examples of research topics in 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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Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2024 2024.

Information Based Approach for Detecting Change Points in Inverse Gaussian Model with Applications , Alexis Anne Wallace

Theses/Projects/Dissertations from 2023 2023

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

An Exposition of the Curvature of Warped Product Manifolds , Angelina Bisson

Jackknife Empirical Likelihood Tests for Equality of Generalized Lorenz Curves , Anton Butenko

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

Statistical Analysis of Health Habits for Incoming College Students , Wendy Isamara Lizarraga Noriega

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

Distance Correlation Based Feature Selection in Random Forest , Jose Munoz-Lopez

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

The Examination of the Arithmetic Surface (3, 5) Over Q , Rachel J. Arguelles

Error Terms for the Trapezoid, Midpoint, and Simpson's Rules , Jessica E. Coen

de Rham Cohomology, Homotopy Invariance and the Mayer-Vietoris Sequence , Stacey Elizabeth Cox

Symmetric Generation , Ana Gonzalez

SYMMETRIC PRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Samar Mikhail Kasouha

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

THE DECOMPOSITION OF THE SPACE OF ALGEBRAIC CURVATURE TENSORS , Katelyn Sage Risinger

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

Non-Abelian Finite Simple Groups as Homomorphic Images , Sandra Bahena

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

Symmetric Presentation of Finite Groups, and Related Topics , Marina Michelle Duchesne

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

Sum of Cubes of the First n Integers , Obiamaka L. Agu

Permutation and Monomial Progenitors , Crystal Diaz

Tile Based Self-Assembly of the Rook's Graph , Ernesto Gonzalez

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

DNA COMPLEXES OF ONE BOND-EDGE TYPE , Andrew Tyler Lavengood-Ryan

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

ASSESSING STUDENT UNDERSTANDING WHILE SOLVING LINEAR EQUATIONS USING FLOWCHARTS AND ALGEBRAIC METHODS , Edima Umanah

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

VANISHING LOCAL SCALAR INVARIANTS ON GENERALIZED PLANE WAVE MANIFOLDS , Brian Matthew Friday

Analogues Between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle , Lacey Taylor James

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle , Antonio Saucedo Jr.

THE EFFECTIVENESS OF DYNAMIC MATHEMATICAL SOFTWARE IN THE INSTRUCTION OF THE UNIT CIRCLE , Edward Simons

CALCULUS REMEDIATION AS AN INDICATOR FOR SUCCESS ON THE CALCULUS AP EXAM , Ty Stockham

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

Progenitors, Symmetric Presentations, and Related Topics , Joana Viridiana Luna

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

THE STRUGGLE WITH INVERSE FUNCTIONS DOING AND UNDOING PROCESS , Jesus Nolasco

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING , Gina Marie Cheney

Solving Absolute Value Equations and Inequalities on a Number Line , Melinda A. Curtis

BIO-MATHEMATICS: INTRODUCTION TO THE MATHEMATICAL MODEL OF THE HEPATITIS C VIRUS , Lucille J. Durfee

ANALYSIS AND SYNTHESIS OF THE LITERATURE REGARDING ACTIVE AND DIRECT INSTRUCTION AND THEIR PROMOTION OF FLEXIBLE THINKING IN MATHEMATICS , Genelle Elizabeth Gonzalez

LIFE EXPECTANCY , Ali R. Hassanzadah

PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS , Sean M. Hearon

A Dual Fano, and Dual Non-Fano Matroidal Network , Stephen Lee Johnson

Mathematical Reasoning and the Inductive Process: An Examination of The Law of Quadratic Reciprocity , Nitish Mittal

The Kauffman Bracket and Genus of Alternating Links , Bryan M. Nguyen

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

Indicators of Future Mathematics Proficiency: Literature Review & Synthesis , Claudia Preciado

Ádám's Conjecture and Arc Reversal Problems , Claudio D. Salas

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

Discovering and Applying Geometric Transformations: Transformations to Show Congruence and Similarity , Tamara V. Bonn

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS , Leonard B. Lamp

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

Unique Prime Factorization of Ideals in the Ring of Algebraic Integers of an Imaginary Quadratic Number Field , Nolberto Rezola

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examples of research topics in mathematics

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.

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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202 Math Research Topics: List To Vary Your Ideas

$202 Math Research Topics$

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

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

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

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

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

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

How to Structure Your Math Essay or Paper

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

Choosing a Topic to Discuss

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

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

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

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

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

Research Topics in Math

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

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

Undergraduate Math Research Topics

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

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

Math Research Topics for High School Students

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

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

Interesting Math Research Topics

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

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

Cool Math Topics to Research

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

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

Math Research Topics for Middle School

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

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

Applied Math Research Topics

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

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

Math Research Topics for College Students

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

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

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

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DEPARTMENT OF MATHEMATICS

Undergraduate

Undergraduate Research Projects

Northwestern undergraduates have opportunities to explore mathematics beyond our undergraduate curriculum by enrolling in math 399-0 independent study, working on a summer project, or writing a senior thesis under the supervision of a faculty member. below are descriptions of projects that our faculty have proposed. students interested in one of these projects should contact the project adviser. this should not be taken to be an exhaustive list of all projects that are availalbe, nor as a list of the only faculty open to supervising such projects. contact the director of undergraduate studies for additional guidance. these projects are only available to northwestern undergraduates., combinatorial structures in symplectic topology, eric zaslow, symplectic and contact geometry describe the mathematics of phase space for particles and light, respectively. they therefore are the mathematical home for dynamical systems arising from physics. a noteworthy structure within contact geometry is that of a legendrian surface, closely related to the wavefront of propagating light. these subspaces sometimes have combinatorial descriptions via graphs. the project explores how well the combinatorial descriptions can distinguish legendrian surfaces, just as in knot theory one might explore whether the jones polynomial can distinguish different knots. , prerequisites: math 330-1 or math 331-1, math 342-0. recommended: math 308-0., complexity and periodicity, the simplest bi-infinite sequences in $\{0, 1\}^{\mathbb z}$ are the periodic sequences, where a single pattern is concatenated with itself infinitely often. at the opposite extreme are bi-infinite sequences containing every possible configuration of $0$'s and $1$'s. for periodic sequences, the number of substrings of length $n$ is bounded, while in the second case, all substrings appear and so there are $2^n$ substrings of length $n$. the growth rate of the possible patterns is a measurement of the complexity of the sequence, giving information about the sequence itself and describing objects encoded by the sequence. symbolic dynamics is the study of such sequences, associated dynamical systems, and their properties. an old theorem of morse and hedlund gives a simple relation between this measurement of complexity and periodicity: a bi-infinite sequence with entries in a finite alphabet $\mathcal a$ is periodic if and only if there exists some $n\in\mathbb n$ such that the sequence contains at most $n$ words of length $n$. however, as soon as we turn to higher dimensions, meaning a sequence in $\mathcal a^{\mathbb z^d}$ for some $d\geq 2$ rather than $d=1$, the relation between complexity and periodicity is no longer clear. even defining what is meant by low complexity or periodicity is not clear. this project will cover what is known in one dimension and then turn to understanding how to generalize these phenomena to higher dimensions. prerequisite: math 320-3 or math 321-3., finite simple groups, ezra getzler, finite simple groups are the building blocks of finite groups. for any finite group $g$, there is a normal subgroup $h$ such that $g/h$ is a simple group: the simple groups are those groups with no nontrivial normal subgroups. the abelian finite simple groups are the cyclic groups of prime order; in this sense, finite simple groups generalize the prime numbers. one of the beautiful theorems of algebra is that the alternating groups $a_n$ (subgroups of the symmetric groups $s_n$) are simple for $n\geq 5$. in fact, $a_5$ is the smallest non-abelian finite simple group (its order is $60$). another series of finite simple groups was discovered by galois. let $\mathbb f$ be a field. the group $sl_2(\mathbb f)$ is the group of all $2\times2$ matrices of determinant $1$. if we take $\mathbb f$ to be a finite field, we get a finite group; for example, we can take $\mathbb f=\mathbb f_p$, the field with $p$ elements. it is a nice exercise to check that $sl_2(\mathbb f_p)$ has $p^3-p$ elements. the center $z(sl_2(\mathbb f_p))$ of $sl_2(\mathbb f_p)$ is the set of matrices $\pm i$; this has two elements unless $p=2$. the group $psl_2(\mathbb f)$ is the quotient of $sl_2(\mathbb f)$ by its center $z(sl_2(\mathbb f))$: we see that $psl_2(\mathbb f_p)$ has order $(p^3-p)/2$ unless $p=2$. it turns out that $psl_2(\mathbb f_2)$ and $psl_2(\mathbb f_3)$ are isomorphic to $s_3$ and $a_4$, which are not simple, but $psl_2(\mathbb f_5)$ is isomorphic to $a_5$, the smallest nonabelian finite simple group, and $psl_2(\mathbb f_7)$, of order $168$, is the second smallest nonabelian finite simple group. (when $\mathbb f$ is the field of complex numbers, the group $psl_2(\mathbb c)$ is also very interesting, though of course it is not finite: it is isomorphic to the lorentz group of special relativity.) the goal of this project is to learn about generalizations of this construction, which together with the alternating groups yield all but a finite number of the finite simple groups. (there are 26 missing ones called the sporadic simple groups that cannot be obtained in this way.) this mysterious link between geometry and algebra is hard to explain, but very important: much of what we know about the finite simple groups comes from the study of matrix groups over the complex numbers. prerequisite: math 330-3 or math 331-3., fourier series and representation theory, fourier series allow you to write a periodic function in terms of a basis of sines and cosines. one way to think of this is to understand sines and cosines as the eigenfunctions of the second derivative operator – so fourier series generalize the spectral theorem of linear algebra in this sense. there is another viewpoint that is useful: periodic functions can be thought of as functions defined on a circle, which is itself a group. the connection between group theory and fourier series runs deeper, and this is the subject of this project. moving up a dimension, functions on a sphere can be described in terms of spherical harmonics. while the sphere is not a group, it is the orbit space of the unit vector in the vertical direction. thus it can be constructed as a homogeneous space: it is the group of rotations modulo the group of rotations around the vertical axis. we can therefore access functions on the sphere via functions on the group of rotations. the peter-weyl theorem describes the vector space of functions on the group in terms of its representation theory. (a representation of a group is a vector space on which group elements act as linear transformations [e.g., matrices], consistent with their relations.) the entries of matrix elements of the irreducible representations of the group play the role that sines and cosines did above. indeed, we can combine sines and cosines into complex exponentials and these are the sole entries of the one-by-one matrices (characters) representing the abelian circle group. finally, we will connect spherical harmonics to polynomial functions relevant to geometric structures described in the borel-weyl-bott theorem. students will explore many examples along with learning the foundations of the theory. prerequisites: math 351-0 or math 381-0., linear poisson geometry, santiago cañez , a poisson bracket is a type of operation which takes as input two functions and outputs some expression obtained by multiplying their derivatives, subject to some constraints. for instance, the standard poisson bracket of two functions $f,g$ on $\mathbb r^2$ is defined by $\{f,g\} =\frac{\partial f}{\partial x} \frac{\partial g}{\partial y} - \frac{\partial f}{\partial y} \frac{\partial g}{\partial x}$. such objects first arose in physics in order to describe the time evolution of mechanical systems, but have now found other uses as well. in particular, a linear poisson bracket on a vector space turns out to encode the same data as that of a lie algebra, another type of algebraic object which is ubiquitous in mathematics. this relation between linear poisson brackets and lie algebra structures allows one to study the same object from different perspectives; in particular, this allows one to better understand the notion of coadjoint orbits and the hidden structure within them., the goal of this project is to understand the relation between linear poisson brackets and lie algebras, and to use this relation to elucidate properties of coadjoint orbits. all of these structures are heavily used in physics, and gaining a deep understanding as to why depends on the relation described above. moreover, this project will bring in topics from many different areas of mathematics – analysis, group theory, and linear algebra – to touch on areas of modern research., prerequisites: math 320-1 or math 321-1, math 330-1 or math 331-1, math 334-0 or math 291-2., noncommutative topology, given a space $x$, one can consider various types of functions defined on $x$, say for instance continuous functions from $x$ to $\mathbb c$. the set $c(x)$ of all such functions often comes equipped with some additional structure itself, which allows for the study of various geometric or topological properties of $x$ in terms of the set of functions $c(x)$ instead. in particular, when $x$ is a compact hausdorff space, the set $c(x)$ of complex-valued continuous functions on $x$ has the structure of what is known as a commutative $c^*$-algebra, and the gelfand-naimark theorem asserts that all knowledge about $x$ can be recovered from that of $c(x)$. this then suggests that arbitrary non-commutative $c^*$-algebras can be viewed as describing functions on "noncommutative spaces," of the type which arise in various formulations of quantum mechanics. the goal of this project is to understand the relation between compact hausdorff spaces and commutative $c^*$-algebras, and see how the topological information encoded within $x$ is reflected in the algebraic information encoded within $c(x)$. this duality between topological and algebraic data is at the core of many aspects of modern mathematics, and beautifully blends together concepts from analysis, algebra, and topology. the ultimate aim in this area is to see how much geometry and topology one can carry out using only algebraic means. prerequisites: math 330-2 or math 331-2, math 344-1., simple lie algebras, a lie algebra is a vector space equipped with a certain type of algebraic operation known as a lie bracket, which gives a way to measure how close two elements are to commuting with one another. for instance, the most basic example is that of the space of all $n \times n$ matrices, where the "bracket" operation takes two $n \times n$ matrices $a$ and $b$ and outputs the difference $ab-ba$; in this case the lie bracket of $a$ and $b$ is zero if and only if $a$ and $b$ commute in the usual sense. lie algebras arise in various contexts, and in particular are used to describe "infinitesimal symmetries" of physical systems. among all lie algebras are those referred to as being simple, which in a sense are the lie algebras from which all other lie algebras can be built. it turns out that one can encode the structure of a simple lie algebra in terms of purely combinatorial data, and that in particular one can classify simple lie algebras in terms of certain pictures known as dynkin diagrams. the goal of this project is to understand the classification of simple lie algebras in terms of dynkin diagrams. there are four main families of such lie algebras which describe matrices with special properties, as well as a few so-called exceptional lie algebras whose existence seems to come out of nowhere. such structures are now commonplace in modern physics, and their study continues to shed new light on various phenomena. prerequisites: math 330-2 or math 331-2, math 334-0 or math 291-2., the spectral theory of polygons, jared wunsch, we can study, for any domain the plane, the eigenfunctions of the laplace-operator (with boundary conditions) on this domain: these are the natural frequencies of vibration of this drum head. students might want to read mark kac's famous paper "can you hear the shape of a drum" as part of this project, and there is lots of fun mathematics associated to this classical question and its negative answer by gordon-webb-wolpert. an ambitious direction that this could possibly head in would be the theory of diffraction of waves on surfaces. in the plane, this is a classical theory, going back to work of sommerfeld in the 1890's, but there's still a remarkable amount that we don't know. the mathematical story is more or less as follows: a wave (i.e. a solution to the wave equation, which could be a sound or electromagnetic wave, or, with a slight change of point of view, the wavefunction of a quantum particle) is known to reflect nicely off a straight interface. at a corner, however, something quite interesting happens, which is that the tip of the corner acts as a new point source of waves. this is the phenomenon of diffraction, and is responsible for many fascinating effects in mathematical physics. the student could learn the classical theory in the 2d context, starting with flat surfaces and possibly (if there is sufficient geometric background) curved ones, and then work on a novel project in one of a number of directions, which would touch current research in the field., prerequisites: math 320-1 or math 321-1, math 325-0 or math 382-0. more ambitious parts of this project might require math 410-1,2,3..

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

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

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

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

Learning from Research, Advancing the Field

The Narcissism of Mathematics Education

Educational Research on Learning and Teaching Mathematics

Avoid common mistakes on your manuscript.

1 An international survey in two rounds

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

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

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

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

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

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

2 Methodological approach

2.1 themes for the coming decade (2019).

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

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

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

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

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

Artistic impression of the future themes

2.2 Has the pandemic changed your view? (2020)

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

3 The themes

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

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

3.1 Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

3.1.1 Teaching strategies

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

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

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

3.1.2 Curriculum

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

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

3.2 Goals of mathematics education

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

3.2.1 Societal goals

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

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

3.2.2 Educational goals

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

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

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

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

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

3.3 Relation of mathematics education to other practices

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

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

3.4 Teacher professional development

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

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

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

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

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

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

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

3.5 Technology

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

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

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

3.6 Equity, diversity, and inclusion

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

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

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

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

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

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

3.8 Assessment

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

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

4 Mathematics education research itself

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

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

4.2 Methodology

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

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

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

4.3 Reflection on our discipline

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

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

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

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

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

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

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

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

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

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

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

4.4 Interdisciplinarity and transdisciplinarity

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

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

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

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

5 The themes in their coherence: an artistic impression

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

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

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

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

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

6 Research challenges

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

6.1 Aligning new goals, curricula, and teaching approaches

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

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

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

6.2 Researching mathematics education across contexts

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

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

6.3 Focusing teacher professional development

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

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

6.4 Using low-tech resources

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

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

6.5 Staying in touch online

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

6.6 Studying and improving equity without perpetuating inequality

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

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

6.7 Assessing online

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

6.8 Doing and publishing interdisciplinary research

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

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

7 Concluding remarks

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

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

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Acknowledgments

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

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Appendix 1: Explanation of Fig. 1

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

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260 Interesting Math Topics for Essays & Research Papers

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

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

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

🔝 Top 10 Interesting Math Topics

✅ branches of mathematics, ✨ fun math topics.

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

🔍 References

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

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

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

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

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

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

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

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

🏫 Math Essay Topics for High School

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

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

🎓 College Math Topics for a Paper

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

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

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

🤔 Advanced Topics in Math to Write a Paper on

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

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

📚 Math Research Topics for a Paper

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

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

Mathematical writing should be well-structured, precise, and easy readable

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

✏️ Math Education Research Topics

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

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

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

🧮 Algebra Topics for a Paper

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

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

📏 Geometry Topics for a Research Paper

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

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

The role of study of non-Euclidean geometry

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

➗ Calculus Topics to Write a Paper on

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

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

Calculus was developed by Isaac Newton and Gottfried Leibnitz.

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

💵 Business Math Topics to Write About

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

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

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

What Is Calculus?: Southern State Community College
What Is Mathematics?: Tennessee Tech University
What Is Geometry?: University of Waterloo
What Is Algebra?: BBC
Ten Simple Rules for Mathematical Writing: Ohio State University
Practical Algebra Lessons: Purplemath
Topics in Geometry: Massachusetts Institute of Technology
The Geometry Junkyard: All Topics: Donald Bren School of Information and Computer Sciences
Calculus I: Lamar University
Business Math for Financial Management: The Balance Small Business
What Is Mathematics: Life Science
What Is Mathematics Education?: University of California, Berkeley
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Similar to the instructions in a recipe book, process essays convey information in a step-by-step format. In this type of paper, you follow a structured chronological process. You can also call it a how-to essay. A closely related type is a process analysis essay. Here you have to carefully consider...

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Examining examples and non-examples can help students understand definitions. While a square may be defined as a quadrilateral with four equal sides and one right angle, seeing concrete examples of squares of various sizes, as well as considering rectangular non-examples, can help children clarify the notion of square. When we teach linear algebra and introduce the concept of subspace, we often provide examples and non-examples for students. We may point out that the polynomials of degree less than or equal to two form a subspace of the space of all polynomials, whereas the polynomials of degree two do not. Is the provision of such examples always desirable? Would it perhaps be better to ask undergraduate students to provide their own examples and non-examples? Would they be able to? Given a false conjecture, would students be able to come up with counterexamples? Several studies shed light on these questions.

Successful Math Majors Generate Their Own Examples

In upper-division courses like abstract algebra and real analysis, students often encounter a host of formal definitions, many new to them. After presenting a few examples and non-examples along with a few proofs of theorems, we hope they will use these definitions to tackle problems, examine conjectures, and construct their own proofs. Is this the best way to proceed? How do such students deal with new definitions?

To answer this question, Randall P. Dahlberg and David L. Housman of Allegheny College conducted an in-depth study of eleven undergraduate students - ten seniors and one junior. All but one, who was in computer science, were math majors. The students had successfully completed introductory real analysis and algebra, as well as courses in linear algebra and foundations and a seminar covering set theory and the foundations of analysis. In individually conducted audio-taped interviews, the authors presented the students with a written definition of a "fine function," which they had made up to see how the students would deal with a formally defined concept. A function was called fine if it had a root (zero) at each integer. When interviewed, students were first asked to study this definition for five to ten minutes, saying or writing as much as possible of what they were thinking, after which they were asked to generate examples and non-examples of "fine functions." Subsequently, they were given functions, such as

and asked to determine whether these were examples and, if so, why. Next, they were asked to determine the truth of four conjectures, such as "No polynomial is a fine function." Finally they were asked about their perceptions of the interview.

Four basic learning strategies were used by the students on being presented with this new definition - example generation, reformulation, decomposition and synthesis, and memorization. Examples generated included the constant zero function and a sinusoidal graph with integer x -intercepts. Reformulations included

Decomposition and synthesis included underlining parts of the definition and asking about the meaning of "root." Two students simply read the definition - they could not provide examples without interviewer help and were the ones who most often misinterpreted the definition. They found the interview quite different from their usual mathematics classes, where examples and explanations were provided.

Of these four strategies, example generation (together with reflection) elicited the most powerful "learning events," i.e., instances where the authors thought students made real progress in understanding the newly introduced concept. Students who initially employed example generation as their learning strategy came up with a variety of discontinuous, periodic continuous, and non-periodic continuous examples and were able to use these in their explanations. Those who employed memorization or decomposition and synthesis as their learning strategies often misinterpreted the definition, e.g., interpreting the phrase "root at each integer" to mean a fine function must vanish at each integer in its domain, but that need not include all integers. Students who employed reformulation as their learning strategy developed algorithms to decide whether functions given them were fine, but had difficulty providing counterexamples to false conjectures. [Cf. "Facilitating Learning Events Through Example Generation," Educ. Studies in Math. 33, 283-299, 1997.]

Finally, Dahlberg and Housman note the relative ineffectualness of their attempted interventions. One student agreed, after a question and answer period with the interviewer, that the zero function was indeed a fine function, but immediately switched her attention to other ideas, not returning until much later when, through self-discovery, she actually realized the zero function was a fine function. Dahlberg and Housman suggest it might be beneficial to introduce students to new concepts by having them generate their own examples or having them decide whether teacher-provided candidates are examples or non-examples, before providing students examples and explanations. However, some of their students were reluctant to engage in either example generation or usage -- a not uncommon phenomenon in such circumstances.

Being Asked For Examples Can Be Disconcerting

Coming up with examples requires different cognitive skills from carrying out algorithms - one needs to look at mathematical objects in terms of their properties. To be asked for an example, whether of a "fine function" or something else, can be disconcerting. Students have no prelearned algorithms to show the "correct way." This is what Orit Hazzan and Rina Zazkis, of the Technion - Israel Institute of Technology, found when they asked three groups of preservice elementary teachers to provide examples of (1) a 6-digit number divisible first by 9, then by 17, (2) a function whose value at x = 3 is -2, and (3) a sample space and an event that has probability 2/7 in that space. In addition, they asked the students to explain how they generated their examples and to provide five additional examples.

The students used a variety of approaches to generate examples, beginning with trial and error, e.g., some simply picked a number at random and checked whether it was divisible by 9. Others picked a number N , and upon dividing by 17 and getting a remainder of 2, would use N -2 for their next trial. Students often found constructing examples and making the necessary choices difficult, e.g., they inquired of the interviewers whether the elements of the sample space were to be numbers, letters, or other objects. Some students designed their own algorithms for generating functions, e.g., one focused on y = ax + b , plugged in (3, -2) to get -2 = a *3 + b , chose a = 2 and solved for b = -8, finally declaring her function to be y = 2 x - 8.

Interestingly, very few students produced "trivial examples," such as 170,000 for a 6-digit number divisible by 17 or y = -2 as their function. Hazzan and Zazkis conjecture that these examples might not be seen as prototypical - a function is expected to involve x and a 6-digit number is seen as having a wider variety of digits. There was also a strong tendency to (directly) check the correctness of examples, e.g., some students who had created a number divisible by 17 by choosing a multiplier and performing the multiplication, verified the correctness of their example by division. Quite a number of students had difficulty dealing with "degrees of freedom," e.g., in order to find a number divisible by 9, one student who knew the sum of the digits needed to be divisible by 9, first chose 18, noted that 8 and 2 make 10, then broke 8 into the sum of 4, 3, and 1, and declared that 82431 should be divisible by 9. When asked for another strategy, she suggested something very similar -- making the initial sum 27, instead of 18.

Constructing examples proved to be more difficult for these students than checking the divisibility of a number, calculating the value of a function, or finding the probability of an event. They were often uncertain how to proceed and were especially troubled by having to make choices in mathematics. The authors suggest that teachers at all levels assign more "give an example" problems. [Cf. "Constructing Knowledge by Constructing Examples for Mathematical Concepts," Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education , Vol. 4, 299-306, 1997]

Furthermore, when students are allowed to discuss mathematical ideas and propose conjectures in class, teachers need to be able to evaluate student-generated examples, as well as to be able to propose counterexamples for their students' consideration. Students quite often fail to see a single counterexample as disproving a conjecture. This can happen when a counterexample is perceived as "the only" one that exists, rather than being seen as generic, e.g., sometimes the square root of 2 is considered the only irrational or | x | is perceived as the only continuous, nondifferentiable function.

Generating Counterexamples That Are Explanatory

Perhaps not surprisingly, experienced secondary mathematics teachers are better at generating explanatory counterexamples than preservice teachers. Irit Peled, University of Haifa, and Orit Zaslavsky, the Technion, asked some of each to generate at least one counterexample for each of the two following unfamiliar, false geometry statements supposedly given by a secondary student. (1) Two rectangles, having congruent diagonals, are congruent. (2) Two parallelograms, having one congruent side and one congruent diagonal, are congruent. They were also asked to explain how they came up with their counterexamples. None generated more than one counterexample for each task.

Two groups participated in the study -- 38 inservice teachers, most of whom had more than five years of teaching experience and a B.Sc. in mathematics and 45 third year student-teachers who had completed several advanced undergraduate mathematics courses. For the first conjecture (Task 1), 97% of the inservice teachers gave adequate counterexamples, i.e., ones that refuted the claim, but only 53% of the student-teachers did so. For the second conjecture (Task 2), 76% of the teachers and 42% of the student-teachers gave adequate counterexamples.

The counterexamples were analyzed for their explanatory power as specific, semi-general, and general. A specific counterexample is one which contradicts the claim, but gives no indication as to how one might construct similar or related counterexamples. For example, for Task 1 one subject carefully drew two rectangles of different dimensions, but with congruent diagonals. A counterexample was called semi-general if it provided some idea how one might generate similar or related counterexamples, but did not tell "the whole story" or did not cover "the whole space" of counterexamples. For instance, on Task 1, one subject drew two rectangles with congruent diagonals, but the angle between the two diagonals of second rectangle was indicated as twice that of the first rectangle. (Here it should be noted that, while some conjectures might not lend themselves to the generation of numerous counterexamples, i.e., they might be correct except for a small number of special "pathological" cases, these two conjectures were chosen to be far from "almost correct.") A general counterexample provides insight as to why a conjecture is false and suggests a way to generate an entire counterexample space. In response to Task 1, one subject specified that the angle between the diagonals could be arbitrary, rather than merely double that of the first rectangle.

Both teachers and student-teachers produced counterexamples of all the above types, but the former produced more semi-general and general counterexamples (92% vs. 38% on Task 1, and 61% vs. 33% on Task 2). Both of these types were labeled explanatory by the authors. The difficulty in suggesting only a specific counterexample lies in its potential for misleading students, whereas the pedagogical value of explanatory counterexamples lies in their ability to provide insight into why a conjecture fails. The authors suggest that both prospective and in-service mathematics teachers could benefit from an analysis and discussion of the pedagogical aspects of counterexamples. [Cf. "Counter-Examples That (Only) Prove and Counter-Examples That (Also) Explain," Focus on Learning Problems in Mathematics 19 (3), 49-61, 1997.]

"If I Don't Know What It Says, How Can I Find an Example of It?"

This hypothetical quote, illustrates the chicken-and-egg quandary some students might typically face when encountering a formal definition, whether of "fine function" or quotient group. A definition asserts the existence of something having certain properties. However, the student has often never seen or considered such a thing. To give an example or non-example, he/she would need at least some understanding of the concept. But how can he/she obtain such understanding? A good, and possibly the best, way seems to be through an examination of examples. Thus, the student is faced with an epistemological dilemma: Mathematical definitions, by themselves, supply few (psychological) meanings. Meanings derive from properties. Properties, in turn, depend on definitions. [This is a paraphrase from Richard Noss' plenary address to the September 1996 Research in Collegiate Mathematics Education Conference, as reported in Focus 17(1), 1&3, February 1997.] For mathematicians, this does not seem to be a dilemma. We suspect they view definitions differently than students - this allows them to search for examples in order to gain understandings of formal definitions.

Not only does such circularity play a role in students' failure to construct examples, so does their limited knowledge of concepts involved in a formal definition. When Zaslavsky and Peled asked 67 preservice and 36 inservice secondary teachers to provide examples of binary operations which were commutative and nonassociative, their subjects had great difficulty. Only 33% of the experienced teachers and 4% of the third-year undergraduate students came up with complete, correct, and well-justified examples. Just 56% of the experienced teachers and 31% of the student teachers were able to provide any kind of example (correct or incorrect). Upon investigating why this might be so, the authors found their subjects' underlying mathematical knowledge was deficient. For example, one subject defined a * b = | a + b | and claimed this was nonassociative because | a + b | + | c | does not equal | a | + | b + c |. Another proposed the operation of subtraction claiming it was commutative because -2 - 3 = -3 - 2, rather than 3 - (-2). Yet another proposed the unary operation

and tried to check commutativity using

The authors suggest their subjects tended to conflate commutativity and associativity due to the way the "issue of order" is treated in schools. For example, when a child is asked to calculate 6 + 7 + 4, he/she is usually encourage to do it more efficiently as (6 + 4 ) + 7 and told "order doesn't matter." [Cf. "Inhibiting Factors in Generating Examples by Mathematics Teachers and Student Teachers: The Case of Binary Operations," JRME 27(1), 67-78, 1996.]

Dahlberg and Housman also noted that their undergraduate subjects had trouble with the underlying concepts, e.g., function and root, making it hard to generate examples and non-examples of "fine functions." One student identified "root" with "continuity," three others initially thought the graph of the zero function was a point, and one did not believe the zero function was periodic. In addition, most students' initially thought in terms of functions which were nonconstant polynomials or continuous.

Since success in mathematics, especially at the advanced undergraduate and graduate levels appears to be associated with the ability to generate examples and counterexamples, what is the best way to develop this ability? One suggestion, given above, is to ask students at all levels to "give me an example of . . . ". Granted the inherent epistemological difficulties of finding examples for oneself, are we, in a well-intentioned attempt to help students understand newly defined concepts, ultimately hobbling them, by providing them with predigested examples of our own? Are we inadvertently denying students the opportunity to learn to generate examples for themselves? Difficulties with the strikingly simple idea of "fine function" suggest some students may be excessively dependent upon explicit instruction. Another in-between suggestion, given above, is to provide students with a list of potential examples (or counterexamples) and ask them to decide whether they are indeed examples (or counterexamples) and why. Are there other ways we might help students become example generators? Finally, a tendency to generate examples is not the same as an ability to do so -- it would be interesting to know how each of these relates to understanding and doing mathematics.

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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.
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bio-mathematics: introduction to the mathematical model of the hepatitis c virus, lucille j. durfee. pdf. analysis and synthesis of the literature regarding active and direct instruction and their promotion of flexible thinking in mathematics, genelle elizabeth gonzalez. pdf. life expectancy, ali r. hassanzadah. pdf
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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.
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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.
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202 Math Research Topics: List To Vary Your Ideas. Mathematics is an exceptional field of study dealing primarily with numbers. It also deals with structures, formulas, shapes, spaces, and quantities of where they are contained. Maths encompasses different types of computations that are applied in the real world. Math requires a lot of analysis.
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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 ...
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Pure Mathematics Research Pure Mathematics Fields The E 8 Lie group. Algebra & Algebraic Geometry; Algebraic Topology; Analysis & PDEs; Geometry & Topology; Mathematical Logic & Foundations; ... Department of Mathematics Headquarters Office Simons Building (Building 2), Room 106 77 Massachusetts Avenue Cambridge, MA 02139-4307 Campus Map (617 ...
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Research in Mathematics is a broad open access journal publishing all aspects of mathematics including pure, applied, and interdisciplinary mathematics, and mathematical education and other fields. The journal primarily publishes research articles, but also welcomes review and survey articles, and case studies. Topics include, but are not limited to:
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This Research Topic is dedicated to the use of digital resources in the context of modeling processes in mathematics education. ... with a sample size of 121 Mathematics teachers selected from a ...
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Moreover, this project will bring in topics from many different areas of mathematics - analysis, group theory, and linear algebra - to touch on areas of modern research. ... to touch on areas of modern research. Prerequisites: MATH 320-1 or MATH 321-1, MATH 330-1 or MATH 331-1, MATH 334-0 or MATH 291-2. ... the most basic example is that of ...
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The mathematics of voting and the mathematics of fair division are two fascinating topics in the field of mathematics and politics. Research questions look at types of voting systems, and the properties that we would want a voting system to satisfy, as well as the idea of fairness when splitting up a single object, like cake, or a collection of ...
Computational Mathematics
Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations ...
Research in Mathematics Education: Vol 26, No 1 (Current issue)
Research in Mathematics Education, Volume 26, Issue 1 (2024) ... Topic specificity of students' conceptual and procedural fraction knowledge and its impact on errors. ... Comparing example generation with classification in the learning of new mathematics concepts. George Kinnear. Pages: 109-132.
[100+] Mathematics Research Topics With Free [Thesis Pdf] 2023
2. Wavelet-Galerkin technique for solving certain numerical differential equations and inverse Ill-posed problems. Download. 3. Some study on edge geodetic number of a graph. Download. 4. Haar wavelet technique for solving certain differential integral and integro differential equations. Download.
List of issues Research in Mathematics Education
Browse the list of issues and latest articles from Research in Mathematics Education. All issues. Special issues. Latest articles. Volume 25 2023. Volume 24 2022. Volume 23 2021. Volume 22 2020. Volume 21 2019.
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.
Research Sampler 5: Examples in Learning Mathematics
Some students designed their own algorithms for generating functions, e.g., one focused on y = ax + b, plugged in (3, -2) to get -2 = a *3 + b, chose a = 2 and solved for b = -8, finally declaring her function to be y = 2 x - 8. Interestingly, very few students produced "trivial examples," such as 170,000 for a 6-digit number divisible by 17 or ...
PDF INCREASING STUDENT LEARNING IN MATHEMATICS WITH THE USE OF ...
school students' achievements in mathematics. The teacher researchers had noticed a trend of low scores on teacher-made chapter tests and non-completion of daily homework. Standardized tests showed that most students scored below average on the mathematics portion, and the number of students having to repeat mathematics courses had increased.
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. Applied Mathematics Fields The mathematics of surface tension. Combinatorics; Computational Biology
Make Math Instruction Better: 3 Tips on How From Researchers
Education Week reporter and data journalist Sarah D. Sparks attended the American Educational Research Association's annual conference in Philadelphia earlier this month. Here, she shares three ...

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Computation and convergence of fixed-point with an RLC-electric circuit model in an extended b-suprametric space

Spectral shifted Chebyshev collocation technique with residual power series algorithm for time fractional problems

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