research topic about mathematics

251+ Math Research Topics [2024 Updated]

$Math research topics$

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

How Do You Write A Math Research Topic?

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

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

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

251+ Math Research Topics: Beginners To Advanced

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

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

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

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181 Mathematics Research Topics From PhD Experts

$math research topics$

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

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

Our Newest Research Topics in Math

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

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

Cool Math Topics to Research

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

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

Undergraduate Math Research Topics

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

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

Number Theory Topics to Research

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

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

Math Research Topics for High School

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

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

Complex Math Topics

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

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

Easy Math Research Paper Topics

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

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

Logic Topics to Research

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

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

Algebra Topics for a Research Paper

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

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

Math Education Research Topics

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

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

Computability Theory Topics to Research

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

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

Interesting Math Research Topics

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

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

Applied Math Research Topics

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

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

Geometry Topics for a Research Paper

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

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

Math Research Topics for Middle School

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

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

Math Research Topics for College Students

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

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

Calculus Topics for a Research Paper

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

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

Simple Math Research Paper Topics for High School

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

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

Business Math Topics

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

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

Probability and Statistics Topics for Research

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

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

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

Algebra, Combinatorics, and Geometry

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

Analysis and Partial Differential Equations

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

Applied Analysis

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

Mathematical Biology

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

Mathematical Finance

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

Numerical Analysis and Scientific Computing

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

Topology and Differential Geometry

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

Mathematics Research Paper Topics

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

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

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

Fields of Mathematics

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

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

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

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

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

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

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

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Articles on Mathematics

Displaying 1 - 20 of 536 articles.

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

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

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

Zachary del Rosario , Olin College of Engineering

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

Colin Foster , Loughborough University

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

Leah McCoy , Wake Forest University

GOP primary elections use flawed math to pick nominees

Ismar Volić , Wellesley College

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

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

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

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

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

Chris Impey , University of Arizona

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

Simon Thorne , Cardiff Metropolitan University

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

Rob Roznowski , Michigan State University

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

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

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

Randolph Grace , University of Canterbury

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

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

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

Ben Zunica , University of Sydney

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

Neil Saunders , University of Greenwich

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

Alma Harris , Cardiff Metropolitan University

20 people, 2.4 quintillion possibilities: the baffling statistics of Secret Santa

Stephen Woodcock , University of Technology Sydney

Low PISA math scores post-pandemic : Policies need to consider both academic excellence and equity

Louis Volante , Brock University

Australian teenagers record steady results in international tests, but about half are not meeting proficiency standards

Lisa De Bortoli , Australian Council for Educational Research

Nuclear bombs, artificial intelligence and the madness of reason – in The Maniac, Benjamin Labatut examines the troubling dawn of the digital age

Charles Barbour , Western Sydney University

Top contributors

Laureate Professor of Mathematics, University of Newcastle

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

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

Professor of Mathematics, University of Florida

Senior Lecturer in Mathematical Biology, University of Bath

PhD Candidate, Australian National University

Professor of Mathematics, Toronto Metropolitan University

Mathematician, University of Portsmouth

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

Lillian Gilbreth Postdoctoral Fellow, Purdue University

Lecturer, London School of Hygiene & Tropical Medicine

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

Senior lecturer, UNSW Sydney

Lecturer, School of Biological Sciences, Monash University

Associate Professor, Department of Physiology, Monash University

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Applied mathematics articles from across Nature Portfolio

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

Latest Research and Reviews

Quantifying in vitro B. anthracis growth and PA production and decay: a mathematical modelling approach

Bevelynn Williams
Jamie Paterson
Martín López-García

Transmission dynamics of Zika virus with multiple infection routes and a case study in Brazil

Liying Wang
Qiaojuan Jia

A new unit distribution: properties, estimation, and regression analysis

Kadir Karakaya
C. S. Rajitha
Ahmed M. Gemeay

Collaborative optimization of depot location, capacity and rolling stock scheduling considering maintenance requirements

Qingwei Zhong

Digital twins in mechanical and aerospace engineering

While there is a clear opportunity for digital twins to bring value in mechanical and aerospace engineering, they must be considered as an asset in their own right so that their full potential can be realized.

Alberto Ferrari
Karen Willcox

Topological assessment of recoverability in public transport networks

Public Transport Networks often suffer from disruption that needs to be reduced and mitigated. The authors study the correlation between topological and recoverability indicators for 42 transport networks in various cities, and find that denser and more efficient networks can better withstand disruptions, while larger networks with additional redundancy can rebound faster to normal performance levels during the recovery process.

Renzo Massobrio

News and Comment

The role of computational science in digital twins

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

Brittany Segundo

Distilling data into code

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

Joseph Bakarji

Why even specialists struggle with black hole proofs

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

Alejandro Penuela Diaz

Packing finite numbers of spheres efficiently

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

Zoe Budrikis

Graph theory captures hard-core exclusion

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

Zoltán Toroczkai

A new chapter in the physics of firefly swarms

Orit Peleg describes how a blend of models and experiments are revealing new insights about the intricate physics of firefly swarms.

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Including number theory, algebraic geometry, and combinatorics

We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-algebras, representation theory, combinatorics, game theory, and coding.

Chairs: George Bergman and Tony Feng

Algebra Faculty, Courses, Dissertations

Senate faculty, graduate students, visiting faculty, meet our faculty, george m. bergman, richard e. borcherds, sylvie corteel, david eisenbud, edward frenkel, vadim gorin, mark d. haiman, robin c. hartshorne, tsit-yuen lam (林節玄), hannah k. larson, hendrik w. lenstra, jr., ralph mckenzie, david nadler, andrew p. ogg, arthur e. ogus, martin olsson, alexander paulin, nicolai reshetikhin, john l. rhodes, marc a. rieffel, thomas scanlon, vera serganova.

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Research Areas

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

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

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

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

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

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

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

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

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

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$research topic about mathematics$

+ - Algebraic geometry Click to collapse

Geometric Invariant Theory : Faculty: Santosha Pattanayak

+ - Commutative Algebra Click to collapse

The research areas are Derivations, Higher Derivations, Differential Ideals, Multiplication Modules and the Radical Formula. Faculty : A. K. Maloo

+ - Complex Analysis & Operator Theory Click to collapse

I mainly consider various analytic function spaces defined on the unit disk or on some half plane of the complex plane and various operators on these spaces such as multiplication operators, composition operators, Cesaro operators. Also, I work on similar operators on some discrete function spaces defined on an infinite rooted tree (graph), in particular, on the discrete analogue of Hardy spaces. I deal with number of other problems which connects geometric function theory with function spaces and operator theory. Faculty : P. Muthukumar

+ - Computational Acoustics and Electromagnetics Click to collapse

The study of interaction of electromagnetic fields with physical objects and the environment constitutes the main subject matter of Computational Electromagnetics. One of the major challenges in this area of research is in the development of efficient, accurate and rapidly-convergent algorithms for the simulation of propagation and scattering of acoustic and electromagnetic fields within and around structures that possess complex geometrical characteristics. These problems are of fundamental importance in diverse fields, with applications ranging from space exploration, medical imaging and oil exploration on the civilian side to aircraft design and decoy detection on the military side - just to name a few. Computational modeling of electromagnetic scattering problems has typically been attempted on the basis of classical, low-order Finite-Difference-Time-Domain (FDTD) or Finite-Element-Method (FEM) approaches. An important computational alternative to these approaches is provided by boundary integral-equation formulations that we have adopted owing to a number of excellent properties that they enjoy. Listed below are some of the key areas of interest in related research: 1. Design of high-order integrators for boundary integral equations arising from surface and volumetric scattering of acoustic and electromagnetic waves from complex engineering structures including from open surfaces and from geometries with singular features like edges and corners. 2. Accurate representation of complex surfaces in three dimensions with applications to enhancement of low quality CAD models and in the development of direct CAD-to-EM tools. 3. High frequency scattering methods in three dimensions with frequency independent cost in the context of multiple scattering configurations. A related field of interest in this regard includes high-order geometrical optics simulator for inverse ray tracing. 4. High performance computing. Faculty : Akash Anand , B. V. Rathish Kumar

+ - Computational Fluid Dynamics Click to collapse

Development of Numerical Schemes for Incompressible Newtonian and Non-Newtonian Fluid Flows based on FDM, FEM, FVM, Wavelets, SEM, BEM etc. Development of Parallel Numerical Methods for Heat & Fluid Flow Analysis on Large Scale Parallel Computing systems based on MPI-OpenMP-Cuda programming concepts, ANN/ML methods for Flow Analysis. Global Climate Modelling on Very Large Scale Parallel Systems. Faculty : B. V. Rathish Kumar , Saktipada Ghorai

+ - Differential Equations Click to collapse

Semigroups of linear operators and their applications, Functional differential equations, Galerkin approximations

Many unsteady state physical problems are governed by partial differential equations of parabolic or hyperbolic types. These problems are mostly prototypes since they represent as members of large classes of such similar problems. So, to make a useful study of these problems we concentrate on their invariant properties which are satisfied by each member of the class. We reformulate these problems as evolution equations in abstract spaces such as Hilbert or more generally Banach spaces. The operators appearing in these equations have the property that they are the generators of semigroups. The theory of semigroups then plays an important role of establishing the well-posedness of these evolution equations. The analysis of functional differential equations enhances the applicability of evolution equations as these include the equations involving finite as well as infinite delays. Equations involving integrals can also be tackled using the techniques of functional differential equations. The Galerkin method and its nonlinear variants are fundamental tools to obtain the approximate solutions of the evolution and functional differential equations.

Faculty : D. Bahuguna

Homogenization and Variational methods for partial differential equation

The main interest is on Aysmptotic Analysis of partial differential equations. This is a technique to understand the macroscopic behaviour of a composite medium through its microscopic properties. The technique is commonly used for PDE with highly oscillating coefficients. The idea is to replace a given heterogeneous medium by a fictitious homogeneous one (the `homogenized' material) for numerical computations. The technique is also known as ``Multi scale analysis''. The known and unknown quantities in the study of physical or mechanical processes in a medium with micro structure depend on a small parameter $\varepsilon$. The study of the limit as $ \varepsilon \rightarrow0 $, is the aim of the mathematical theory of homogenization. The notion of $G$-convergence, $H$-convergence, two-scale convergence are some examples of the techniques employed for specific cases. The variational characterization of the technique for problems in calculus of variations is given by $\Gamma$- convergence.

Faculty : T. Muthukumar , B.V. Rathish Kumar

Functional inequalities on Sobolev space

Sobolev spaces are the natural spaces where one looks for solutions of Partial differential equations (PDEs). Functional inequalities on this spaces ( for example Moser-Trudinger Inequality, Poincare Inequality, Hardy- Sobolev Inequality and many other) plays a very significant role in establishing existence of solutions for various PDEs. Existence of extremal function for such inequalities is another key aspect that is investigated

Asymptotic analysis on changing domains

Study of differential equations on long cylinders appears naturally in various branches of Physics, Engineering applications and real life problems. Problems (not necessarily PDEs, can be purely variational in nature) set on cylindrical domains whose length tends to infinity, is analysed. Faculty : Prosenjit Roy , Kaushik Bal + - Functional Analysis & Operator Theory Click to collapse

Banach space theory

Geometric and proximinality aspects in Banach spaces. Faculty: P. Shunmugaraj

Function-theoretic and graph-theoretic operator theory

The primary goal is to implement methods from the complex function theory and the graph theory into the multivariable operator theory. The topics of interests include de Branges-Rovnyak spaces and weighted shifts on directed graphs. Faculty: Sameer Chavan

Non-commutative geometry

The main emphasis is on the metric aspect of noncommutative geometry. Faculty: Satyajit Guin

Bounded linear operators

A central theme in operator theory is the study of B(H), the algebra of bounded linear operators on a separable complex Hilbert space. We focus on operator ideals, subideals and commutators of compact operators in B(H). There is also a continuing interest in semigroups of operators in B(H) from different perspectives. We work in operator semigroups involve characterization of special classes of semigroups which relate to solving certain operator equations. Faculty: Sasmita Patnaik

+ - Harmonic Analysis Click to collapse

Operator spaces

The main emphasis is on operator space techniques in abstract Harmonic Analysis.

In the Euclidean setting

Analysis, boundedness and weighted boundedness of singular integral operators are major thrust areas in the department. In abstract Harmonic analysis we do work in studying Lacunary sets in the noncommutative Lp spaces.

Faculty : Parasar Mohanty

On Lie groups

Problems related to integral geometry on Lie groups are being studied.

Faculty : Rama Rawat

+ - Homological Algebra Click to collapse

Cohomology and Deformation theory of algebraic structures

Research work in this area encompasses cohomology and deformation theory of algebraic structures, mainly focusing on Lie and Leibniz algebras arising out of topology and geometry. In particular, one is interested in the cohomology and Versal deformation for Lie and Leibniz brackets on the space of sections of vector bundles e.g. Lie algebroids and Courant algebroids.

This study naturally relate questions about other algebraic structures which include Lie-Rinehart algebras, hom-Lie-Rinehart algebras, Hom-Gerstenhaber algebras, homotopy algebras associated to Courant algebras, higher categories and related fields.

Faculty : Ashis Mandal + - Image Processing Click to collapse

TPDE based Image processing for Denoising, Inpainting, Classification, Compression, Registration, Optical flow analysis etc. Bio-Medical Image Analysis based on CT/MRI/US clinical data, ANN/ML methods in Image Analysis, Wavelet methods for Image processing.

Faculty : B. V. Rathish Kumar + - Mathematical Biology Click to collapse

There is an active group working in the area of Mathematical Biology. The research is carried out in the following directions.

Mathematical ecology

1. Research in this area is focused on the local and global stability analysis, detection of possible bifurcation scenario and derivation of normal form, chaotic dynamics for the ordinary as well as delay differential equation models, stochastic stability analysis for stochastic differential equation model systems and analysis of noise induced phenomena. Also the possible spatio-temporal pattern formation is studied for the models of interacting populations dispersing over two dimensional landscape.

2. Mathematical Modeling of the survival of species in polluted water bodies; depletion of dissolved oxygen in water bodies due to organic pollutants.

Mathematical epidemiology

1. Mathematical Modeling of epidemics using stability analysis; effects of environmental, demographic and ecological factors.

2. Mathematical Modeling of HIV Dynamics in vivo

Bioconvection

Bioconvection is the process of spontaneous pattern formation in a suspension of swimming microorganisms. These patterns are associated with up- and down-welling of the fluid. Bioconvection is due to the individual and collective behaviours of the micro-organisms suspended in a fluid. The physical and biological mechanisms of bioconvection are investigated by developing mathematical models and analysing them using a variety of linear, nonlinear and computational techniques.

Bio-fluid dynamics

Mathematical Models for blood flow in cardiovascular system; renal flows; Peristaltic transport; mucus transport; synovial joint lubrication.

Faculty : Malay Banerjee , Saktipada Ghorai , B.V. Rathish Kumar

Cardiac electrophysiology

Theory, Modeling & Simulation of Cardiac Electrical Activity (CEA) in Human Cardiac Tissue based on PDEODE models such as Monodomain Model, Biodomain model, Cardiac Arrhythmia, pace makers etc

Faculty : B.V. Rathish Kumar + - Number Theory & Arithmetic Geometry Click to collapse

Algebraic number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Galois representations, Congruences between special values of L-functions.

Faculty : Sudhanshu Shekhar

Analytic number theory

L-functions, sub-convexity problems, Sieve method

Faculty : Saurabh Kumar Singh

Number theory and Arithmetic geometry

Iwasawa Theory of elliptic curves and modular forms, Selmer groups

Faculty : Somnath Jha

Number theory, Dynamical systems, Random walks on groups

During the last four decades, it has been realized that some problems in number theory and, in particular, in Diophantine approximation, can be solved using techniques from the theory of homogeneous dynamics, random walks on homogeneous spaces etc. Indeed, one translates such problems into a problem on the behavior of certain trajectories in homogeneous spaces of Lie groups under flows or random walks; and subsequently resolves using very powerful techniques from the theory of dynamics on homogeneous spaces, random walk etc. I undertake this theme.

Faculty : Arijit Ganguly + - Numerical Analysis and Scientific Computing Click to collapse

The faculty group in the area of Numerical Analysis & Scientific Computing are very actively engaged in high-quality research in the areas that include (but are not limited to): Singular Perturbation problems, Multiscale Phenomena, Hyperbolic Conservation Laws, Elliptic and Parabolic PDEs, Integral Equations, Computational Acoustics and Electromagnetics, Computational Fluid Dynamics, Computer-Aided Tomography and Parallel Computing. The faculty group is involved in the development, analysis, and application of efficient and robust algorithms for solving challenging problems arising in several applied areas. There is expertise in several discretization methods that include: Finite Difference Methods, Finite Element Methods, Spectral Element Methods, Boundary Element Methods, Nyström Method, Spline and Wavelet approximations, etc. This encompasses a very high level of computation that requires software skills of the highest order and parallel computing as well.

Faculty : B. V. Rathish Kumar , Akash Anand + - Operator Algebra Click to collapse

Broadly speaking, I work with topics in C*-algebras and von Neumann algebras. More precisely, my work involves Jones theory of subfactors and planar algebras.

Faculty : Keshab Chandra Bakshi + - Representation Theory Click to collapse

Representation of Lie and linear algebraic groups over local fields, Representation-theoretic methods, automorphic representations over local and global fields, Linear algebraic groups and related topics MSC classification (22E50, 11F70, 20Gxx:)

Faculty : Santosh Nadimpalli

Representations of finite and arithmetic groups

Current research interests: Representations of Linear groups over local rings, Projective representations of finite and arithmetic groups, Applications of representation theory.

Faculty : Pooja Singla

Representation theory of Lie algebras and algebraic groups

Faculty : Santosha Pattanayak

Representation theory of infinite dimensional Lie algebras

Current research interest: Representation theory of Kac-Moody algebras; Toroidal Lie algebras and extended affine Lie algebras.

Faculty : Sachin S. Sharma

Representation theory and Invariant theory

Current research interest: Representation and structure theory of algebraic groups, Classical invariant theory of reductive algebraic groups and associated Weyl groups.

Faculty : Preena Samuel

Combinatorial representation theory

String algebras form a class of tame representation type algebras that are presented combinatorially using quivers and relations. Currently I am interested in studying the combinatorics of strings to understand the Auslander-Reiten quiver that encodes the generators for the category of finite length R-modules as well as the Ziegler spectrum associated with string algebras whose topology is described model-theoretically

Faculty : Amit Kuber + - Set Theory and Logic Click to collapse

Set theory (MSC Classification 03Exx)

We apply tools from set theory to problems from other areas of mathematics like measure theory and topology. Most of these applications involve the use of forcing to establish independence results. For examples of such results see https://home.iitk.ac.in/~krashu/

Faculty : Ashutosh Kumar

Rough set theory and Modal logic

Algebraic studies of structures and corresponding logics that have arisen in the course of investigations in Rough Set Theory (RST) constitute a primary part of my research. Currently, we are working on algebras and logics stemming from a combination of formal concept analysis and RST, and also from different approaches to paraconsistency.

Faculty : Mohua Banerjee + - Several Complex Variables Click to collapse

Broadly speaking, my work lies in the theory of functions of several complex variables. Two major themes of my work till now are related to _Pick-Nevanlinna interpolation problem_ and on the _Kobayashi geometry of bounded domains_. I am also interested in complex potential theory and complex dynamics in one variable setting.

Faculty : Vikramjeet Singh Chandel + - Topology and Geometry Click to collapse

Algebraic topology and Homotopy theory

The primary interest is in studying equivariant algebraic topology and homotopy theory with emphasis on unstable homotopy. Specific topics include higher operations such as Toda bracket, pi-algebras, Bredon cohomology, simplicial/ cosimplicial methods, homotopical algebra.

Faculty : Debasis Sen

Algebraic topology, Combinatorial topology

I apply tools from algebraic topology and combinatorics to address problems in topology and graph theory.

Faculty : Nandini Nilakantan

Differential geometry

Geometric Analysis and Geometric PDEs. Interested in geometry of the eigenvalues of Laplace operator, Geometry of geodesics.

Faculty : G. Santhanam

Low dimensional topology

The main interest is in Knot Theory and its Applications. This includes the study of amphicheirality, the study of closed braids, and the knot polynomials, specially the Jones polynomial.

Faculty : Aparna Dar

Geometric group theory and Hyperbolic geometry

Work in this area involves relatively hyperbolic groups and Cannon-Thurston maps between relatively hyperbolic boundaries. Mapping Class Groups are also explored. Faculty: Abhijit Pal

Faculty : Abhijit Pal

Manifolds and Characteristic classes

We are interested in the construction of new examples of non-Kahler complex manifolds. We aim also at answering the question of existence of almost-complex structures on certain even dimension real manifolds. Characteristic classes of vector bundles over certain spaces are also studied.

Faculty : Ajay Singh Thakur

Moduli spaces of hyperbolic surfaces

The central question we study here to find combinatorial descriptions of moduli spaces of closed and oriented hyperbolic surfaces. Also, we study isometric embedding of metric graphs on surfaces of following types: (a) quasi-essential on closed and oriented hyperbolic surfaces (b) non-compact surfaces, where complementary regions are punctured discs, (c) on half-translation surfaces etc.

Faculty : Bidyut Sanki

Systolic topology and Geometry

We are interested to study the configuration of systolic geodesics (i.e., shortest closed geodesics) on oriented hyperbolic surfaces. Also, we are interested in studying the maximal surfaces and deformations on hyperbolic surfaces of finite type to increase systolic lengths.

Topological graph theory

We study configuration of graphs, curves, arcs on surfaces, fillings, action of mapping class groups on graphs on surfaces, minimal graphs of higher genera.

Faculty : Bidyut Sanki + - Tribology Click to collapse

Active work has been going on in the area of "Tribology". Tribology deals with the issues related to lubrication, friction and wear in moving machine parts. Work is going in the direction of hydrodynamic and elastohydrodynamic lubrication, including thermal, roughness and non-newtonian effects. The work is purely theoretical in nature leading to a system on non-linear partial differential equations, which are solved using high speed computers.

Faculty : B. V. Rathish Kumar

Research Areas in Statistics and Probability Theory

Here are the areas of Statistics in which research is being done currently.

$research topic about mathematics$

+ - Bayesian Nonparametric Methods Click to collapse

Exponential growth in computing power in the past few decades has made Bayesian methods for infinitedimensional models possible, which is termed as the Bayesian nonparametric (BN) methods. BN is a vast area dealing with modelling and making inference in various fields of Statistics, including, and not restricted to density estimation, regression, variable selection, classification, clustering. Irrespective of the field of execution, a BN method deals with prior construction on an infinite-dimensional parameter space, posterior computation and thereby making posterior predictive inference. Finally, the method is validated by supportive asymptotic properties to show the closeness of the proposed method to the true underlying data generating process.

Faculty member: Minerva Mukhopadhyay

+ - Data Mining in Finance Click to collapse

Economic globalization and evolution of information technology has in recent times accounted for huge volume of financial data being generated and accumulated at an unprecedented pace. Effective and efficient utilization of massive amount of financial data using automated data driven analysis and modelling to help in strategic planning, investment, risk management and other decision-making goals is of critical importance. Data mining techniques have been used to extract hidden patterns and predict future trends and behaviours in financial markets. Data mining is an interdisciplinary field bringing together techniques from machine learning, pattern recognition, statistics, databases and visualization to address the issue of information extraction from such large databases. Advanced statistical, mathematical and artificial intelligence techniques are typically required for mining such data, especially the high frequency financial data. Solving complex financial problems using wavelets, neural networks, genetic algorithms and statistical computational techniques is thus an active area of research for researchers and practitioners.

Faculty: Amit Mitra , Sharmishtha Mitra

+ - Econometric Modelling Click to collapse

Econometric modelling involves analytical study of complex economic phenomena with the help of sophisticated mathematical and statistical tools. The size of a model typically varies with the number of relationships and variables it is applying to replicate and simulate in a regional, national or international level economic system. On the other hand, the methodologies and techniques address the issues of its basic purpose – understanding the relationship, forecasting the future horizon and/or building "what-if" type scenarios. Econometric modelling techniques are not only confined to macro-economic theory, but also are widely applied to model building in micro-economics, finance and various other basic and social sciences. The successful estimation and validation part of the model-building relies heavily on the proper understanding of the asymptotic theory of statistical inference. A challenging area of econometric

Faculty: Shalabh , Sharmishtha Mitra

+ - Entropy Estimation and Applications Click to collapse

Estimation of entropies of molecules is an important problem in molecular sciences. A commonly used method by molecular scientist is based on the assumption of a multivariate normal distribution for the internal molecular coordinates. For the multivariate normal distribution, we have proposed various estimators of entropy and established their optimum properties. The assumption of a multivariate normal distribution for the internal coordinates of molecules is adequate when the temperature at which the molecule is studied is low, and thus the fluctuations in internal coordinates are small. However, at higher temperatures, the multivariate normal distribution is inadequate as the dihedral angles at higher temperatures exhibit multimodes and skewness in their distribution. Moreover the internal coordinates of molecules are circular variables and thus the assumption of multivariate normality is inappropriate. Therefore a nonparametric and circular statistic approach to the problem of estimation of entropy is desirable. We have adopted a circular nonparametric approach for estimating entropy of a molecule. This approach is getting a lot of attention among molecular scientists.

Faculty: Neeraj Misra

+ - Environmental Statistics Click to collapse

The main goal of environmental statistics is to build sophisticated modelling techniques that are necessary for analysing temperature, precipitation, ozone concentration in air, salinity in seawater, fire weather index, etc. There are multiple sources of such observations, like weather stations, satellites, ships, and buoys, as well as climate models. While station-based data are generally available for long time periods, the geographical coverage of such stations is mostly sparse. On the other hand, satellite-derived data are available only for the last few decades, but they are generally of much higher spatial resolution. While the current statistical literature has already explored various techniques for station-based data, methods available for modelling high-resolution satellite-based datasets are relatively scarce and there is ample opportunity for building statistical methods to handle such datasets. Here, the data are not only huge in volume, but they are also spatially dependent. Modelling such complex dependencies is challenging also due to the high nonstationary often present in the data. The sophisticated methods also need suitable computational tools and thus provide scopes for novel research directions in computational statistics. Apart from real datasets, statistical modelling of climate model outputs is a new area of research, particularly keeping in mind the issue of climate change. Under different representative concentration pathways (RCPs) of the Intergovernmental Panel for Climate Change (IPCC), different carbon emission

Faculty: Arnab Hazra

+ - Estimation in Restricted Parameter Space Click to collapse

In many practical situations, it is natural to restrict the parameter space. This additional information of restricted parameter space can be intelligently used to derive estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We deal with the problems of estimation parameters of one or more populations when it is known apriori that some or all of them satisfy certain restrictions, leading to the consideration of restricted parameter space. The goal is to find estimators that improve upon the standard (natural) estimators, meant for the case of unrestricted parameter space. We also deal with the decision theoretic aspects of this problem.

+ - Game Theory Click to collapse

The mathematical discipline of Game theory models and analyses interactions between competing and cooperative players. Some research areas in game theory are choice theory, mechanism design, differential games, stochastic games, graphon games, combinatorial games, evolutionary games, cooperative games, Bayesian games, algorithmic games - and this list is certainly not exhaustive. Gametheoretic models are used in many real-life problems such as decision making, voting, matching, auctioning, bargaining/negotiating, queuing, distributing/dividing wealth, dealing with cheap talks, the evolution of living organisms, disease propagation, cancer treatment, and many more. 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 measure-theoretic), analysis (real and functional), algebra (linear and abstract), etc., are used in solving game-theoretic problems.

Faculty: Soumyarup Sadhukhan

+ - Machine Learning and Statistical Pattern Recognition Click to collapse

Build machine learning algorithms based on statistical modeling of data. With a statistical model in hand, we apply probability theory to get a sound understanding of the algorithms.

Faculty: Subhajit Dutta

+ - Markov chain Monte Carlo Click to collapse

Markov chain Monte Carlo (MCMC) algorithms produce correlated samples from a desired target distribution, using an ergodic Markov chain. Due to the lack of independence of the samples, and the challenges of working with Markov chains, many theoretical and practical questions arise. Much of the research in this area can be divided into three broad topics: (1) development of new sampling algorithms for complicated target distributions, (2) studying rates of convergence of the Markov chains employed in various applications like variable selection, regression, survival analysis etc, and (3) measuring the quality of MCMC samples in an effort to quantify the variability in the final estimators of the features of the target.

Faculty: Dootika Vats

+ - Non-Parametric and Robust Statistical methods Click to collapse

Detection of different features (in terms of shape) of non-parametric regression functions are studied; asymptotic distributions of the proposed estimators (along with their robustness properties) of the shaperestricted regression function are also investigated. Apart from this, work on the test of independence for more than two random variables is pursued. Statistical Signal Processing and Statistical Pattern Recognition are the other areas of interest.

Faculty: Subhra Sankar Dhar

+ - Optimal Experimental Design Click to collapse

The area of optimal experimental design has been an integral part of many scientific investigation including agriculture and animal husbandry, biology, medicine, physical and chemical sciences, and industrial research. A well-designed experiment utilizes the limited recourse (cost, time, experimental units, etc) optimally to answer the underlying scientific question. For example, optimal cluster/crossover designs may be applied to cluster/cross randomized trials to efficiently estimates the treatment effects. Optimal standard ANOVA designs can be utilized to test the equality of several experimental groups. Most popular categories of optimal designs include Bayesian designs, longitudinal designs, designs for ordered experiments and factorial designs to name a few.

Faculty: Satya Prakash Singh

+ - Ranking and Selection Problems Click to collapse

About fifty years ago statistical inference problems were first formulated in the now-familiar "Ranking and Selection" framework. Ranking and selection problems broadly deal with the goal of ordering of different populations in terms of unknown parameters associated with them. We deal with the following aspects of Ranking and Selection Problems:1. Obtaining optimal ranking and selection procedures using decision theoretic approach;2. Obtaining optimal ranking and selection procedures under heteroscedasticity;3. Simultaneous confidence intervals for all distances from the best and/or worst populations, where the best (worst) population is the one corresponding to the largest (smallest) value of the parameter;4. Estimation of ranked parameters when the ranking between parameters is not known apriori;5. Estimation of (random) parameters of the populations selected using a given decision rule for ranking and selection problems.

+ - Regression Modelling Click to collapse

The outcome of any experiment depends on several variables and such dependence involves some randomness which can be characterized by a statistical model. The statistical tools in regression analysis help in determining such relationships based on the sample experimental data. This helps further in describing the behaviour of the process involved in experiment. The tools in regression analysis can be applied in social sciences, basic sciences, engineering sciences, medical sciences etc. The unknown and unspecified form of relationship among the variables can be linear as well as nonlinear which is to be determined on the basis of a sample of experimental data only. The tools in regression analysis help in the determination of such relationships under some standard statistical assumptions. In many experimental situations, the data do not satisfy the standard assumptions of statistical tools, e.g. the input variables may be linearly related leading to the problem of multicollinearity, the output data may not have constant variance giving rise to the hetroskedasticity problem, parameters of the model may have some restrictions, the output data may be auto correlated, some data on input and/or output variables may be missing, the data on input and output variables may not be correctly observable but contaminated with measurement errors etc. Different types of models including the econometric models, e.g., multiple regression models, restricted regression models, missing data models, panel data models, time series models, measurement error models, simultaneous equation models, seemingly unrelated regression equation models etc. are employed in such situations. So the need of development of new statistical tools arises for the detection of problem, analysis of such non-standard data in different models and to find the relationship among different variables under nonstandard statistical conditions. The development of such tools and the study of their theoretical statistical properties using finite sample theory and asymptotic theory supplemented with numerical studies based on simulation and real data are the objectives of the research work in this area.

Faculty: Shalabh

+ - Robust Estimation in Nonlinear Models Click to collapse

Efficient estimation of parameters of nonlinear regression models is a fundamental problem in applied statistics. Isolated large values in the random noise associated with model, which is referred to as an outliers or an atypical observation, while of interest, should ideally not influence estimation of the regular pattern exhibited by the model and the statistical method of estimation should be robust against outliers. The nonlinear least squares estimators are sensitive to presence of outliers in the data and other departures from the underlying distributional assumptions. The natural choice of estimation technique in such a scenario is the robust M-estimation approach. Study of the asymptotic theoretical properties of Mestimators under different possibilities of the M-estimation function and noise distribution assumptions is an interesting problem. It is further observed that a number of important nonlinear models used to model real life phenomena have a nested superimposed structure. It is thus desirable also to have robust order estimation techniques and study the corresponding theoretical asymptotic properties. Theoretical asymptotic properties of robust model selection techniques for linear regression models are well established in the literature, it is an important and challenging problem to design robust order estimation techniques for nonlinear nested models and establish their asymptotic optimality properties. Furthermore, study of the asymptotic properties of robust M-estimators as the number of nested superimposing terms increase is also an important problem. Huber and Portnoy established asymptotic behavior of the M-estimators when the number of components in a linear regression model is large and established conditions under which consistency and asymptotic normality results are valid. It is possible to derive conditions under which similar results hold for different nested nonlinear models.

Faculty: Debasis Kundu , Amit Mitra

+ - Rough Paths and Regularity structures Click to collapse

The seminal works of Terry Lyons on extensions of Young integration, the latter being an extension of Riemann integration, to functions with Holder regular paths (or those with finite p-variation for some 0 < p < 1) lead to the study of Rough Paths and Rough Differential Equations. Martin Hairer, Massimiliano Gubinelli and their collaborators developed fundamental results in this area of research. Extensions of these ideas to functions with negative regularity (read as "distributions") opened up the area of Regularity structures. Important applications of these topics include constructions of `pathwise' solutions of stochastic differential equations and stochastic partial differential equations.

Faculty: Suprio Bhar

+ - Spatial statistics Click to collapse

The branch of statistics that focuses on the methods for analysing data observed across some spatial locations in 2-D or 3-D (most common), is called spatial statistics. The spatial datasets can be broadly divided into three types: point-referenced data, areal data, and point patterns. Temperature data collected by a few monitoring stations spread across a city on some specific day is an example of the first type. When data are obtained as summaries of some geographical regions, they are of the second type, crime rate dataset from the different states of India on a specific year is an example. An example of the third type is the IED attack locations in Afghanistan during a year, where the geographical coordinates are themselves the data. Because of the natural dependence among the observations obtained from two close locations, the data cannot be assumed to be independent. When the study domain is large, often we have a large number of observational sites and at the same time, those sites are possibly distributed across a nonhomogeneous area. This leads to the necessity of models that can handle a large number of sites as well as the nonstationary dependence structure and this is a very active area of research. Apart from common geostatistical models, a very active area of research is focused on spatial extreme value theory where max-stable stochastic processes are the natural models to explain the tail-dependence. While the available methods for such spatial extremes are highly scarce, specifically for moderately highdimensional problems, different future research directions are being explored currently in the literature. For better uncertainty quantification and computational flexibility using hierarchically defined models, the Bayesian paradigm is often a natural choice.

+ - Statistical Signal Processing Click to collapse

Signal processing may broadly be considered to involve the recovery of information from physical observations. The received signals are usually disturbed by thermal, electrical, atmospheric or intentional interferences. Due to the random nature of the signal, statistical techniques play an important role in signal processing. Statistics is used in the formulation of appropriate models to describe the behaviour of the system, the development of appropriate techniques for estimation of model parameters, and the assessment of model performances. Statistical Signal Processing basically refers to the analysis of random signals using appropriate statistical techniques. Different one and multidimensional models have been used in analyzing various one and multidimensional signals. For example ECG and EEG signals, or different grey and white or colour textures can be modelled quite effectively, using different non-linear models. Effective modelling are very important for compression as well as for prediction purposes. The important issues are to develop efficient estimation procedures and to study their properties. Due to non-linearity, finite sample properties of the estimators cannot be derived; most of the results are asymptotic in nature. Extensive Monte Carlo simulations are generally used to study the finite sample behaviour of the different estimators.

+ - Step-Stress Modelling Click to collapse

Traditionally, life-data analysis involves analysing the time-to-failure data obtained under normal operating conditions. However, such data are difficult to obtain due to long durability of modern days. products, lack of time-gap in designing, manufacturing and actually releasing such products in market, etc. Given these difficulties as well as the ever-increasing need to observe failures of products to better understand their failure modes and their life characteristics in today's competitive scenario, attempts have been made to devise methods to force these products to fail more quickly than they would under normal use conditions. Various methods have been developed to study this type of "accelerated life testing" (ALT) models. Step-stress modelling is a special case of ALT, where one or more stress factors are applied in a life-testing experiment, which are changed according to pre-decided design. The failure data observed as order statistics are used to estimate parameters of the distribution of failure times under normal operating conditions. The process requires a model relating the level of stress and the parameters of the failure distribution at that stress level. The difficulty level of estimation procedure depends on several factors like, the lifetime distribution and number of parameters thereof, the uncensored or various censoring (Type I, Type II, Hybrid, Progressive, etc.) schemes adopted, the application of non-Bayesian or Bayesian estimation procedures, etc.

Faculty: Debasis Kundu , Sharmishtha Mitra

+ - Stochastic Partial Differential Equations Click to collapse

The study of Stochastic calculus, more specifically, that of stochastic differential equations and stochastic partial differential equations, has a broad range of applications across various disciplines or branches of Mathematics, such as Partial Differential Equations, Evolution systems, Interacting particle systems, Finance, Mathematical Biology. Theoretical understanding for such equations was first obtained in finite dimensional Euclidean spaces. Later on, to describe various natural phenomena, models were constructed (and analyzed) with values in Banach spaces, Hilbert spaces and in the duals of nuclear spaces. Important topics/questions in this area of research include existence and uniqueness of solutions, Stability, Stationarity, Stochastic flows, Stochastic Filtering theory and Stochastic Control Theory, to name a few.

+ - Theory of Stochastic Orders and Aging and Applications Click to collapse

The manner in which a component (or system) improves or deteriorates with time can be described by concepts of aging. Various aging notions have been proposed in the literature. Similarly lifetimes of two different systems can be compared using the concepts of stochastic orders between the probability distributions of corresponding (random) lifetimes. Various stochastic orders between probability distributions have been defined in the literature. We study the concepts of aging and stochastic orders for various coherent systems. In many situations, the performance of a system can be improved by introducing some kind of redundancy into the system. The problem of allocating redundant components to the components of a coherent system, in order to optimize its reliability or some other system performance characteristic, is of considerable interest in reliability engineering. These problems often lead to interesting theoretical results in Probability Theory. We study the problem of optimally allocating spares to the components of various coherent systems, in order to optimize their reliability or some other system performance characteristic. Performances of systems arising out of different allocations are studied using concepts of aging and stochastic orders.

DEPARTMENT OF Mathematics & Statistics

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

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11 Real World Math Activities That Engage Students

Bridging the gap between abstract math concepts and real life experiences can make the subject accessible and relevant for kids.

During a unit on slope, José Vilson’s students just weren’t getting it, and their frustration was growing. The former middle school math teacher began brainstorming creative ways to illustrate the concept. “I kept thinking, ‘My students already understand how this works—they just don’t know that they know,’” Vilson writes in a recent article for Teacher2Teacher . “How can I activate knowledge they don’t believe they have?”

Then he thought about a hill a couple of blocks from school that his students “walk up every day to get to the subway.” He tacked up paper and began sketching stick figures on the hill. “One was at the top of the hill, one was halfway up, one was near the bottom skating on flat ground, and one was on a cliff,” writes Vilson, now the executive director of EduColor. “Which of these figures will go faster and why?” he asked his students. “That got my kids laughing because, of course, my stick figures weren’t going to hang in the MoMA.” Still, his sketch got them thinking and talking, and it provided a simple stepping stone that “gave that math relevance and belonging in their own lives,” Vilson concludes.

“It’s not unusual for students to walk into our classrooms thinking that math belongs to people who are smarter, who are older, or who aren’t in their immediate circle,” Vilson writes. “But every time I teach math in a way that’s accessible and real for my students, I’m teaching them: ‘The math is yours.’”

To build on Vilson’s idea, we posted on our social channels asking teachers to share their favorite strategies for connecting math to students’ experiences and lives outside of school. We received hundreds of responses from math educators across grade levels. Here are 11 teacher-tested ideas that get students seeing and interacting with the math that surrounds them each day.

Hunt for clues

Coordinate systems can feel abstract to some students—but using coordinates to navigate a familiar space can solidify the concept in a relevant and fun way. “Before starting a unit on coordinates, I make gridded maps of the school—I make them look old using tea staining —and send my students off on a treasure hunt using the grid references to locate clues,” says Kolbe Burgoyne, an educator in Australia. “It’s meaningful, it’s fun, and definitely gets them engaged.”

Budget a trip

Students enjoy planning and budgeting for imaginary trips, teachers tell us, offering ample opportunities to practice adding, subtracting, and multiplying large numbers. In Miranda Henry’s resource classroom, for example, students are assigned a budget for a fictional spring break trip; then they find flights, hotels, food, and whatever else they’ll need, while staying within budget.

Math teacher Alicia Wimberley has her Texas students plan and budget a hypothetical trip to the Grand Canyon. “They love the real world context of it and start to see the relevance of the digits after the decimal—including how the .00 at the end of a price was relevant when adding.” One of Wimberley’s students, she writes, mixed up his decimals and nearly planned a $25,000 trip, but found his mistake and dialed back his expenses to under $3,000.

Tap into pizza love

Educators in our audience are big fans of “pizza math”—that is, any kind of math problem that involves pizza. “Pizza math was always a favorite when teaching area of a circle,” notes Shane Capps. If a store is selling a 10-inch pizza, for example, and we know that’s referring to its diameter, what is its total area? “Pizza math is a great tool for addition, subtraction, multiplication, word problems, fractions, and geometry,” another educator writes on our Instagram. There are endless pizza-based word problems online. Here’s a simple one to start, from Jump2Math : “The medium pizza had six slices. Mom and Dad each ate one slice. How much pizza is left?”

Break out the measuring cups

Lindsey Allan has her third-grade students break into pairs, find a recipe they like online, and use multiplication to calculate how much of each ingredient they’d need in order to feed the whole class. The class then votes on a favorite recipe, and they write up a shopping list—“which involves more math, because we have to decide, ‘OK, if we need this much butter for the doubled recipe, will we need three or four sticks, and then how much will be left over?’” Allan writes. “And then it turns out students were also doing division without even realizing!”

Sometimes, a cooking mistake teaches students about proportions the hard way. “Nobody wants a sad chocolate chip cookie where you doubled the dough but not the chocolate chips,” adds teacher Holly Satter.

Heading outdoors is good for kids’ bodies , of course, but it can also be a rich mathematical experience. In second grade, kids can head out to measure perimeters, teacher Jenna McCann suggests—perhaps of the flower boxes in the school garden. If outdoors isn’t an option, there’s plenty of math to be found by walking around inside school—like measuring the perimeter of the tables in the cafeteria or the diameters of circles taped off on the gym floor.

In Maricris Lamigo’s eighth-grade geometry class, “I let [students] roam around the school and take photos of things where congruent triangles were applied,” says Lamigo. “I have students find distances in our indoor courtyard between two stickers that I place on the floor using the Pythagorean theorem,” adds Christopher Morrone, another eighth-grade teacher. In trigonometry, Cathee Cullison sends students outside “with tape measures and homemade clinometers to find heights, lengths, and areas using learned formulas for right and non-right triangles.” Students can make their own clinometers , devices that measure angles of elevation, using protractors and a few other household items.

Plan for adult life

To keep her math lessons both rigorous and engaging, Pamela Kranz runs a monthlong project-based learning activity where her middle school students choose an occupation and receive a salary based on government data. Then they have to budget their earnings to “pay rent, figure out transportation, buy groceries,” and navigate any number of unexpected financial dilemmas, such as medical expenses or car repairs. While learning about personal finance, they develop their mathematical understanding of fractions, decimals, and percents, Kranz writes.

Dig into sports stats

To help students learn how to draw conclusions from data and boost their comfort with decimals and percentages, fourth-grade teacher Kyle Pisselmyer has his students compare the win-loss ratio of the local sports team to that of Pisselmyer’s hometown team. While students can struggle to grasp the relevance of decimals—or to care about how 0.3 differs from 0.305—the details snap into place when they look at baseball players’ stats, educator Maggierose Bennion says.

March Madness is a great source of real world data for students to analyze in math class, says sixth-grade math teacher Jeff Norris. Last March, Norris decorated his classroom like a basketball court, then had his students do basic statistical analysis—like calculating mean, median, and mode—using March Madness data, including individual game scores and the total win rate of each team. “We also did some data collection through our own basketball stations to make it personally relevant,” Norris says; students lined up in teams to shoot paper balls into a basket in a set amount of time, recorded their scores in a worksheet, and then examined the scoring data of the entire class to answer questions about mean, median, mode, range, and outliers.

Go on a (pretend) shopping spree

“My students love any activities that include SHOPPING!” says Jessie, a sixth-grade teacher who creates shopping-related problems using fake (or sometimes real) store ads and receipts. Her students practice solving percentage problems, and the exercise includes opportunities to work with fractions and decimals.

To get students more engaged with the work, math educator Rachel Aleo-Cha zeroes in on objects she knows students are excited about. “I make questions that incorporate items like AirPods, Nike shoes, makeup, etc.,” Aleo-Cha says. She also has students calculate sales tax and prompts them to figure out “what a 50% off plus 20% off discount is—it’s not 70% off.”

Capture math on the fly

Math is everywhere, and whipping out a smartphone when opportunities arise can lead to excellent content for math class. At the foot of Mount Elbert in Colorado, for example, math teacher Ryan Walker recorded a short word problem for his fourth- and fifth-grade students. In the video, he revealed that it was 4:42 a.m., and it would probably take him 249 minutes to reach the summit. What time would he reach the summit, he asked his students—and, assuming it took two-thirds as long to descend, what time would he get back down?

Everyday examples can be especially relatable. At the gas station, “I record a video that tells the size of my gas tank, shows the current price of gas per gallon, and shows how empty my gas tank is,” says Walker. “Students then use a variety of skills (estimation, division, multiplying fractions, multiplying decimals, etc.) to make their estimate on how much money it will cost to fill my tank.”

Connect to social issues

It can be a powerful exercise to connect math to compelling social issues that students care about. In a unit on ratios and proportions, middle school teacher Jennifer Schmerler starts by having students design the “most unfair and unjust city”—where resources and public services like fire departments are distributed extremely unevenly. Using tables and graphs that reflect the distribution of the city’s population and the distribution of its resources, students then design a more equitable city.

Play entrepreneur

Each year, educator Karen Hanson has her fourth- and fifth-grade students brainstorm a list of potential business ideas and survey the school about which venture is most popular. Then the math begins: “We graph the survey results and explore all sorts of questions,” Hanson writes, like whether student preferences vary with age. Winning ideas in the past included selling T-shirts and wallets made of duct tape.

Next, students develop a resource list for the business, research prices, and tally everything up. They calculate a fair price point for the good they’re selling and the sales quantity needed to turn a profit. As a wrap-up, they generate financial statements examining how their profits stack up against the sales figures they had projected.

HELP OTHER TEACHERS OUT!

We’d love this article to be an evolving document of lesson ideas that make math relevant to kids. So, teachers, please tell us about your go-to activities that connect math to kids’ real world experiences.

Department of Mathematics

2024 cube program announced.

The 2024 CUBE Program is an in-person summer research experience for undergraduates. This year’s program will be held on the beautiful campus of Vanderbilt University in Nashville, TN. Participants will conduct research in groups of 2-3 under the supervision of research faculty, postdoctoral scholars, and graduate students. The topics will be in the field of geometric group theory. The program will run for 7 weeks, from June 3 to July 19. The compensation includes room and board, travel to/from Nashville, and a stipend.

See https://margalit.droppages.net/reu.html for information about CUBE.

The application consists of a cover letter, an undergraduate transcript, a Curriculum Vita or resume, a personal statement, and two letters of recommendation. Applications from women and students from other groups that are under-represented in mathematics are especially encouraged.

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Four Penn undergrads are 2024 Goldwater Scholars

Goldwater scholarships are awarded to students planning research careers in mathematics, the natural sciences, or engineering..

Four University of Pennsylvania undergraduates have received 2024 Goldwater Scholarships , awarded to second- or third-year students planning research careers in mathematics, the natural sciences, or engineering.

Penn’s 2024 Goldwater Scholars are third-years Hayle Kim, Eric Myzelev, and Eric Tao in the College of Arts and Sciences , and Kaitlin Mrksich in the School of Engineering and Applied Science .

They are among the 438 students named 2024 Goldwater Scholars from 1,353 undergraduates students nominated by 446 academic institutions in the United States, according to the Barry Goldwater Scholarship & Excellence in Education Foundation . Each scholarship provides as much as $7,500 each year for as many as two years of undergraduate study.

The students applied for the Goldwater Scholarship with assistance from Penn’s Center for Undergraduate Research and Fellowships . Penn has had 63 Goldwater Scholars named since Congress established the scholarship in 1986 to honor U.S. Senator Barry Goldwater.

Kim, from Knoxville, Tennessee, is majoring in neuroscience. She works in the lab of Matthew Kayser at Penn Medicine , where she studies the molecular basis of sleep maturation using Drosophila melanogaster as a model organism. At Penn, Kim is the co-founder and co-president of the undergraduate chapter of the Asian Pacific American Medical Student Association , and was the internal vice president of the Penn Korean Student Association . She is a teaching assistant for the course Chronobiology and Sleep taught by David Raizen , professor of neurology, and has been a learning assistant and peer tutor for general chemistry. Kim is a University Scholar and CURF Research Peer Advisor . She volunteers for the nonprofit One House at a Time in its Beds for Kids program, and in the emergency department of the Children's Hospital of Philadelphia . After graduating, Kim plans to pursue an M.D./Ph.D. in neuroscience.

Mrksich, from Hinsdale, Illinois, is majoring in bioengineering. She is interested in developing drug delivery systems that can serve as novel therapeutics for a variety of diseases. Mrksich works in the lab of Michael J. Mitchell where she investigates the ionizable lipid component of lipid nanoparticles for mRNA delivery. At Penn, Mrksich is the president of the Biomedical Engineering Society , where she plans community building and professional development events for bioengineering majors. She is a member of the Kite and Key Society , where she organizes virtual programming to introduce prospective students to Penn. She is a member of Tau Beta Pi engineering honor society, and the Sigma Kappa sorority. She also teaches chemistry to high schoolers as a volunteer in the West Philadelphia Tutoring Project through the Civic House . After graduating, Mrksich plans to pursue an M.D./Ph.D. in bioengineering.

Myzelev, from Toronto, is majoring in mathematics, with a minor in computer and information science, and is submatriculating for a master’s degree. His research interests include algebra, combinatorics, and using deep learning to solve partial differential equations. Myzelev has worked on numerous research projects in combinatorics and deep learning, has both a co-authored publication and an accepted paper, and has presented his work at several international conferences. Myzelev has been a research assistant for Sasha Indarte , assistant professor of finance at the Wharton School, using recurrent neural networks to identify racial biases in personal bankruptcy outcomes, and he was an intern with the Penn Wharton Budget Model . He is a problem writer for the Canadian Astronomy and Astrophysics Olympiad and an event supervisor for Science Olympiad at Penn . After graduating, Myzelev plans to pursue a Ph.D. in math and research algebraic geometry and combinatorics.

Tao, from Wallingford, Pennsylvania, is majoring in cognitive science, mathematics, and logic, with a focus on language and the mind. They are interested in related fields including physics, linguistics, and logic. Tao studies the neuroscience of social behavior under Marc Schmidt , professor of biology, using multimodal mating displays in the brown-headed cowbird as a model system. Tao co-founded the Penn Math Contest to promote enthusiasm for mathematics among high schoolers, and volunteers with similar organizations such as Science Olympiad at Penn . Tao also works as a teaching assistant in Penn’s Mathematics Department . After graduating, Tao plans to pursue a graduate degree in computational neuroscience.

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Interventions and diversity, equity, and inclusion: Two current directions in research on the teaching and learning of calculus

Original Paper
Published: 25 March 2024

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Tenchita Alzaga Elizondo 2 &
Sean Larsen ORCID: orcid.org/0000-0002-6179-3783 1

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Calculus continues to be an important topic of discussion among mathematics education researchers given how it often acts as a gatekeeper for students in STEM. In their extensive 2017 review of calculus literature, Larsen and colleagues identified two main areas of applied research that had largely been neglected: research related (1) to efforts to improve calculus teaching and learning and (2) to equity and social justice. In this review we investigate how scholars have answered this call by reviewing recent literature related to these two themes. First we identified some promising intervention studies that investigated changes at the course level (e.g., calculus courses intended for engineering students) and at the level of specific calculus topics (e.g. using digital tools to help students understand the Fundamental Theorem). Second, we identified several studies on diversity, equity, and inclusion. We found that some studies in this collection still approached this research through traditional methods (e.g., so called achievement gaps) but we also identified promising new directions for research in which scholars utilize critical theories and provide counter-narratives that highlight the strengths of calculus students from historically marginalized groups. We conclude our review by discussing future directions we hope to see in the field that we argue will strengthen current work.

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Activities for Enacting Equity in Mathematics Education Research

Shandy Hauk, Allison F. Toney, … Katie Salguero

A story of the national calculus curriculum: how culture, research, and policy compete and compromise in shaping the calculus curriculum in South Korea

Hyunkyoung Yoon, Younggon Bae, … Oh Nam Kwon

Second-Order Models as Acts of Equity

We mark with ** papers from the review with annotated bibliographies and with * the rest of the papers included in the review.

Acker, J. (1990). Hierarchies, jobs, bodies: A theory of gendered organizations. Gender & Society, 4 (2), 139–158.

Article Google Scholar

**Adiredja, A. P. (2019). Anti-deficit narratives: Engaging the politics of research on mathematical sense making. Journal for Research in Mathematics Education, 50 (4), 401–435. https://doi.org/10.5951/jresematheduc.50.4.0401 . This paper presents a call for more scholars to take an anti-deficit perspective to cognitive research on students’ sense-making. The author introduced counter-narratives as a way to counteract deficit master-narratives that can be especially harmful to students from marginalized groups. To illustrate his point, the author presented a counter-narrative of a Latina student making sense of the formal definition of function limit.

*Adiredja, A. P. (2021). The pancake story and the epsilon-delta definition. Primus, 31 (6), 662–677. https://doi.org/10.1080/10511970.2019.1669231

Adiredja, A. P., & Andrews-Larson, C. (2017). Taking the sociopolitical turn in postsecondary mathematics education research. International Journal of Research in Undergraduate Mathematics Education, 3 (3), 444–465. https://doi.org/10.1007/s40753-017-0054-5

*Alessio, F., Demeio, L., & Telloni, A. I. (2022). Promoting a meaningful learning of double integrals through routes of digital tasks. Teaching Mathematics and Computer Science , 20 (1), Article 1. https://doi.org/10.5485/TMCS.2022.0539

*Arnold, E. G., Burroughs, E. A., & Deshler, J. M. (2020). Investigating classroom implementation of research-based interventions for reducing stereotype threat in calculus. International Journal of Research & Method in Education, 43 (1), 67–77. https://doi.org/10.1080/1743727X.2019.1575352

*Bakri, S. R. A., Liew, C. Y., Chen, C. K., Tuh, M. H., & Ling, S. C. (2021). Bridging the gap between the derivatives and graph sketching in calculus: An innovative game-based learning approach. Asian Journal of University Education, 16 (4), 121–136.

**Battey, D., Amman, K., Leyva, L. A., Hyland, N., & McMichael, E. W. (2022). Racialized and gendered labor in students’ responses to precalculus and calculus instruction. Journal for Research in Mathematics Education, 53 (2), 94–113. https://doi.org/10.5951/jresematheduc-2020-0170 . This study investigated the type of labor, and associated coping mechanisms, women, Black, and Latino/a calculus students described in response to instructional events they perceive as racialized or gendered. Results indicated that students engaged in both cognitive and emotional labor in response to these events and mitigated their participation in class in order to cope with that labor. The authors described implications of the students’ coping mechanisms.

Biancani, S., & McFarland, D. A. (2013). Social networks research in higher education. In M. B. Paulsen (Ed.), Higher education: Handbook of theory and research (Vol. 28, pp. 151–215). Springer Netherlands. https://doi.org/10.1007/978-94-007-5836-0_4

Chapter Google Scholar

**Bos, R., Doorman, M., & Piroi, M. (2020). Emergent models in a reinvention activity for learning the slope of a curve. The Journal of Mathematical Behavior, 59 , 100773. https://doi.org/10.1016/j.jmathb.2020.100773 . This paper reports on an instructional design study that is supported by the theory of Realistic Mathematics Education and focused on the concept of slope of a curve. Within an a-didactical context students begin the process of reinventing the concept and the authors analyzed their mathematical activity to connect it to various approaches identified in their a priori analysis. They identified a number of informal approaches that could be productively developed during a subsequent institutionalization phase directed by a teacher.

Bressoud, D. M., Carlson, M. P., Mesa, V., & Rasmussen, C. (2013). The calculus student: insights from the Mathematical Association of America national study. International Journal of Mathematical Education in Science and Technology , 44 (5), 685–698.

Cai, J. (2017). Compendium for research in mathematics education . National Council of Teachers of Mathematics.

Google Scholar

Celedón-Pattichis, S., Borden, L. L., Pape, S. J., Clements, D. H., Peters, S. A., Males, J. R., Chapman, O., & Leonard, J. (2018). Asset-based approaches to equitable mathematics education research and practice. Journal for Research in Mathematics Education, 49 (4), 373–389. https://doi.org/10.5951/jresematheduc.49.4.0373

Cetina-Vázquez, M., Cabañas-Sánchez, G., & Sosa-Moguel, L. (2019). Collective mathematical progress in an introductory calculus course during the treatment of the quadratic function. International Journal of Education in Mathematics, Science and Technology, 7 (2), 155–169.

**Champion, J., & Mesa, V. (2018). Pathways to calculus in U.S. high schools. Primus, 28 (6), 508–527. https://doi.org/10.1080/10511970.2017.1315473 . This study investigated the different pathways students take to get to calculus in high school. As part of this investigation, the authors explored correlations between pathways and multiple variables. The results of this study found that self-efficacy, students’ ethnicity, socioeconomic status, and 9th grade course placement were strongly associated with calculus completion among high school students.

Chase, J. P. (2012). From STEM to stern: A review and test of stereotype threat interventions on women’s math performance and motivation [Thesis, Montana State University-Bozeman, College of Letters and Science]. https://scholarworks.montana.edu/xmlui/handle/1/1062 . Accessed 4 Apr 2023.

Crespo, S., Herbst, P., Lichtenstein, E. K., Matthews, P. G., & Chazan, D. (2022). Challenges to and opportunities for sustaining an equity focus in mathematics education research. Journal for Research in Mathematics Education, 53 (2), 88–93. https://doi.org/10.5951/jresematheduc-2021-0215

Devine, A., Fawcett, K., Szűcs, D., & Dowker, A. (2012). Gender differences in mathematics anxiety and the relation to mathematics performance while controlling for test anxiety. Behavioral and Brain Functions, 8 (1), 33. https://doi.org/10.1186/1744-9081-8-33

*Ellis, B., Larsen, S., Voigt, M., & Vroom, K. (2021). Where calculus and engineering converge: An analysis of curricular change in calculus for engineers. International Journal of Research in Undergraduate Mathematics Education, 7 (2), 379–399. https://doi.org/10.1007/s40753-020-00130-9

Fairclough, N. (2013). Critical discourse analysis: The critical study of language . Routledge.

Book Google Scholar

González-Martín, A. S., Nardi, E., & Biza, I. (2018). From resource to document: Scaffolding content and organising student learning in teachers’ documentation work on the teaching of series. Educational Studies in Mathematics, 98 (3), 231–252. https://doi.org/10.1007/s10649-018-9813-8

Guay, R. (1977). Purdue spatial visualization tests . Purdue Research Foundation.

Gutiérrez, R. (2009). Framing equity: Helping students “play the game” and “change the game.” Teaching for Excellence and Equity in Mathematics, 1 (1), 4–8.

Gutiérrez, R. (2013). The sociopolitical turn in mathematics education. Journal for Research in Mathematics Education , 44 (1), 37–68. JSTOR. https://doi.org/10.5951/jresematheduc.44.1.0037

Hauk, S., Toney, A. F., Brown, A., & Salguero, K. (2021). Activities for enacting equity in mathematics education research. International Journal of Research in Undergraduate Mathematics Education, 7 (1), 61–76. https://doi.org/10.1007/s40753-020-00122-9

*Henderson, R., Hewagallage, D., Follmer, J., Michaluk, L., Deshler, J., Fuller, E., & Stewart, J. (2022). Mediating role of personality in the relation of gender to self-efficacy in physics and mathematics. Physical Review Physics Education Research, 18 (1), 010143. https://doi.org/10.1103/PhysRevPhysEducRes.18.010143

Hong, D. S., & Lee, J. K. (2022). Contrasting cases of college calculus instructors: Their preferences and potential pedagogy in teaching derivative graphs. International Journal of Mathematical Education in Science and Technology . https://doi.org/10.1080/0020739X.2022.2120838

Inzlicht, M., & Ben-Zeev, T. (2000). A threatening intellectual environment: Why females are susceptible to experiencing problem-solving deficits in the presence of males. Psychological Science, 11 (5), 365–371. https://doi.org/10.1111/1467-9280.00272

Jaremus, F., Gore, J., Prieto-Rodriguez, E., & Fray, L. (2020). Girls are still being ‘counted out’: Teacher expectations of high-level mathematics students. Educational Studies in Mathematics, 105 (2), 219–236. https://doi.org/10.1007/s10649-020-09986-9

*Jett, C. C. (2021). The qualms and quarrels with online undergraduate mathematics: The experiences of African American male STEM majors. Investigations in Mathematics Learning, 13 (1), 18–28. https://doi.org/10.1080/19477503.2020.1827663

Joseph, N. M., Frank, T. J., & Elliott, T. Y. (2021). A call for a critical-historical framework in addressing the mathematical experiences of black teachers and students. Journal for Research in Mathematics Education, 52 (4), 476–490. https://doi.org/10.5951/jresematheduc-2020-0013

*Kouropatov, A., & Ovodenko, R. (2022). An explorative digital tool as a pathway to meaning: The case of the inflection point. Teaching Mathematics and Its Applications: An International Journal of the IMA, 41 (2), 142–166. https://doi.org/10.1093/teamat/hrac007

*Lagrange, J.-B., & Laval, D. (2023). Connecting algorithmics to mathematics learning: A design study of the intermediate value theorem and the bisection algorithm. Educational Studies in Mathematics, 112 (2), 225–245. https://doi.org/10.1007/s10649-022-10192-y

Larsen, S., Marrongelle, K., Bressoud, D., & Graham, K. (2017). Understanding the concepts of calculus: Frameworks and roadmaps emerging from educational research. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 526–550). National Council of Teachers of Mathematics.

Levine, S. C., Huttenlocher, J., Taylor, A., & Langrock, A. (1999). Early sex differences in spatial skill. Developmental Psychology, 35 (4), 940.

*Lewis, D. (2020). Gender effects on re-assessment attempts in a standards-based grading implementation. Primus, 30 (5), 539–551. https://doi.org/10.1080/10511970.2019.1616636

*Leyva, L. A., McNeill, R. T., Marshall, B. L., & Guzmán, O. A. (2021a). “It seems like they purposefully try to make as many kids drop”: An analysis of logics and mechanisms of racial-gendered inequality in introductory mathematics instruction. The Journal of Higher Education, 92 (5), 784–814. https://doi.org/10.1080/00221546.2021.1879586

**Leyva, L. A., Quea, R., Weber, K., Battey, D., & López, D. (2021b). Detailing racialized and gendered mechanisms of undergraduate precalculus and calculus classroom instruction. Cognition and Instruction, 39 (1), 1–34. https://doi.org/10.1080/07370008.2020.1849218 . This study investigated mechanisms in instruction that women, Black, and Latino/a students identify as sources of racial and gendered marginalization. Three mechanisms of inequities were found: 1) creating differential opportunities for classroom participation and instructor support, 2) limiting within-group peer support and 3) activating racialized and gendered ideas of who belongs in STEM. Based on these results, the authors cautioned that instructors’ behaviors cannot be decontextualized from the sociohistorical realities in which they occur.

*Leyva, L. A., Amman, K., Wolf McMichael, E. A., Igbinosun, J., & Khan, N. (2022). Support for all? Confronting racism and patriarchy to promote equitable learning opportunities through undergraduate calculus instruction. International Journal of Research in Undergraduate Mathematics Education, 8 (2), 339–364. https://doi.org/10.1007/s40753-022-00177-w

Liakos, Y., Gerami, S., Mesa, V., Judson, T., & Ma, Y. (2022). How an inquiry-oriented textbook shaped a calculus instructor’s planning. International Journal of Mathematical Education in Science and Technology, 53 (1), 131–150. https://doi.org/10.1080/0020739X.2021.1961171

Linn, M. C., & Petersen, A. C. (1985). Emergence and characterization of sex differences in spatial ability: A meta-analysis. Child Development , 56 (6), 1479–1498. https://doi.org/10.2307/1130467

*Maciejewski, W., & Star, J. R. (2016). Developing flexible procedural knowledge in undergraduate calculus. Research in Mathematics Education, 18 (3), 299–316. https://doi.org/10.1080/14794802.2016.1148626

*McCunn, L., & Cilli-Turner, E. (2020). Spatial training and calculus ability: Investigating impacts on student performance and cognitive style. Journal of Educational Research and Practice . https://doi.org/10.5590/JERAP.2020.10.1.20

*McNeill, R. T., Leyva, L. A., & Marshall, B. (2022). “They’re just students. There’s no clear distinction”: A critical discourse analysis of color-evasive, gender-neutral faculty discourses in undergraduate calculus instruction. Journal of the Learning Sciences, 31 (4–5), 630–672. https://doi.org/10.1080/10508406.2022.2073233

*Nagle, C., Tracy, T., Adams, G., & Scutella, D. (2017). The notion of motion: Covariational reasoning and the limit concept. International Journal of Mathematical Education in Science and Technology, 48 (4), 573–586. https://doi.org/10.1080/0020739X.2016.1262469

Naidoo, J., & Likwambe, B. (2018). Exploring the nature of dialogue within South African pre-service teachers’ calculus lecture rooms. African Journal of Research in Mathematics, Science and Technology Education, 22 (3), 374–385. https://doi.org/10.1080/18117295.2018.1533612

Park, J., & Rizzolo, D. (2022). Use of variables in calculus class: Focusing on Teaching Assistants’ discussion of variables. International Journal of Mathematical Education in Science and Technology, 53 (1), 165–189. https://doi.org/10.1080/0020739X.2021.1971314

Peters, T., Johnston, E., Bolles, H., Ogilvie, C., Knaub, A., & Holme, T. (2020). Benefits to students of team-based learning in large enrollment calculus. Primus, 30 (2), 211–229. https://doi.org/10.1080/10511970.2018.1542417

Pierce, J. L. (1996). Gender trials: Emotional lives in contemporary law firms . University of California Press.

*Rasmussen, C., Apkarian, N., Hagman, J. E., Johnson, E., Larsen, S., & Bressoud, D. (2019). Brief report: Characteristics of precalculus through calculus 2 programs: Insights From a national census survey. Journal for Research in Mathematics Education, 50 (1), 98–111. https://doi.org/10.5951/jresematheduc.50.1.0098

Reinholz, D. (2017a). Co-calculus: Integrating the academic and the social. International Journal of Research in Education and Science, 3 (2), 521–542.

Reinholz, D. (2017b). Peer conferences in calculus: The impact of systematic training. Assessment & Evaluation in Higher Education, 42 (1), 1–17. https://doi.org/10.1080/02602938.2015.1077197

Reinholz, D. L. (2018). Three approaches to focusing peer feedback. International Journal for the Scholarship of Teaching and Learning , 12 (2), 10.

**Ryberg, U. (2018). Generating different lesson designs and analyzing their effects: The impact of representations when discerning aspects of the derivative. The Journal of Mathematical Behavior, 51 , 1–14. https://doi.org/10.1016/j.jmathb.2018.03.012 . The authors report on a two part investigation consisting of a qualitative design study and a follow-up quantitative study to compare the relative impact of two designs. The resulting findings suggest that it may be beneficial to delay the introduction of symbols in derivative instruction as that representation seems to dominate (and limit) student activity once it is introduced.

*Sadler, P., & Sonnert, G. (2018). The path to college calculus: The impact of high school mathematics coursework. Journal for Research in Mathematics Education, 49 (3), 292–329.

Schoenfeld, A. H. (2000). Purposes and methods of research in mathematics education. Notices of the AMS , 47 (6), 641–649.

Schoenfeld, A. H. (2010). Research methods in (mathematics) education. In L. English, & D. Kirshner (Eds.), Handbook of International Research in Mathematics Education (pp. 467–519). Routledge.

Sealey, V., Infante, N., Campbell, M. P., & Bolyard, J. (2020). The generation and use of graphical examples in calculus classrooms: The case of the mean value theorem. The Journal of Mathematical Behavior, 57 , 100743. https://doi.org/10.1016/j.jmathb.2019.100743

Seymour, E., & Hunter, A.-B. (2019). Talking about leaving revisited: Persistence, relocation, and loss in undergraduate STEM education . Springer Nature.

*Sickle, J. V., Schuler, K. R., Holcomb, J. P., Carver, S. D., Resnick, A., Quinn, C., Jackson, D. K., Duffy, S. F., & Sridhar, N. (2020). Closing the achievement gap for underrepresented minority students in STEM: A deep look at a comprehensive intervention. Journal of STEM Education: Innovations and Research , 21 (2), 5–18. https://jstem.org/jstem/index.php/JSTEM/article/view/2452

Solórzano, D. G., & Yosso, T. J. (2002). Critical race methodology: Counter-storytelling as an analytical framework for education research. Qualitative Inquiry, 8 (1), 23–44. https://doi.org/10.1177/107780040200800103

Sorby, S., Casey, B., Veurink, N., & Dulaney, A. (2013). The role of spatial training in improving spatial and calculus performance in engineering students. Learning and Individual Differences, 26 , 20–29.

Stokes, D. E. (1997). Pasteur’s quadrant: Basic science and technological innovation . Brookings Institution Press.

**Swidan, O. (2020). A learning trajectory for the fundamental theorem of calculus using digital tools. International Journal of Mathematical Education in Science and Technology, 51 (4), 542–562. https://doi.org/10.1080/0020739X.2019.1593531 . The author described a learning trajectory delineated by a set of nine “learning focuses” identified as 11 pairs of students engaged with a digital tool that allowed them to explore connections between the graph of a function and the graph of its accumulation function. Nine such focuses are described and situated with respect to a mathematical description of the Fundamental Theorem of Calculus (informed by prior research).

*Swidan, O., & Fried, M. (2021). Focuses of awareness in the process of learning the fundamental theorem of calculus with digital technologies. The Journal of Mathematical Behavior, 62 , 100847. https://doi.org/10.1016/j.jmathb.2021.100847

Toulmin, S. E. (2003). The uses of argument . Cambridge University Press.

**Tremaine, R., Hagman, J. E., Voigt, M., Damas, S., & Gehrtz, J. (2022). You don’t want to come into a broken system: Perspectives for increasing diversity in STEM among undergraduate calculus program stakeholders. International Journal of Research in Undergraduate Mathematics Education, 8 (2), 365–388. https://doi.org/10.1007/s40753-022-00184-x . Drawing on critical race theories, the authors of this study investigated calculus stakeholders’ motivations for increasing diversity in STEM. The authors presented a framework of these motivations, comparing four motivation themes along both a critical and dominant axes. This framework can be used by institutions to highlight areas of strength and growth in stakeholders regarding equity.

Turra, H., Carrasco, V., González, C., Sandoval, V., & Yáñez, S. (2019). Flipped classroom experiences and their impact on engineering students’ attitudes towards university-level mathematics. Higher Education Pedagogies, 4 (1), 136–155. https://doi.org/10.1080/23752696.2019.1644963

*Voigt, M., Apkarian, N., & Rasmussen, C. (2020). Undergraduate course variations in precalculus through Calculus 2. International Journal of Mathematical Education in Science and Technology, 51 (6), 858–875. https://doi.org/10.1080/0020739X.2019.1636148

Williams, S. R., & Leatham, K. R. (2017). Journal quality in mathematics education. Journal for Research in Mathematics Education, 48 (4), 369–396. https://doi.org/10.5951/jresematheduc.48.4.0369

*Yang, T.-C., Fu, H.-T., Hwang, G.-J., & Yang, S. J. H. (2017). Development of an interactive mathematics learning system based on a two-tier test diagnostic and guiding strategy. Australasian Journal of Educational Technology , 33 (1), Article 1. https://doi.org/10.14742/ajet.2154

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Alzaga Elizondo, T., Larsen, S. Interventions and diversity, equity, and inclusion: Two current directions in research on the teaching and learning of calculus. ZDM Mathematics Education (2024). https://doi.org/10.1007/s11858-024-01553-3

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

Arthur bakker.

1 Utrecht University, Utrecht, Netherlands

2 University of Delaware, Newark, DE USA

Linda Zenger

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

An international survey in two rounds

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

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

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

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

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

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

Methodological approach

Themes for the coming decade (2019).

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

Numbers of responses per continent (2019)

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

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

On September 20, 2019, the three authors met physically at Utrecht University to analyze the responses. After each individual proposal, we settled on a joint list of seven main themes (the first seven in Table Table2), 2 ), which were neither mutually exclusive nor exhaustive. The third author (Zenger, then still a student in educational science) next color coded all parts of responses belonging to a category. These formed the basis for the frequencies and percentages presented in the tables and text. The first author (Bakker) then read all responses categorized by a particular code to identify and synthesize the main topics addressed within each code. The second author (Cai) read all of the survey responses and the response categories, and commented. After the initial round of analysis, we realized it was useful to add an eighth theme: assessment (including evaluation).

Percentages of responses mentioned in each theme (2019)

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

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

It has been a daunting and humbling experience to study the huge coverage and diversity of topics that our colleagues care about. Any categorization felt like a reduction of the wealth of ideas, and we are aware of the risks of “sorting things out” (Bowker & Star, 2000 ), which come with foregrounding particular challenges rather than others (Stephan et al., 2015 ). Yet the best way to summarize the bigger picture seemed by means of clustering themes and pointing to their relationships. As we identified these eight themes of mathematics education research for the future, a recurring question during the analysis was how to represent them. A list such as Table Table2 2 does not do justice to the interrelations between the themes. Some relationships are very clear, for example, educational approaches (theme 2) working toward educational or societal goals (theme 1). Some themes are pervasive; for example, equity and (positive) affect are both things that educators want to achieve but also phenomena that are at stake during every single moment of learning and teaching. Diagrams we considered to represent such interrelationships were either too specific (limiting the many relevant options, e.g., a star with eight vertices that only link pairs of themes) or not specific enough (e.g., a Venn diagram with eight leaves such as the iPhone symbol for photos). In the end, we decided to use an image and collaborated with Elisabeth Angerer (student assistant in an educational sciences program), who eventually made the drawing in Fig. Fig.1 1 to capture themes in their relationships.

An external file that holds a picture, illustration, etc.
Object name is 10649_2021_10049_Fig1_HTML.jpg

Artistic impression of the future themes

Has the pandemic changed your view? (2020)

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

The most frequently mentioned theme was what we labeled approaches to teaching (64% of the respondents, see Table Table2). 2 ). Next was the theme of goals of mathematics education on which research should shed more light in the coming decade (54%). These goals ranged from specific educational goals to very broad societal ones. Many colleagues referred to mathematics education’s relationships with other practices (communities, institutions…) such as home, continuing education, and work. Teacher professional development is a key area for research in which the other themes return (what should students learn, how, how to assess that, how to use technology and ensure that students are interested?). Technology constitutes its own theme but also plays a key role in many other themes, just like affect. Another theme permeating other ones is what can be summarized as equity, diversity, and inclusion (also social justice, anti-racism, democratic values, and several other values were mentioned). These values are not just societal and educational goals but also drivers for redesigning teaching approaches, using technology, working on more just assessment, and helping learners gain access, become confident, develop interest, or even love for mathematics. To evaluate if approaches are successful and if goals have been achieved, assessment (including evaluation) is also mentioned as a key topic of research.

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

Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

Teaching strategies

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

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

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

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

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

Goals of mathematics education

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

Societal goals

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

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

Educational goals

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

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

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

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

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

Relation of mathematics education to other practices

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

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

Teacher professional development

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

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

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

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

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

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

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

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

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

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

Equity, diversity, and inclusion

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

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

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

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

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

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

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

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

Mathematics education research itself

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

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

Methodology

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

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

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

Reflection on our discipline

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

We must normalize (and expect) the full taking up the philosophical and theoretical underpinnings of all of our work (even work that is not considered “philosophical”). Not doing so leads to uncritical analysis and implications.
We must develop norms wherein it is considered embarrassing to do “uncritical” research.
There is no such thing as “neutral.” Amongst other things, this means that we should be cultivating norms that recognize the inherent political nature of all work, and norms that acknowledge how superficially “neutral” work tends to empower the oppressor.
We must recognize the existence of but not cater to the fragility of privilege.

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

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

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

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

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

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

Interdisciplinarity and transdisciplinarity

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

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

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

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

The themes in their coherence: an artistic impression

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

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

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

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

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

Research challenges

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

Aligning new goals, curricula, and teaching approaches

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

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

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

Researching mathematics education across contexts

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

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

Focusing teacher professional development

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

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

Using low-tech resources

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

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

Staying in touch online

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

Studying and improving equity without perpetuating inequality

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

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

Assessing online

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

Doing and publishing interdisciplinary research

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

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

Concluding remarks

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

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

Acknowledgments

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

Appendix 1: Explanation of Fig. Fig.1 1

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

Declarations

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

Publisher’s note

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

Akkerman SF, Bakker A. Boundary crossing and boundary objects. Review of Educational Research. 2011; 81 (2):132–169. doi: 10.3102/0034654311404435. [ CrossRef ] [ Google Scholar ]
Arendt, H. (1958/1998). The human condition (2nd ed.). University of Chicago Press.
Backes B, Cowan J. Is the pen mightier than the keyboard? The effect of online testing on measured student achievement. Economics of Education Review. 2019; 68 :89–103. doi: 10.1016/j.econedurev.2018.12.007. [ CrossRef ] [ Google Scholar ]
Bakkenes, I., Vermunt, J. D., & Wubbels, T. (2010). Teacher learning in the context of educational innovation: Learning activities and learning outcomes of experienced teachers. Learning and Instruction , 20 (6), 533–548. 10.1016/j.learninstruc.2009.09.001
Bakker A. What is worth publishing? A response to Niss. For the Learning of Mathematics. 2019; 39 (3):43–45. [ Google Scholar ]
Bakker A, Gravemeijer KP. An historical phenomenology of mean and median. Educational Studies in Mathematics. 2006; 62 (2):149–168. doi: 10.1007/s10649-006-7099-8. [ CrossRef ] [ Google Scholar ]
Bakx A, Bakker A, Koopman M, Beijaard D. Boundary crossing by science teacher researchers in a PhD program. Teaching and Teacher Education. 2016; 60 :76–87. doi: 10.1016/j.tate.2016.08.003. [ CrossRef ] [ Google Scholar ]
Battey, D. (2013). Access to mathematics: “A possessive investment in whiteness”. Curriculum Inquiry , 43 (3), 332–359.
Bawa, P. (2020). Learning in the age of SARS-COV-2: A quantitative study of learners’ performance in the age of emergency remote teaching. Computers and Education Open , 1 , 100016. 10.1016/j.caeo.2020.100016
Beckers D, Beckers A. ‘Newton was heel exact wetenschappelijk – ook in zijn chemische werk’. Nederlandse wetenschapsgeschiedenis in niet-wetenschapshistorische tijdschriften, 1977–2017. Studium. 2019; 12 (4):185–197. doi: 10.18352/studium.10203. [ CrossRef ] [ Google Scholar ]
Bessot, A., & Ridgway, J. (Eds.). (2000). Education for mathematics in the workplace . Springer.
Bickerton, R. T., & Sangwin, C. (2020). Practical online assessment of mathematical proof. arXiv preprint:2006.01581 . https://arxiv.org/pdf/2006.01581.pdf .
Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education . Springer.
Bini, G., Robutti, O., & Bikner-Ahsbahs, A. (2020). Maths in the time of social media: Conceptualizing the Internet phenomenon of mathematical memes. International Journal of Mathematical Education in Science and Technology , 1–40. 10.1080/0020739x.2020.1807069
Bosch, M., Dreyfus, T., Primi, C., & Shiel, G. (2017, February). Solid findings in mathematics education: What are they and what are they good for? CERME 10 . Ireland: Dublin https://hal.archives-ouvertes.fr/hal-01849607
Bowker, G. C., & Star, S. L. (2000). Sorting things out: Classification and its consequences . MIT Press. 10.7551/mitpress/6352.001.0001
Burkhardt, H. (2019). Improving policy and practice. Educational Designer , 3 (12) http://www.educationaldesigner.org/ed/volume3/issue12/article46/
Cai J, Hwang S. Constructing and employing theoretical frameworks in (mathematics) education research. For the Learning of Mathematics. 2019; 39 (3):44–47. [ Google Scholar ]
Cai J, Jiang C. An analysis of problem-posing tasks in Chinese and U.S. elementary mathematics textbooks. International Journal of Science and Mathematics Education. 2017; 15 (8):1521–1540. doi: 10.1007/s10763-016-9758-2. [ CrossRef ] [ Google Scholar ]
Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics , 105 , 287–301. 10.1007/s10649-020-10008-x
Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., … Hiebert, J. (2020). Improving the impact of research on practice: Capitalizing on technological advances for research. Journal for Research in Mathematics Education , 51 (5), 518–529 https://pubs.nctm.org/view/journals/jrme/51/5/article-p518.xml
Chronaki, A. (2019). Affective bodying of mathematics, children and difference: Choreographing ‘sad affects’ as affirmative politics in early mathematics teacher education. ZDM-Mathematics Education , 51 (2), 319–330. 10.1007/s11858-019-01045-9
Civil, M., & Bernier, E. (2006). Exploring images of parental participation in mathematics education: Challenges and possibilities. Mathematical Thinking and Learning , 8 (3), 309–330. 10.1207/s15327833mtl0803_6
Cobb P, Gresalfi M, Hodge LL. An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education. 2009; 40 (1):40–68. [ Google Scholar ]
Darragh L. Identity research in mathematics education. Educational Studies in Mathematics. 2016; 93 (1):19–33. doi: 10.1007/s10649-016-9696-5. [ CrossRef ] [ Google Scholar ]
de Abreu, G., Bishop, A., & Presmeg, N. C. (Eds.). (2006). Transitions between contexts of mathematical practices . Kluwer.
de Freitas, E., Ferrara, F., & Ferrari, G. (2019). The coordinated movements of collaborative mathematical tasks: The role of affect in transindividual sympathy. ZDM-Mathematics Education , 51 (2), 305–318. 10.1007/s11858-018-1007-4
Deng, Z. (2018). Contemporary curriculum theorizing: Crisis and resolution. Journal of Curriculum Studies , 50 (6), 691–710. 10.1080/00220272.2018.1537376
Dobie, T. E., & Sherin, B. (2021). The language of mathematics teaching: A text mining approach to explore the zeitgeist of US mathematics education. Educational Studies in Mathematics . 10.1007/s10649-020-10019-8
Eames, C., & Eames, R. (1977). Powers of Ten [Film]. YouTube. https://www.youtube.com/watch?v=0fKBhvDjuy0
Engelbrecht, J., Borba, M. C., Llinares, S., & Kaiser, G. (2020). Will 2020 be remembered as the year in which education was changed? ZDM-Mathematics Education , 52 (5), 821–824. 10.1007/s11858-020-01185-3 [ PMC free article ] [ PubMed ]
English, L. (2008). Setting an agenda for international research in mathematics education. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 3–19). Routledge.
Ernest, P. (2020). Unpicking the meaning of the deceptive mathematics behind the COVID alert levels. Philosophy of Mathematics Education Journal , 36 http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/pome36/index.html
Freudenthal, H. (1986). Didactical phenomenology of mathematical structures . Springer.
Gilmore, C., Göbel, S. M., & Inglis, M. (2018). An introduction to mathematical cognition . Routledge.
Goos, M., & Beswick, K. (Eds.). (2021). The learning and development of mathematics teacher educators: International perspectives and challenges . Springer. 10.1007/978-3-030-62408-8
Gorard, S. (Ed.). (2020). Getting evidence into education. Evaluating the routes to policy and practice . Routledge.
Gravemeijer, K., Stephan, M., Julie, C., Lin, F.-L., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education , 15 (1), 105–123. 10.1007/s10763-017-9814-6
Hannula, M. S. (2019). Young learners’ mathematics-related affect: A commentary on concepts, methods, and developmental trends. Educational Studies in Mathematics , 100 (3), 309–316. 10.1007/s10649-018-9865-9
Hilt, L. T. (2015). Included as excluded and excluded as included: Minority language pupils in Norwegian inclusion policy. International Journal of Inclusive Education , 19 (2), 165–182.
Hodgen, J., Taylor, B., Jacques, L., Tereshchenko, A., Kwok, R., & Cockerill, M. (2020). Remote mathematics teaching during COVID-19: Intentions, practices and equity . UCL Institute of Education https://discovery.ucl.ac.uk/id/eprint/10110311/
Horn, I. S. (2017). Motivated: Designing math classrooms where students want to join in . Heinemann.
Hoyles C, Noss R, Pozzi S. Proportional reasoning in nursing practice. Journal for Research in Mathematics Education. 2001; 32 (1):4–27. doi: 10.2307/749619. [ CrossRef ] [ Google Scholar ]
Ito, M., Martin, C., Pfister, R. C., Rafalow, M. H., Salen, K., & Wortman, A. (2018). Affinity online: How connection and shared interest fuel learning . NYU Press.
Jackson K. Approaching participation in school-based mathematics as a cross-setting phenomenon. The Journal of the Learning Sciences. 2011; 20 (1):111–150. doi: 10.1080/10508406.2011.528319. [ CrossRef ] [ Google Scholar ]
Jansen, A., Herbel-Eisenmann, B., & Smith III, J. P. (2012). Detecting students’ experiences of discontinuities between middle school and high school mathematics programs: Learning during boundary crossing. Mathematical Thinking and Learning , 14 (4), 285–309. 10.1080/10986065.2012.717379
Johnson, L. F., Smith, R. S., Smythe, J. T., & Varon, R. K. (2009). Challenge-based learning: An approach for our time (pp. 1–38). The New Media Consortium https://www.learntechlib.org/p/182083
Jullien, F. (2018). Living off landscape: Or the unthought-of in reason . Rowman & Littlefield.
Kazima, M. (2019). What is proven to work in successful countries should be implemented in other countries: The case of Malawi and Zambia. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 73–78). PME.
Kim, H. (2019). Ask again, “why should we implement what works in successful countries?” In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 79–82). PME.
Kolovou, A., Van Den Heuvel-Panhuizen, M., & Bakker, A. (2009). Non-routine problem solving tasks in primary school mathematics textbooks—a needle in a haystack. Mediterranean Journal for Research in Mathematics Education , 8 (2), 29–66.
Kwon, O. N., Han, C., Lee, C., Lee, K., Kim, K., Jo, G., & Yoon, G. (2021). Graphs in the COVID-19 news: A mathematics audit of newspapers in Korea. Educational Studies in Mathematics . 10.1007/s10649-021-10029-0 [ PMC free article ] [ PubMed ]
Lefebvre, H. (2004). Rhythmanalysis: Space, time and everyday life (Original 1992; Translation by S. Elden & G. Moore) . Bloomsbury Academic. 10.5040/9781472547385.
Li, Y. (2019). Should what works in successful countries be implemented in other countries? In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 67–72). PME.
Martin, D., Gholson, M., & Leonard, J. (2010). Mathematics as gatekeeper: Power and privilege in the production of power. Journal of Urban Mathematics Education , 3 (2), 12–24.
Masschelein, J., & Simons, M. (2019). Bringing more ‘school’ into our educational institutions. Reclaiming school as pedagogic form. In A. Bikner-Ahsbahs & M. Peters (Eds.), Unterrichtsentwicklung macht Schule (pp. 11–26) . Springer. 10.1007/978-3-658-20487-7_2
Meeter, M., Bele, T., den Hartogh, C., Bakker, T., de Vries, R. E., & Plak, S. (2020). College students’ motivation and study results after COVID-19 stay-at-home orders. https://psyarxiv.com .
Nemirovsky, R., Kelton, M. L., & Civil, M. (2017). Toward a vibrant and socially significant informal mathematics education. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 968–979). National Council of Teachers of Mathematics.
Niss M. The very multi-faceted nature of mathematics education research. For the Learning of Mathematics. 2019; 39 (2):2–7. [ Google Scholar ]
OECD. (2020). Back to the Future of Education: Four OECD Scenarios for Schooling. Educational Research and Innovation . OECD Publishing. 10.1787/20769679
Potari, D., Psycharis, G., Sakonidis, C., & Zachariades, T. (2019). Collaborative design of a reform-oriented mathematics curriculum: Contradictions and boundaries across teaching, research, and policy. Educational Studies in Mathematics , 102 (3), 417–434. 10.1007/s10649-018-9834-3
Proulx, J., & Maheux, J. F. (2019). Effect sizes, epistemological issues, and identity of mathematics education research: A commentary on editorial 102(1). Educational Studies in Mathematics , 102 (2), 299–302. 10.1007/s10649-019-09913-7
Roos, H. (2019). Inclusion in mathematics education: An ideology, A way of teaching, or both? Educational Studies in Mathematics , 100 (1), 25–41. 10.1007/s10649-018-9854-z
Saenz, M., Medina, A., & Urbine Holguin, B. (2020). Colombia: La prender al onda (to turn on the wave). Education continuity stories series . OECD Publishing https://oecdedutoday.com/wp-content/uploads/2020/12/Colombia-a-prender-la-onda.pdf
Schindler, M., & Bakker, A. (2020). Affective field during collaborative problem posing and problem solving: A case study. Educational Studies in Mathematics , 105 , 303–324. 10.1007/s10649-020-09973-0
Schoenfeld, A. H. (1999). Looking toward the 21st century: Challenges of educational theory and practice. Educational Researcher , 28 (7), 4–14. 10.3102/0013189x028007004
Schukajlow, S., Rakoczy, K., & Pekrun, R. (2017). Emotions and motivation in mathematics education: Theoretical considerations and empirical contributions. ZDM-Mathematics Education , 49 (3), 307–322. 10.1007/s11858-017-0864-6
Sfard A. What could be more practical than good research? Educational Studies in Mathematics. 2005; 58 (3):393–413. doi: 10.1007/s10649-005-4818-5. [ CrossRef ] [ Google Scholar ]
Shimizu, Y., & Vithal, R. (Eds.). (2019). ICMI Study 24 Conference Proceedings. School mathematics curriculum reforms: Challenges, changes and opportunities . ICMI: University of Tsukuba & ICMI http://www.human.tsukuba.ac.jp/~icmi24/
Sierpinska A. Some remarks on understanding in mathematics. For the Learning of Mathematics. 1990; 10 (3):24–41. [ Google Scholar ]
Stephan, M. L., Chval, K. B., Wanko, J. J., Civil, M., Fish, M. C., Herbel-Eisenmann, B., … Wilkerson, T. L. (2015). Grand challenges and opportunities in mathematics education research. Journal for Research in Mathematics Education , 46 (2), 134–146. 10.5951/jresematheduc.46.2.0134
Suazo-Flores, E., Alyami, H., Walker, W. S., Aqazade, M., & Kastberg, S. E. (2021). A call for exploring mathematics education researchers’ interdisciplinary research practices. Mathematics Education Research Journal , 1–10. 10.1007/s13394-021-00371-0
Svensson, P., Meaney, T., & Norén, E. (2014). Immigrant students’ perceptions of their possibilities to learn mathematics: The case of homework. For the Learning of Mathematics , 34 (3), 32–37.
UNESCO. (2015). Teacher policy development guide . UNESCO, International Task Force on Teachers for Education 2030. https://teachertaskforce.org/sites/default/files/2020-09/370966eng_0_1.pdf .
Van den Heuvel-Panhuizen M. Can scientific research answer the ‘what’ question of mathematics education? Cambridge Journal of Education. 2005; 35 (1):35–53. doi: 10.1080/0305764042000332489. [ CrossRef ] [ Google Scholar ]
Wittmann EC. Mathematics education as a ‘design science’ Educational Studies in Mathematics. 1995; 29 (4):355–374. doi: 10.1007/BF01273911. [ CrossRef ] [ Google Scholar ]
Yoon, H., Byerley, C. O. N., Joshua, S., Moore, K., Park, M. S., Musgrave, S., Valaas, L., & Drimalla, J. (2021). United States and South Korean citizens’ interpretation and assessment of COVID-19 quantitative data. The Journal of Mathematical Behavior . 10.1016/j.jmathb.2021.100865.

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251+ Math Research Topics [2024 Updated]
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.
181 Math Research Topics
If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics: Methods to count discrete objects. The origins of Greek symbols in mathematics. Methods to solve simultaneous equations. Real-world applications of the theorem of Pythagoras.
Pure mathematics
Pure mathematics uses mathematics to explore abstract ideas, mathematics that does not necessarily describe a real physical system. This can include developing the fundamental tools used by ...
Research Areas
Department members engage in cutting-edge research on a wide variety of topics in mathematics and its applications. Topics continually evolve to reflect emerging interests and developments, but can roughly grouped into the following areas. Algebra, Combinatorics, and Geometry Algebra, combinatorics, and geometry are areas of very active research at the University of Pittsburgh.
Mathematics Research Paper Topics
Mathematics Research Paper Topics. Mathematics undoubtedly began as an entirely practical activity— measuring fields, determining the volume of liquids, counting out coins, and the like. During the golden era of Greek science, between about the sixth and third centuries B.C., however, mathematicians introduced a new concept to their study of ...
200 Brilliant Math Research Topics For Students To Consider
Math Research Paper Topics for Undergraduate Students. Explain the different theories of mathematical logic. Discuss the origins of Greek symbols in mathematics. Explain the significance of circles. Analyze predictive models. Explain the emergence of patterns in chaos theory. Define abstract algebra.
Mathematics News, Research and Analysis
Top contributors. Jonathan Borwein (Jon) Laureate Professor of Mathematics, University of Newcastle David H. Bailey PhD; Lawrence Berkeley Laboratory (retired) and Research Fellow, University of ...
Future themes of mathematics education research: an international
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 ...
Frontiers in Applied Mathematics and Statistics
Numerical Simulations and Algorithms for High-Performance Computing. Andrea Franceschini. Hussam Al Daas. Yuanzhe Xi. 5,257 views. 1 article. Explores how the application of mathematics and statistics can drive scientific developments across data science, engineering, finance, physics, biology, ecology, business, medicine, and beyond.
Applied Mathematics
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 ...
Research
In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications. Combinatorics. Computational Biology. Physical Applied Mathematics. Computational Science & Numerical Analysis.
Applied mathematics
Applied mathematics is the application of mathematical techniques to describe real-world systems and solve technologically relevant problems. This can include the mechanics of a moving body, the ...
Algebra
We have large groups of researchers active in number theory and algebraic geometry, as well as many individuals who work in other areas of algebra: groups, noncommutative rings, Lie algebras and Lie super-algebras, representation theory, combinatorics, game theory, and coding. Chairs: George Bergman and Tony Feng.
Posing Researchable Questions in Mathematics and Science ...
In research related to mathematics and science education, there is no shortage of evidence for the impact of posing important and researchable questions: Posing new, researchable questions marks real advances in mathematics and science education (Cai et al., 2019a).Although research in mathematics and science education begins with researchable questions, only recently have researchers begun to ...
Mathematical Analysis
Mathematical Analysis. In a rough division of mathematics, mathematical analysis deals with inequalities and limits. In some of its branches, such as asymptotic analysis, these aspects of the subject matter are readily apparent. in others, such as operator algebras, they are concealed in the topology of an algebra or its structure as an ...
A Project-Based Guide to Undergraduate Research in Mathematics
The chapters in this work span a variety of topical areas of pure and applied mathematics and mathematics education. Each chapter gives a self-contained introduction on a research topic with an emphasis on the specific tools and knowledge needed to create and maintain fruitful research programs for undergraduates.
Research Areas
Mathematics Research Center; Robin Li and Melissa Ma Science Library; Contact. Department of Mathematics Building 380, Stanford, California 94305 Phone: (650) 725-6284 mathwebsite [at] lists.stanford.edu (Email) Giving to the Department of Mathematics ...
Mathematics
Mathematics teachers' knowledge is considered one of the most critical factors in instruction and student achievement. As such, various studies have focused on mathematics teachers' knowledge. Despite the expansion of the field, however, a systematic review was rarely implemented. Therefore, this study aimed to identify major research topics and trends on mathematics teachers' knowledge ...
Research in Mathematics: Vol 11, No 1 (Current issue)
Research in Mathematics, Volume 11, Issue 1 (2024) See all volumes and issues. Volume 11, 2024 Vol 10, 2023 Vol 9, 2022 Vol 8, 2021 Vol 7, 2020 Vol 6, 2019 Vol 5, 2018 Vol 4, 2017 Vol 3, 2016 Vol 2, 2015 Vol 1, 2014. Download citations Download PDFs Download issue. Browse by section (All)
Research Areas in Mathematics
The topics of interests include de Branges-Rovnyak spaces and weighted shifts on directed graphs. ... Mathematical ecology. 1. Research in this area is focused on the local and global stability analysis, detection of possible bifurcation scenario and derivation of normal form, chaotic dynamics for the ordinary as well as delay differential ...
Applied Mathematics Research
Applied Mathematics Research. In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications. ... Department of Mathematics Headquarters Office Simons Building (Building 2), Room 106 77 ...
11 Real World Math Activities That Engage Students
March Madness is a great source of real world data for students to analyze in math class, says sixth-grade math teacher Jeff Norris. Last March, Norris decorated his classroom like a basketball court, then had his students do basic statistical analysis—like calculating mean, median, and mode—using March Madness data, including individual ...
2024 CUBE program announced
The 2024 CUBE Program is an in-person summer research experience for undergraduates. This year's program will be held on the beautiful campus of Vanderbilt University in Nashville, TN. Participants will conduct research in groups of 2-3 under the supervision of research faculty, postdoctoral scholars, and graduate students. The topics will be in the field of...
Four Penn undergrads are 2024 Goldwater Scholars
Four University of Pennsylvania undergraduates have received 2024 Goldwater Scholarships, awarded to second- or third-year students planning research careers in mathematics, the natural sciences, or engineering.. Penn's 2024 Goldwater Scholars are third-years Hayle Kim, Eric Myzelev, and Eric Tao in the College of Arts and Sciences, and Kaitlin Mrksich in the School of Engineering and ...
Interventions and diversity, equity, and inclusion: Two current
Calculus is a foundational topic for science and engineering disciplines around the world. At the same time, students often struggle in calculus courses discouraging them from pursuing degrees in these disciplines (Bressoud et al., 2013).The mathematics education research community has the responsibility to guide efforts to improve the teaching and learning of calculus.
Development of neomaths module and its usability for teaching
The main purpose of this research were to develop and to determine the validity of Nearpod-Based Neomaths Module for teaching Differentiation topic in Additional Mathematics, as well as to measure the usability of this module from the perspective of the trainee teachers from the Department of Mathematics, Universiti Pendidikan Sultan Idris (UPSI). Development model of the module was based on ...
Future themes of mathematics education research: an international
An international survey in two rounds. Around the time when Educational Studies in Mathematics (ESM) and the Journal for Research in Mathematics Education (JRME) were celebrating their 50th anniversaries, Arthur Bakker (editor of ESM) and Jinfa Cai (editor of JRME) saw a need to raise the following future-oriented question for the field of mathematics education research:
‎The Teaching Toolbox
The research about math manipulatives really makes us reflect on HOW we are using these tools in the classroom. Pull a chair up to our teacher table and let's chat about this topic together! Topics Discussed What are math manipulatives?Why are they effective in the middle school math classroom?How t…
Apply for 2024 Foerderer Grants
Applications are open for the 2024 Foerderer Grants internal award competition at Children's Hospital of Philadelphia Research Institute. Foerderer Grants are designed to allow ongoing research to move into new and productive areas, or to allow investigators to apply new research techniques toward novel investigations. Eligible applicants include:
Women in Science: Infectious Diseases: Epidemiology and ...
The proportion of women and men in science, technology, engineering, and mathematics (STEM) at undergraduate levels is relatively equal. However, there is a lack of representation of women in senior positions in Public Health. According to the UNESCO Institute for Statistics (UIS) data in 2016, less than 30% of researchers in STEM are women.In the field of Infectious Diseases, there are many ...

251+ Math Research Topics [2024 Updated]

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251+ Math Research Topics: Beginners To Advanced

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Research Areas

Algebra, Combinatorics, and Geometry

Analysis and Partial Differential Equations

Applied Analysis

Mathematical Biology

Mathematical Finance

Numerical Analysis and Scientific Computing

Topology and Differential Geometry

Mathematics Research Paper Topics

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