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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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StatAnalytica

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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sample research title in mathematics

166 Extraordinary Math Research Topics For Your Papers

math research topics

Math research topics cover various genres from which students can choose. Many people think that a research project on a math topic is dull. However, mathematics can be a wonderful and vivid field. Since it’s a universal language, mathematics can describe anything and everything, from galaxies that orbit each other to music. However, the broad nature of this study field also makes selecting a research paper difficult. That’s because learners want to pick interesting topics that will impress educators to award them top scores. This article lists the best math research paper topics. It’s useful because it inspires students to select or customize topics for their academic essays without much struggle.

What Are The Different Types Of Math?

As hinted, math covers several genres. Here are the primary types of mathematics:

Geometry: It’s a math branch that deals with the shapes, size, and relative position of figures. Many people consider geometry a practical math branch because it examines figures, shapes, sizes, and features of various entities, including parts like solids, lines, surfaces, lines, and angles. Algebra: It assists in solving equations and manipulating symbols. This branch helps students represent unknown quantities with alphabets and use them alongside numbers. Calculus: This area is vital in determining rates of change, such as velocity and acceleration. Arithmetic: Arithmetic is the most common and oldest math branch, encompassing basis number operations. These operations include subtraction, addition, divisions, and multiplications, and some schools shorten it as BODMAS. Statistics and Probability: They help analyze numerical data to make predictions. Probability is about chances, while statistics entails handling different data using various techniques. Trigonometry: It assists in calculating angles and distances between points. It mainly deals with triangles’ relationships, sides, and curves.

Now that you understand the types of mathematics, it’s easier to select a suitable research topic. The following are some of the best topic ideas in math. 

 Undergraduate Math Research Topics

Maybe you’re pursuing your undergraduate studies. However, you have challenges comprehending math topics, yet the professor expects you to write a superior paper. In that case, here’s a list of engaging research topics in math to consider for your essays.

  • An in-depth comprehension of the meaning of discrete random variables in math and their identification
  • Math evolution- Comprehending the Gauss-Markov
  • Primary math theorems- Investigating how they work
  • Continuous stochastic process- Exploring its role in the math process
  • Analyzing the Dempster-Shafer theory
  • The application of the transferable belief model
  • Exploring the use of math in artificial intelligence
  • The application of mathematics in daily life
  • Algebra and its history
  • Math and culture- What’s the relationship?
  • How drawing and painting could help with mathematics
  • Ways to boost math interest among learners
  • The social and political significance of learning mathematics
  • Circles and their relevance in mathematics
  • Challenges to math learning in public schools
  • Prove the use of F-Algebras
  • Understanding the meaning of abstract algebra
  • Discuss geometry and algebra
  • How acute square triangulation works
  • Discuss the essence of right triangles
  • Why non-Euclidean geometry should be compulsory for math students
  • Investigating number problems
  • Discuss the meaning of Dirac manifolds
  • How geometry influences chemistry and physics
  • Riemannian manifolds’ application in the Euclidean space

These are exciting math topics for undergraduate students. Nevertheless, prepare adequate time and resources to investigate any of these titles to draft a winning essay. You might have to provide theoretical and practical assessments when writing your essay.

Math Research Topics for High School Learners

Maybe your high school teacher asked you to write a research paper. Choosing a familiar topic is an excellent way to get a high grade. Here are some of the best math research paper topics for high school.

  • How to draw a chart representing the financial analysis of a prominent company over the last five years
  • How to solve a matrix- The vital principles and formulas to embrace
  • Exploring various techniques for solving finance and mathematical gaps
  • Discount factor- Why it’s crucial for learners and ways to achieve it
  • Calculating the interest rate and its essence in the banking industry
  • Why imaginary numbers are important
  • Investigating the application of math in the workplace
  • Explain why learners hate mathematics teachers
  • What makes math a complex subject?
  • Is making math compulsory in high school a good thing?
  • How to solve a dice question from a probability perspective
  • Understanding the Binomial theorem and its essence
  • Investigating Egyptian mathematics
  • Hyperbola- Understanding it and its use in math
  • When should students use calculators in class?
  • How to solve linear equations
  • Is the Pythagoras theorem important in math?
  • The interdependence between math and art
  • Philosophy’s role in math
  • Numerical data overview

High school learners can pick any of these titles and develop them into an essay. Nevertheless, they should prepare to spend some time investigating their topics to write pieces that will impress their educators. Titles that address math history and its influence on education can also suit high school students. However, learners should select titles that fulfil the academic requirements set by the educators.

Applied Math Research Topics

As a branch, applied math deals with mathematical methods and their real-life applications. These methods are manifest in engineering, finance, medicine, biology, physics, and others. Here are some of the exciting topics in this field.

  • Dimensions for examining fingerprints
  • Computer tomography and its significance
  • Step-stress modelling- What is its importance?
  • Explain the essence of data mining- How does it benefit the banking sector?
  • A detailed examination of nonlinear models
  • How genes discovery helps determine unhealthy and healthy patients
  • Algorithms and their role in probabilistic modelling
  • Mathematicians and their importance in robots’ development
  • Mathematicians’ role in crime prevention and data analysis
  • The essence of Law of Motion by Isaac in real life
  • The importance of math in energy conservation
  • Math and its role in quantum theory
  • Analyzing the Lorentz symmetry features
  • Evaluating the processing of the statistical signal in detail
  • Explain the achievement of Galilean Transformation

These are exciting ideas to explore when writing a research paper in applied math. Nevertheless, take your time to carefully and extensively research your preferred title to write a high-quality essay. Students should also note that some topics in this category require specialized knowledge to write superior papers.

It’s a challenge to write a paper for a high grade. Sometimes every student need a professional help with college paper writing. Therefore, don’t be afraid to hire a writer to complete your assignment. Just write a message “Please, write custom research paper for me” and get time to relax. Contact us today and get a 100% original paper. 

Interesting Math Research Topics

Maybe you’re among the learners that prefer working with exciting ideas. In that case, this category has topics that will interest you.

  • The uses of numerical analysis in machine learning
  • Foundations and philosophical problems
  • Convex versus Concave in geometry
  • Homological algebra- What is its purpose?
  • Is math useful in cryptography
  • Probability theory and random variable
  • Functional analysis- What are its four conditions?
  • Vector calculus versus multivariable
  • Mathematics and logicist definitions
  • Ways to apply the number theory in daily life
  • Studying complex math equations
  • How to calculate mode, median, and mean
  • Understanding the meaning of the Scholz conjecture
  • The definition of the past correspondence problem
  • Computational maths- What are its classes?
  • Multiplication table and its importance
  • What the Boolean satisfiability problem means for a learner
  • Understanding the linear speedup theory in mathematics
  • The Turing machine description
  • Understanding the Markov algorithm
  • Investigating the similarities and differences between Buchi automation and Pushdown automation
  • What is the meaning of Tree automation?
  • Describing the enclosing sphere method and its use in combinations
  • Egyptian pyramids and calculus
  • Analyzing De Finetti theorem in statistics and probability
  • Examining the congruence meaning in math
  • Application and purpose of calculus in the banking industry
  • Jean d’Alembert’s most famous works
  • Boolean algebra- What are its essential elements
  • Isaac Newton- His contribution, life, and time in math
  • Understanding the meaning of Sphericon
  • What is the purpose of Martingales?
  • Gauss times, energy, and contributions to math
  • Jakob Bernoulli- Exploring his famous works
  • A brief history of math

Some learners think writing a math essay is complex and tedious. However, you can find a topic you will enjoy working with throughout the project. These are exciting ideas to explore in research papers. However, prepare to spend sufficient time investigating your chosen title to write a winning paper, although these are generally relaxing titles for math papers and essays.

Math Research Topics for Middle School

Some middle school students worry about the math topics for their research. However, they can choose unique titles that will impress their teachers. Here are some of these ideas.

  • The impacts of standard exam curriculum on math education
  • Why is learning math so tricky?
  • What is the meaning of the commutative ring in algebra?
  • The Artin-Wedderburn theorem and its meaning
  • How monopolists and epimorphisms differ
  • Understanding the Jacobson density theorem
  • How linear approximations work
  • Root and ratio test definition
  • Statistics role in business
  • Economic lot scheduling- What does it mean?
  • Causes of the stock market crash
  • How many traders contribute to the New York Stock Exchange
  • The history of revenue management
  • Financial signs of an excellent investment
  • Depreciation and its odds
  • How a poor currency can benefit a country
  • How math helps with debt amortization
  • Ways to calculate a person’s net worth
  • Distinctions in algebra, trigonometry, and calculus
  • Discussing the beginning of calculus
  • The essence of stochastic in math
  • The meaning of limits in math
  • Ways to identify a critical point in a graph
  • Nonstandard analysis- What does it mean in the probability theory?
  • Continuous function description and meaning
  • Calculus- What are its primary principles?
  • Pythagoras theorem- What are its central tenets?
  • Calculus applications in finance
  • Theorem value in math
  • The application of linear approximations

This list has some of the best titles for middle school learners. But they also require some research to write superior essays. However, finding information on such topics is relatively easy, making them suitable for middle school students.

Math Research Topics for College Students

Maybe you’re pursuing college studies and need a title for a math research paper. In that case, here are exciting titles to consider for your essay.

  • What is the purpose of n-dimensional spaces?
  • Card counting- How does it work?
  • How continuous probability and discrete distribution differ
  • Understanding encryption- How Does it work?
  • Extremal problems- Investigating them in discrete geometry
  • The Mobius strip- Examining the topology
  • Why can a math problem be unsolvable?
  • Comparing different statistical methods
  • Explain the vital number theory concepts
  • Analyzing the polynomial functions’ degrees
  • Ways to divide complex numbers
  • Describe the prize problems with the millennium
  • The reasons for the unsolved Riemann hypothesis
  • Methods of solving Sudoku with math
  • Explain the fractals formation
  • Describe the evolution of math
  • Explore different types of Tower of Hanoi solutions
  • Discuss the uses of Napier’s bones
  • With examples, explain the chaos theory
  • Why are mathematical equations important all the time?
  • Fisher’s fundamental theorem and natural selection- Why are they important?

College professors expect students to draft papers with relevant and valuable information. These are relevant titles for college students. However, they require extensive research to write winning papers.

Cool Math Topics to Research

Maybe you don’t need a complex topic for your research paper. In that case, consider any of these ideas for your essay. If you have a problem writing even with these topics and you’re thinking: “solve my math for me,” you can always reach out to our service.

  • How contemporary architectural designs use geometry
  • What makes some math equations complex?
  • Ways to solve the Rubik’s cube
  • Discuss the meaning of prescriptive statistical and predictive analysis
  • Understanding the purpose of the chaos theory
  • What limits calculus?- Provide relevant examples
  • A comparison of universal and abstract algebra- How do they differ?
  • The relationship between probability and card tricks
  • Pascal’s Triangle- What does it mean?
  • Mobius strip- What are its features in geometry?
  • Multiple probability ideas- A brief overview
  • Discuss the meaning of the Golden Ration in Renaissance period paintings
  • How checkers and chess matter in understanding mathematics
  • Ways to measure infinity
  • Evaluating the Georg Contor theory
  • Are hexagons the most balanced shapes in the world?
  • The Koch snowflake- Explain the iterations
  • The history of various number types and their use
  • Game theory use in social science
  • Five math types with significant benefits in computer science

These are some of the most excellent math education research topics. However, they also require extensive research to write high-quality papers.

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Perhaps, you have a topic for your paper but not the time to write a winning piece. Maybe you’re not confident in your research, analytical, and writing skills. Thus, you’re unsure that you can write an essay that will compel your educator to award you the highest grade in your class. Well, you’re not the only one. Many students seek cheap research papers due to varied reasons. Whether it’s limited time and resources or a lack of the necessary skills and experience in academic paper writing, our crew can help you. We offer affordable college paper writing services and help in various math branches. Our experts can assist you if you need help with math research topics for high school students, college, or undergraduates. We are a professional team with a reputation for providing the best-rated academic writing assistance. Whether in university, college, or high school, our crew will offer the service you need to excel academically. Contact us now for cheap and reliable help with your academic essays.

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Home > College of Natural Sciences > Mathematics > Mathematics Theses, Projects, and Dissertations

Mathematics Theses, Projects, and Dissertations

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

ALGEBRA 1 STUDENTS’ ABILITY TO RELATE THE DEFINITION OF A FUNCTION TO ITS REPRESENTATIONS , Sarah A. Thomson

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25 Maths Research Paper Topics

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A List of 25 Research Topics in Mathematics

  • The history of mathematics
  • Greek math theories and inventions
  • The math of ancient great buildings
  • The math of Universe. Universal constants
  • Astronomy and math. How can we know about what we have never seen?
  • The greatest mathematicians and their role in the history
  • The greatest math invention in history
  • Euclidean and non-Euclidean geometries. Going beyond the space we used to
  • Unsolved problems in math
  • Math and Quantum theory
  • Math and Philosophy
  • Art and math: are they connected?
  • Number theory
  • Math and the AI concepts
  • Math in nature
  • Math in our daily life
  • Math and Probabilities
  • Fundamental mathematical theorems
  • Sports and math
  • Is Math unique for humans only?
  • STEM Education: why math is so important?
  • Math and formal logic
  • The math of time
  • Math and gender

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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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Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

St Andrews Research Repository

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Main areas of research activity are algebra, including group theory, semigroup theory, lattice theory, and computational group theory, and analysis, including fractal geometry, multifractal analysis, complex dynamical systems, Kleinian groups, and diophantine approximations.

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Rearrangement groups of connected spaces , modern computational methods for finitely presented monoids , finiteness conditions on semigroups relating to their actions and one-sided congruences , on constructing topology from algebra , interpolating between hausdorff and box dimension .

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sample research title 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.

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500+ Quantitative Research Titles and Topics

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Quantitative Research Topics

Quantitative research involves collecting and analyzing numerical data to identify patterns, trends, and relationships among variables. This method is widely used in social sciences, psychology , economics , and other fields where researchers aim to understand human behavior and phenomena through statistical analysis. If you are looking for a quantitative research topic, there are numerous areas to explore, from analyzing data on a specific population to studying the effects of a particular intervention or treatment. In this post, we will provide some ideas for quantitative research topics that may inspire you and help you narrow down your interests.

Quantitative Research Titles

Quantitative Research Titles are as follows:

Business and Economics

  • “Statistical Analysis of Supply Chain Disruptions on Retail Sales”
  • “Quantitative Examination of Consumer Loyalty Programs in the Fast Food Industry”
  • “Predicting Stock Market Trends Using Machine Learning Algorithms”
  • “Influence of Workplace Environment on Employee Productivity: A Quantitative Study”
  • “Impact of Economic Policies on Small Businesses: A Regression Analysis”
  • “Customer Satisfaction and Profit Margins: A Quantitative Correlation Study”
  • “Analyzing the Role of Marketing in Brand Recognition: A Statistical Overview”
  • “Quantitative Effects of Corporate Social Responsibility on Consumer Trust”
  • “Price Elasticity of Demand for Luxury Goods: A Case Study”
  • “The Relationship Between Fiscal Policy and Inflation Rates: A Time-Series Analysis”
  • “Factors Influencing E-commerce Conversion Rates: A Quantitative Exploration”
  • “Examining the Correlation Between Interest Rates and Consumer Spending”
  • “Standardized Testing and Academic Performance: A Quantitative Evaluation”
  • “Teaching Strategies and Student Learning Outcomes in Secondary Schools: A Quantitative Study”
  • “The Relationship Between Extracurricular Activities and Academic Success”
  • “Influence of Parental Involvement on Children’s Educational Achievements”
  • “Digital Literacy in Primary Schools: A Quantitative Assessment”
  • “Learning Outcomes in Blended vs. Traditional Classrooms: A Comparative Analysis”
  • “Correlation Between Teacher Experience and Student Success Rates”
  • “Analyzing the Impact of Classroom Technology on Reading Comprehension”
  • “Gender Differences in STEM Fields: A Quantitative Analysis of Enrollment Data”
  • “The Relationship Between Homework Load and Academic Burnout”
  • “Assessment of Special Education Programs in Public Schools”
  • “Role of Peer Tutoring in Improving Academic Performance: A Quantitative Study”

Medicine and Health Sciences

  • “The Impact of Sleep Duration on Cardiovascular Health: A Cross-sectional Study”
  • “Analyzing the Efficacy of Various Antidepressants: A Meta-Analysis”
  • “Patient Satisfaction in Telehealth Services: A Quantitative Assessment”
  • “Dietary Habits and Incidence of Heart Disease: A Quantitative Review”
  • “Correlations Between Stress Levels and Immune System Functioning”
  • “Smoking and Lung Function: A Quantitative Analysis”
  • “Influence of Physical Activity on Mental Health in Older Adults”
  • “Antibiotic Resistance Patterns in Community Hospitals: A Quantitative Study”
  • “The Efficacy of Vaccination Programs in Controlling Disease Spread: A Time-Series Analysis”
  • “Role of Social Determinants in Health Outcomes: A Quantitative Exploration”
  • “Impact of Hospital Design on Patient Recovery Rates”
  • “Quantitative Analysis of Dietary Choices and Obesity Rates in Children”

Social Sciences

  • “Examining Social Inequality through Wage Distribution: A Quantitative Study”
  • “Impact of Parental Divorce on Child Development: A Longitudinal Study”
  • “Social Media and its Effect on Political Polarization: A Quantitative Analysis”
  • “The Relationship Between Religion and Social Attitudes: A Statistical Overview”
  • “Influence of Socioeconomic Status on Educational Achievement”
  • “Quantifying the Effects of Community Programs on Crime Reduction”
  • “Public Opinion and Immigration Policies: A Quantitative Exploration”
  • “Analyzing the Gender Representation in Political Offices: A Quantitative Study”
  • “Impact of Mass Media on Public Opinion: A Regression Analysis”
  • “Influence of Urban Design on Social Interactions in Communities”
  • “The Role of Social Support in Mental Health Outcomes: A Quantitative Analysis”
  • “Examining the Relationship Between Substance Abuse and Employment Status”

Engineering and Technology

  • “Performance Evaluation of Different Machine Learning Algorithms in Autonomous Vehicles”
  • “Material Science: A Quantitative Analysis of Stress-Strain Properties in Various Alloys”
  • “Impacts of Data Center Cooling Solutions on Energy Consumption”
  • “Analyzing the Reliability of Renewable Energy Sources in Grid Management”
  • “Optimization of 5G Network Performance: A Quantitative Assessment”
  • “Quantifying the Effects of Aerodynamics on Fuel Efficiency in Commercial Airplanes”
  • “The Relationship Between Software Complexity and Bug Frequency”
  • “Machine Learning in Predictive Maintenance: A Quantitative Analysis”
  • “Wearable Technologies and their Impact on Healthcare Monitoring”
  • “Quantitative Assessment of Cybersecurity Measures in Financial Institutions”
  • “Analysis of Noise Pollution from Urban Transportation Systems”
  • “The Influence of Architectural Design on Energy Efficiency in Buildings”

Quantitative Research Topics

Quantitative Research Topics are as follows:

  • The effects of social media on self-esteem among teenagers.
  • A comparative study of academic achievement among students of single-sex and co-educational schools.
  • The impact of gender on leadership styles in the workplace.
  • The correlation between parental involvement and academic performance of students.
  • The effect of mindfulness meditation on stress levels in college students.
  • The relationship between employee motivation and job satisfaction.
  • The effectiveness of online learning compared to traditional classroom learning.
  • The correlation between sleep duration and academic performance among college students.
  • The impact of exercise on mental health among adults.
  • The relationship between social support and psychological well-being among cancer patients.
  • The effect of caffeine consumption on sleep quality.
  • A comparative study of the effectiveness of cognitive-behavioral therapy and pharmacotherapy in treating depression.
  • The relationship between physical attractiveness and job opportunities.
  • The correlation between smartphone addiction and academic performance among high school students.
  • The impact of music on memory recall among adults.
  • The effectiveness of parental control software in limiting children’s online activity.
  • The relationship between social media use and body image dissatisfaction among young adults.
  • The correlation between academic achievement and parental involvement among minority students.
  • The impact of early childhood education on academic performance in later years.
  • The effectiveness of employee training and development programs in improving organizational performance.
  • The relationship between socioeconomic status and access to healthcare services.
  • The correlation between social support and academic achievement among college students.
  • The impact of technology on communication skills among children.
  • The effectiveness of mindfulness-based stress reduction programs in reducing symptoms of anxiety and depression.
  • The relationship between employee turnover and organizational culture.
  • The correlation between job satisfaction and employee engagement.
  • The impact of video game violence on aggressive behavior among children.
  • The effectiveness of nutritional education in promoting healthy eating habits among adolescents.
  • The relationship between bullying and academic performance among middle school students.
  • The correlation between teacher expectations and student achievement.
  • The impact of gender stereotypes on career choices among high school students.
  • The effectiveness of anger management programs in reducing violent behavior.
  • The relationship between social support and recovery from substance abuse.
  • The correlation between parent-child communication and adolescent drug use.
  • The impact of technology on family relationships.
  • The effectiveness of smoking cessation programs in promoting long-term abstinence.
  • The relationship between personality traits and academic achievement.
  • The correlation between stress and job performance among healthcare professionals.
  • The impact of online privacy concerns on social media use.
  • The effectiveness of cognitive-behavioral therapy in treating anxiety disorders.
  • The relationship between teacher feedback and student motivation.
  • The correlation between physical activity and academic performance among elementary school students.
  • The impact of parental divorce on academic achievement among children.
  • The effectiveness of diversity training in improving workplace relationships.
  • The relationship between childhood trauma and adult mental health.
  • The correlation between parental involvement and substance abuse among adolescents.
  • The impact of social media use on romantic relationships among young adults.
  • The effectiveness of assertiveness training in improving communication skills.
  • The relationship between parental expectations and academic achievement among high school students.
  • The correlation between sleep quality and mood among adults.
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Home > Physical and Mathematical Sciences > Mathematics Education > Theses and Dissertations

Mathematics Education Theses and Dissertations

Theses/dissertations from 2024 2024.

New Mathematics Teachers' Goals, Orientations, and Resources that Influence Implementation of Principles Learned in Brigham Young University's Teacher Preparation Program , Caroline S. Gneiting

Theses/Dissertations from 2023 2023

Impact of Applying Visual Design Principles to Boardwork in a Mathematics Classroom , Jennifer Rose Canizales

Practicing Mathematics Teachers' Perspectives of Public Records in Their Classrooms , Sini Nicole White Graff

Parents' Perceptions of the Importance of Teaching Mathematics: A Q-Study , Ashlynn M. Holley

Engagement in Secondary Mathematics Group Work: A Student Perspective , Rachel H. Jorgenson

Theses/Dissertations from 2022 2022

Understanding College Students' Use of Written Feedback in Mathematics , Erin Loraine Carroll

Identity Work to Teach Mathematics for Social Justice , Navy B. Dixon

Developing a Quantitative Understanding of U-Substitution in First-Semester Calculus , Leilani Camille Heaton Fonbuena

The Perception of At-Risk Students on Caring Student-Teacher Relationships and Its Impact on Their Productive Disposition , Brittany Hopper

Variational and Covariational Reasoning of Students with Disabilities , Lauren Rigby

Structural Reasoning with Rational Expressions , Dana Steinhorst

Student-Created Learning Objects for Mathematics Renewable Assignments: The Potential Value They Bring to the Broader Community , Webster Wong

Theses/Dissertations from 2021 2021

Emotional Geographies of Beginning and Veteran Reformed Teachers in Mentor/Mentee Relationships , Emily Joan Adams

You Do Math Like a Girl: How Women Reason Mathematically Outside of Formal and School Mathematics Contexts , Katelyn C. Pyfer

Developing the Definite Integral and Accumulation Function Through Adding Up Pieces: A Hypothetical Learning Trajectory , Brinley Nichole Stevens

Theses/Dissertations from 2020 2020

Mathematical Identities of Students with Mathematics Learning Dis/abilities , Emma Lynn Holdaway

Teachers' Mathematical Meanings: Decisions for Teaching Geometric Reflections and Orientation of Figures , Porter Peterson Nielsen

Student Use of Mathematical Content Knowledge During Proof Production , Chelsey Lynn Van de Merwe

Theses/Dissertations from 2019 2019

Making Sense of the Equal Sign in Middle School Mathematics , Chelsea Lynn Dickson

Developing Understanding of the Chain Rule, Implicit Differentiation, and Related Rates: Towards a Hypothetical Learning Trajectory Rooted in Nested Multivariation , Haley Paige Jeppson

Secondary Preservice Mathematics Teachers' Curricular Reasoning , Kimber Anne Mathis

“Don’t Say Gay. We Say Dumb or Stupid”: Queering ProspectiveMathematics Teachers’ Discussions , Amy Saunders Ross

Aspects of Engaging Problem Contexts From Students' Perspectives , Tamara Kay Stark

Theses/Dissertations from 2018 2018

Addressing Pre-Service Teachers' Misconceptions About Confidence Intervals , Kiya Lynn Eliason

How Teacher Questions Affect the Development of a Potential Hybrid Space in a Classroom with Latina/o Students , Casandra Helen Job

Teacher Graphing Practices for Linear Functions in a Covariation-Based College Algebra Classroom , Konda Jo Luckau

Principles of Productivity Revealed from Secondary Mathematics Teachers' Discussions Around the Productiveness of Teacher Moves in Response to Teachable Moments , Kylie Victoria Palsky

Theses/Dissertations from 2017 2017

Curriculum Decisions and Reasoning of Middle School Teachers , Anand Mikel Bernard

Teacher Response to Instances of Student Thinking During Whole Class Discussion , Rachel Marie Bernard

Kyozaikenkyu: An In-Depth Look into Japanese Educators' Daily Planning Practices , Matthew David Melville

Analysis of Differential Equations Applications from the Coordination Class Perspective , Omar Antonio Naranjo Mayorga

Theses/Dissertations from 2016 2016

The Principles of Effective Teaching Student Teachershave the Opportunity to Learn in an AlternativeStudent Teaching Structure , Danielle Rose Divis

Insight into Student Conceptions of Proof , Steven Daniel Lauzon

Theses/Dissertations from 2015 2015

Teacher Participation and Motivation inProfessional Development , Krystal A. Hill

Student Evaluation of Mathematical Explanations in anInquiry-Based Mathematics Classroom , Ashley Burgess Hulet

English Learners' Participation in Mathematical Discourse , Lindsay Marie Merrill

Mathematical Interactions between Teachers and Students in the Finnish Mathematics Classroom , Paula Jeffery Prestwich

Parents and the Common Core State Standards for Mathematics , Rebecca Anne Roberts

Examining the Effects of College Algebra on Students' Mathematical Dispositions , Kevin Lee Watson

Problems Faced by Reform Oriented Novice Mathematics Teachers Utilizing a Traditional Curriculum , Tyler Joseph Winiecke

Academic and Peer Status in the Mathematical Life Stories of Students , Carol Ann Wise

Theses/Dissertations from 2014 2014

The Effect of Students' Mathematical Beliefs on Knowledge Transfer , Kristen Adams

Language Use in Mathematics Textbooks Written in English and Spanish , Kailie Ann Bertoch

Teachers' Curricular Reasoning and MKT in the Context of Algebra and Statistics , Kolby J. Gadd

Mathematical Telling in the Context of Teacher Interventions with Collaborative Groups , Brandon Kyle Singleton

An Investigation of How Preservice Teachers Design Mathematical Tasks , Elizabeth Karen Zwahlen

Theses/Dissertations from 2013 2013

Student Understanding of Limit and Continuity at a Point: A Look into Four Potentially Problematic Conceptions , Miriam Lynne Amatangelo

Exploring the Mathematical Knowledge for Teaching of Japanese Teachers , Ratu Jared R. T. Bukarau

Comparing Two Different Student Teaching Structures by Analyzing Conversations Between Student Teachers and Their Cooperating Teachers , Niccole Suzette Franc

Professional Development as a Community of Practice and Its Associated Influence on the Induction of a Beginning Mathematics Teacher , Savannah O. Steele

Types of Questions that Comprise a Teacher's Questioning Discourse in a Conceptually-Oriented Classroom , Keilani Stolk

Theses/Dissertations from 2012 2012

Student Teachers' Interactive Decisions with Respect to Student Mathematics Thinking , Jonathan J. Call

Manipulatives and the Growth of Mathematical Understanding , Stacie Joyce Gibbons

Learning Within a Computer-Assisted Instructional Environment: Effects on Multiplication Math Fact Mastery and Self-Efficacy in Elementary-Age Students , Loraine Jones Hanson

Mathematics Teacher Time Allocation , Ashley Martin Jones

Theses/Dissertations from 2011 2011

How Student Positioning Can Lead to Failure in Inquiry-based Classrooms , Kelly Beatrice Campbell

Teachers' Decisions to Use Student Input During Class Discussion , Heather Taylor Toponce

A Conceptual Framework for Student Understanding of Logarithms , Heather Rebecca Ambler Williams

Theses/Dissertations from 2010 2010

Growth in Students' Conceptions of Mathematical Induction , John David Gruver

Contextualized Motivation Theory (CMT): Intellectual Passion, Mathematical Need, Social Responsibility, and Personal Agency in Learning Mathematics , Janelle Marie Hart

Thinking on the Brink: Facilitating Student Teachers' Learning Through In-the-Moment Interjections , Travis L. Lemon

Understanding Teachers' Change Towards a Reform-Oriented Mathematics Classroom , Linnae Denise Williams

Theses/Dissertations from 2009 2009

A Comparison of Mathematical Discourse in Online and Face-to-Face Environments , Shawn D. Broderick

The Influence of Risk Taking on Student Creation of Mathematical Meaning: Contextual Risk Theory , Erin Nicole Houghtaling

Uncovering Transformative Experiences: A Case Study of the Transformations Made by one Teacher in a Mathematics Professional Development Program , Rachelle Myler Orsak

Theses/Dissertations from 2008 2008

Student Teacher Knowledge and Its Impact on Task Design , Tenille Cannon

How Eighth-Grade Students Estimate with Fractions , Audrey Linford Hanks

Similar but Different: The Complexities of Students' Mathematical Identities , Diane Skillicorn Hill

Choose Your Words: Refining What Counts as Mathematical Discourse in Students' Negotiation of Meaning for Rate of Change of Volume , Christine Johnson

Mathematics Student Teaching in Japan: A Multi-Case Study , Allison Turley Shwalb

Theses/Dissertations from 2007 2007

Applying Toulmin's Argumentation Framework to Explanations in a Reform Oriented Mathematics Class , Jennifer Alder Brinkerhoff

What Are Some of the Common Traits in the Thought Processes of Undergraduate Students Capable of Creating Proof? , Karen Malina Duff

Probing for Reasons: Presentations, Questions, Phases , Kellyn Nicole Farlow

One Problem, Two Contexts , Danielle L. Gigger

The Main Challenges that a Teacher-in-Transition Faces When Teaching a High School Geometry Class , Greg Brough Henry

Discovering the Derivative Can Be "Invigorating:" Mark's Journey to Understanding Instantaneous Velocity , Charity Ann Gardner Hyer

Theses/Dissertations from 2006 2006

How a Master Teacher Uses Questioning Within a Mathematical Discourse Community , Omel Angel Contreras

Determining High School Geometry Students' Geometric Understanding Using van Hiele Levels: Is There a Difference Between Standards-based Curriculum Students and NonStandards-based Curriculum Students? , Rebekah Loraine Genz

The Nature and Frequency of Mathematical Discussion During Lesson Study That Implemented the CMI Framework , Andrew Ray Glaze

Second Graders' Solution Strategies and Understanding of a Combination Problem , Tiffany Marie Hessing

What Does It Mean To Preservice Mathematics Teachers To Anticipate Student Responses? , Matthew M. Webb

Theses/Dissertations from 2005 2005

Fraction Multiplication and Division Image Change in Pre-Service Elementary Teachers , Jennifer J. Cluff

An Examination of the Role of Writing in Mathematics Instruction , Amy Jeppsen

Theses/Dissertations from 2004 2004

Reasoning About Motion: A Case Study , Tiffini Lynn Glaze

Theses/Dissertations from 2003 2003

An Analysis of the Influence of Lesson Study on Preservice Secondary Mathematics Teachers' View of Self-As Mathematics Expert , Julie Stafford

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Research sampler 5: examples in learning mathematics, by annie and john selden.

Successful Math Majors Generate Their Own Examples Being Asked For Examples Can Be Disconcerting Generating Counterexamples That Are Explanatory "If I Don't Know What It Says, How Can I Find an Example of It?" Coda

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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Department of Mathematics

Title-finite quotients of four-punctured sphere bundle groups- location sc-1432.

The finite quotients of the fundamental group of a 3-manifold are the deck groups of its finite regular covers. We often pass to these finite-sheeted covers for different reasons, and these deck groups are organized into a topological group called the profinite completion of a 3-manifold group. In this talk, we will discuss how to leverage certain properties of the mapping class group of the four-punctured sphere to study the profinite completions of the fundamental groups of fibered hyperbolic four-punctured sphere bundles over the circle.

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