prof. U B Tewari Distinguished Lecture Series
 Conferences
 Student Seminars
COMMENTS
251+ Math Research Topics [2024 Updated] General / By admin / 2nd March 2024. 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 ...
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. Realworld applications of the theorem of Pythagoras.
Applied Math. Applied mathematics at the Stanford Department of Mathematics focuses, very broadly, on the areas of scientific computing, stochastic modeling, and applied analysis.
Department members engage in cuttingedge 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.
Research Areas. Ranked among the top 20 math graduate programs by U.S. News & World Report, our faculty conduct more than $3.7 million in research each year for industry, the Department of Defense, the National Science Foundation, and the National Institutes of Health. Our faculty of 35 includes three National Academy of Science members and two ...
biomathematics: introduction to the mathematical model of the hepatitis c virus, lucille j. durfee. pdf. analysis and synthesis of the literature regarding active and direct instruction and their promotion of flexible thinking in mathematics, genelle elizabeth gonzalez. pdf. life expectancy, ali r. hassanzadah. pdf
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 ...
Applied Mathematics. Faculty and students interested in the applications of mathematics are an integral part of the Department of Mathematics; there is no formal separation between pure and applied mathematics, and the Department takes pride in the many ways in which they enrich each other. We also benefit tremendously from close collaborations ...
Department of Mathematics. Headquarters Office. Simons Building (Building 2), Room 106. 77 Massachusetts Avenue. Cambridge, MA 021394307. Campus Map. (617) 2534381. Website Questions: [email protected]. Undergraduate Admissions: [email protected].
Explores how the application of mathematics and statistics can drive scientific developments across data science, engineering, finance, physics, biology, ecology, business, medicine, and beyond ... 119 Research Topics Guest edit your own article collection Suggest a topic. Submission. null. Submission
About the course. The MSc by Research is an advanced research degree which provides the opportunity to investigate a project in depth and write a thesis which makes a significant contribution in the field. The research project is however designed to take less time than a Doctorate degree (normally two years, though it is possible to complete ...
Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.Current research topics in CFD include: high resolution methods for solving nonlinear conservation laws with shock wave solutions.
Research field projects. In addition to individual projects listed on FindAPhD, we are also looking for postgraduate researchers for potential projects within a number of other research fields. Browse these fields below and get in contact with the named supervisor to find out more. Applied Mathematics and Numerical Analysis. Continuum mechanics.
Duke's Mathematics Department has a large group of mathematicians whose research involves scientific computing, numerical analysis, machine learning, computational topology, and algorithmic algebraic geometry. The computational mathematics research of our faculty has applications in data analysis and signal processing, fluid and solid mechanics ...
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 ...
The department has strong research programs in: Control and Dynamical Systems (including differential equations) Fluid Mechanics. Mathematical Medicine and Biology. Mathematical Physics. Mathematics of Data Science and Machine Learning. Scientific Computing. Researchers in our department are at the forefront of a number of exciting research areas.
Research topics Continuum and Fluid Mechanics students AMATH 361 ... Recent Master's research papers Conference research posters Graduate Student Profiles ... Department of Applied Mathematics University of Waterloo Waterloo, Ontario Canada N2L 3G1 Phone: 5198884567, ext. 45098 ...
Our department offers Masters degrees in Mathematics, Applied Mathematics, and Statistics as well as a Ph.D. Degree in Mathematics, which can have an emphasis in any of the three areas mentioned. ... Math 77107790: Special Topics Courses. More information on the courses . Faculty and Areas of Research. ... The department also contains the A. H ...
Game Theory is also a popular research area in computer science where equilibrium structures are explored using computer algorithms. Mathematical topics such as combinatorics, graph theory, probability (discrete and measuretheoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving gametheoretic problems.
Basic mathematics. This branch is typically taught in secondary education or in the first year of university. Outline of arithmetic. Outline of discrete mathematics. List of calculus topics. List of geometry topics. Outline of geometry. List of trigonometry topics. Outline of trigonometry.
Updated: April 2024 Math/Stats Thesis and Colloquium Topics 2024 2025 The degree with honors in Mathematics or Statistics is awarded to the student who has demonstrated outstanding intellectual achievement in a program of study which extends beyond the requirements of the major. The principal considerations for recommending a student for the degree with honors will be: Mastery of core ...
Research Mapping of Conducted Mathematics Graduate Researches in Region I (20072016) Figure 3 also presents a research map of every research agenda conducted in the region along mathematics ...
Drastically oversimplified, a mathematics research project goes like this: Develop, or select from the existing literature, a mathematical statement ("conjecture") that you think will be of interest to other mathematicians, and whose truth or falsity is not known. (For example, "There are infinitely many pairs of prime numbers that differ by 2.")
Department of Mathematics. Headquarters Office. Simons Building (Building 2), Room 106. 77 Massachusetts Avenue. Cambridge, MA 021394307. Campus Map. (617) 2534381. Website Questions: [email protected]. Undergraduate Admissions: [email protected].
Scientists are using sophisticated laser techniques to research quantum entanglement between the states of a chemical reaction. Quantum entanglement is a key concept at the heart of quantum information science, whereby two particles can occupy a shared quantum state.
The empowerment of educational reform and innovation through AI technology has become a topic of increasing interest in the field of education. The advent of AI technology has made comprehensive and indepth teaching evaluation possible, serving as a significant driving force for efficient and precise teaching. There were few empirical studies on the application of highquality precision ...