research topics in mathematics education for undergraduate

Research Topics & Ideas: Education

170+ Research Ideas To Fast-Track Your Project

If you’re just starting out exploring education-related topics for your dissertation, thesis or research project, you’ve come to the right place. In this post, we’ll help kickstart your research topic ideation process by providing a hearty list of research topics and ideas , including examples from actual dissertations and theses..

PS – This is just the start…

We know it’s exciting to run through a list of research topics, but please keep in mind that this list is just a starting point . To develop a suitable education-related research topic, you’ll need to identify a clear and convincing research gap , and a viable plan of action to fill that gap.

If this sounds foreign to you, check out our free research topic webinar that explores how to find and refine a high-quality research topic, from scratch. Alternatively, if you’d like hands-on help, consider our 1-on-1 coaching service .

Overview: Education Research Topics

• How to find a research topic (video)
• List of 50+ education-related research topics/ideas
• List of 120+ level-specific research topics 
• Examples of actual dissertation topics in education
• Tips to fast-track your topic ideation (video)
• Free Webinar : Topic Ideation 101
• Where to get extra help

Education-Related Research Topics & Ideas

Below you’ll find a list of education-related research topics and idea kickstarters. These are fairly broad and flexible to various contexts, so keep in mind that you will need to refine them a little. Nevertheless, they should inspire some ideas for your project.

• The impact of school funding on student achievement
• The effects of social and emotional learning on student well-being
• The effects of parental involvement on student behaviour
• The impact of teacher training on student learning
• The impact of classroom design on student learning
• The impact of poverty on education
• The use of student data to inform instruction
• The role of parental involvement in education
• The effects of mindfulness practices in the classroom
• The use of technology in the classroom
• The role of critical thinking in education
• The use of formative and summative assessments in the classroom
• The use of differentiated instruction in the classroom
• The use of gamification in education
• The effects of teacher burnout on student learning
• The impact of school leadership on student achievement
• The effects of teacher diversity on student outcomes
• The role of teacher collaboration in improving student outcomes
• The implementation of blended and online learning
• The effects of teacher accountability on student achievement
• The effects of standardized testing on student learning
• The effects of classroom management on student behaviour
• The effects of school culture on student achievement
• The use of student-centred learning in the classroom
• The impact of teacher-student relationships on student outcomes
• The achievement gap in minority and low-income students
• The use of culturally responsive teaching in the classroom
• The impact of teacher professional development on student learning
• The use of project-based learning in the classroom
• The effects of teacher expectations on student achievement
• The use of adaptive learning technology in the classroom
• The impact of teacher turnover on student learning
• The effects of teacher recruitment and retention on student learning
• The impact of early childhood education on later academic success
• The impact of parental involvement on student engagement
• The use of positive reinforcement in education
• The impact of school climate on student engagement
• The role of STEM education in preparing students for the workforce
• The effects of school choice on student achievement
• The use of technology in the form of online tutoring

Level-Specific Research Topics

Looking for research topics for a specific level of education? We’ve got you covered. Below you can find research topic ideas for primary, secondary and tertiary-level education contexts. Click the relevant level to view the respective list.

Research Topics: Pick An Education Level

Primary education.

• Investigating the effects of peer tutoring on academic achievement in primary school
• Exploring the benefits of mindfulness practices in primary school classrooms
• Examining the effects of different teaching strategies on primary school students’ problem-solving skills
• The use of storytelling as a teaching strategy in primary school literacy instruction
• The role of cultural diversity in promoting tolerance and understanding in primary schools
• The impact of character education programs on moral development in primary school students
• Investigating the use of technology in enhancing primary school mathematics education
• The impact of inclusive curriculum on promoting equity and diversity in primary schools
• The impact of outdoor education programs on environmental awareness in primary school students
• The influence of school climate on student motivation and engagement in primary schools
• Investigating the effects of early literacy interventions on reading comprehension in primary school students
• The impact of parental involvement in school decision-making processes on student achievement in primary schools
• Exploring the benefits of inclusive education for students with special needs in primary schools
• Investigating the effects of teacher-student feedback on academic motivation in primary schools
• The role of technology in developing digital literacy skills in primary school students
• Effective strategies for fostering a growth mindset in primary school students
• Investigating the role of parental support in reducing academic stress in primary school children
• The role of arts education in fostering creativity and self-expression in primary school students
• Examining the effects of early childhood education programs on primary school readiness
• Examining the effects of homework on primary school students’ academic performance
• The role of formative assessment in improving learning outcomes in primary school classrooms
• The impact of teacher-student relationships on academic outcomes in primary school
• Investigating the effects of classroom environment on student behavior and learning outcomes in primary schools
• Investigating the role of creativity and imagination in primary school curriculum
• The impact of nutrition and healthy eating programs on academic performance in primary schools
• The impact of social-emotional learning programs on primary school students’ well-being and academic performance
• The role of parental involvement in academic achievement of primary school children
• Examining the effects of classroom management strategies on student behavior in primary school
• The role of school leadership in creating a positive school climate Exploring the benefits of bilingual education in primary schools
• The effectiveness of project-based learning in developing critical thinking skills in primary school students
• The role of inquiry-based learning in fostering curiosity and critical thinking in primary school students
• The effects of class size on student engagement and achievement in primary schools
• Investigating the effects of recess and physical activity breaks on attention and learning in primary school
• Exploring the benefits of outdoor play in developing gross motor skills in primary school children
• The effects of educational field trips on knowledge retention in primary school students
• Examining the effects of inclusive classroom practices on students’ attitudes towards diversity in primary schools
• The impact of parental involvement in homework on primary school students’ academic achievement
• Investigating the effectiveness of different assessment methods in primary school classrooms
• The influence of physical activity and exercise on cognitive development in primary school children
• Exploring the benefits of cooperative learning in promoting social skills in primary school students

Secondary Education

• Investigating the effects of school discipline policies on student behavior and academic success in secondary education
• The role of social media in enhancing communication and collaboration among secondary school students
• The impact of school leadership on teacher effectiveness and student outcomes in secondary schools
• Investigating the effects of technology integration on teaching and learning in secondary education
• Exploring the benefits of interdisciplinary instruction in promoting critical thinking skills in secondary schools
• The impact of arts education on creativity and self-expression in secondary school students
• The effectiveness of flipped classrooms in promoting student learning in secondary education
• The role of career guidance programs in preparing secondary school students for future employment
• Investigating the effects of student-centered learning approaches on student autonomy and academic success in secondary schools
• The impact of socio-economic factors on educational attainment in secondary education
• Investigating the impact of project-based learning on student engagement and academic achievement in secondary schools
• Investigating the effects of multicultural education on cultural understanding and tolerance in secondary schools
• The influence of standardized testing on teaching practices and student learning in secondary education
• Investigating the effects of classroom management strategies on student behavior and academic engagement in secondary education
• The influence of teacher professional development on instructional practices and student outcomes in secondary schools
• The role of extracurricular activities in promoting holistic development and well-roundedness in secondary school students
• Investigating the effects of blended learning models on student engagement and achievement in secondary education
• The role of physical education in promoting physical health and well-being among secondary school students
• Investigating the effects of gender on academic achievement and career aspirations in secondary education
• Exploring the benefits of multicultural literature in promoting cultural awareness and empathy among secondary school students
• The impact of school counseling services on student mental health and well-being in secondary schools
• Exploring the benefits of vocational education and training in preparing secondary school students for the workforce
• The role of digital literacy in preparing secondary school students for the digital age
• The influence of parental involvement on academic success and well-being of secondary school students
• The impact of social-emotional learning programs on secondary school students’ well-being and academic success
• The role of character education in fostering ethical and responsible behavior in secondary school students
• Examining the effects of digital citizenship education on responsible and ethical technology use among secondary school students
• The impact of parental involvement in school decision-making processes on student outcomes in secondary schools
• The role of educational technology in promoting personalized learning experiences in secondary schools
• The impact of inclusive education on the social and academic outcomes of students with disabilities in secondary schools
• The influence of parental support on academic motivation and achievement in secondary education
• The role of school climate in promoting positive behavior and well-being among secondary school students
• Examining the effects of peer mentoring programs on academic achievement and social-emotional development in secondary schools
• Examining the effects of teacher-student relationships on student motivation and achievement in secondary schools
• Exploring the benefits of service-learning programs in promoting civic engagement among secondary school students
• The impact of educational policies on educational equity and access in secondary education
• Examining the effects of homework on academic achievement and student well-being in secondary education
• Investigating the effects of different assessment methods on student performance in secondary schools
• Examining the effects of single-sex education on academic performance and gender stereotypes in secondary schools
• The role of mentoring programs in supporting the transition from secondary to post-secondary education

Tertiary Education

• The role of student support services in promoting academic success and well-being in higher education
• The impact of internationalization initiatives on students’ intercultural competence and global perspectives in tertiary education
• Investigating the effects of active learning classrooms and learning spaces on student engagement and learning outcomes in tertiary education
• Exploring the benefits of service-learning experiences in fostering civic engagement and social responsibility in higher education
• The influence of learning communities and collaborative learning environments on student academic and social integration in higher education
• Exploring the benefits of undergraduate research experiences in fostering critical thinking and scientific inquiry skills
• Investigating the effects of academic advising and mentoring on student retention and degree completion in higher education
• The role of student engagement and involvement in co-curricular activities on holistic student development in higher education
• The impact of multicultural education on fostering cultural competence and diversity appreciation in higher education
• The role of internships and work-integrated learning experiences in enhancing students’ employability and career outcomes
• Examining the effects of assessment and feedback practices on student learning and academic achievement in tertiary education
• The influence of faculty professional development on instructional practices and student outcomes in tertiary education
• The influence of faculty-student relationships on student success and well-being in tertiary education
• The impact of college transition programs on students’ academic and social adjustment to higher education
• The impact of online learning platforms on student learning outcomes in higher education
• The impact of financial aid and scholarships on access and persistence in higher education
• The influence of student leadership and involvement in extracurricular activities on personal development and campus engagement
• Exploring the benefits of competency-based education in developing job-specific skills in tertiary students
• Examining the effects of flipped classroom models on student learning and retention in higher education
• Exploring the benefits of online collaboration and virtual team projects in developing teamwork skills in tertiary students
• Investigating the effects of diversity and inclusion initiatives on campus climate and student experiences in tertiary education
• The influence of study abroad programs on intercultural competence and global perspectives of college students
• Investigating the effects of peer mentoring and tutoring programs on student retention and academic performance in tertiary education
• Investigating the effectiveness of active learning strategies in promoting student engagement and achievement in tertiary education
• Investigating the effects of blended learning models and hybrid courses on student learning and satisfaction in higher education
• The role of digital literacy and information literacy skills in supporting student success in the digital age
• Investigating the effects of experiential learning opportunities on career readiness and employability of college students
• The impact of e-portfolios on student reflection, self-assessment, and showcasing of learning in higher education
• The role of technology in enhancing collaborative learning experiences in tertiary classrooms
• The impact of research opportunities on undergraduate student engagement and pursuit of advanced degrees
• Examining the effects of competency-based assessment on measuring student learning and achievement in tertiary education
• Examining the effects of interdisciplinary programs and courses on critical thinking and problem-solving skills in college students
• The role of inclusive education and accessibility in promoting equitable learning experiences for diverse student populations
• The role of career counseling and guidance in supporting students’ career decision-making in tertiary education
• The influence of faculty diversity and representation on student success and inclusive learning environments in higher education

Education-Related Dissertations & Theses

While the ideas we’ve presented above are a decent starting point for finding a research topic in education, they are fairly generic and non-specific. So, it helps to look at actual dissertations and theses in the education space to see how this all comes together in practice.

Below, we’ve included a selection of education-related research projects to help refine your thinking. These are actual dissertations and theses, written as part of Master’s and PhD-level programs, so they can provide some useful insight as to what a research topic looks like in practice.

• From Rural to Urban: Education Conditions of Migrant Children in China (Wang, 2019)
• Energy Renovation While Learning English: A Guidebook for Elementary ESL Teachers (Yang, 2019)
• A Reanalyses of Intercorrelational Matrices of Visual and Verbal Learners’ Abilities, Cognitive Styles, and Learning Preferences (Fox, 2020)
• A study of the elementary math program utilized by a mid-Missouri school district (Barabas, 2020)
• Instructor formative assessment practices in virtual learning environments : a posthumanist sociomaterial perspective (Burcks, 2019)
• Higher education students services: a qualitative study of two mid-size universities’ direct exchange programs (Kinde, 2020)
• Exploring editorial leadership : a qualitative study of scholastic journalism advisers teaching leadership in Missouri secondary schools (Lewis, 2020)
• Selling the virtual university: a multimodal discourse analysis of marketing for online learning (Ludwig, 2020)
• Advocacy and accountability in school counselling: assessing the use of data as related to professional self-efficacy (Matthews, 2020)
• The use of an application screening assessment as a predictor of teaching retention at a midwestern, K-12, public school district (Scarbrough, 2020)
• Core values driving sustained elite performance cultures (Beiner, 2020)
• Educative features of upper elementary Eureka math curriculum (Dwiggins, 2020)
• How female principals nurture adult learning opportunities in successful high schools with challenging student demographics (Woodward, 2020)
• The disproportionality of Black Males in Special Education: A Case Study Analysis of Educator Perceptions in a Southeastern Urban High School (McCrae, 2021)

As you can see, these research topics are a lot more focused than the generic topic ideas we presented earlier. So, in order for you to develop a high-quality research topic, you’ll need to get specific and laser-focused on a specific context with specific variables of interest.  In the video below, we explore some other important things you’ll need to consider when crafting your research topic.

Get 1-On-1 Help

If you’re still unsure about how to find a quality research topic within education, check out our Research Topic Kickstarter service, which is the perfect starting point for developing a unique, well-justified research topic.

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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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• Industrial mathematics curriculum development for Ethiopian Science and Technology universities  Zewdie Woldeamanuel Habte ( 2023-09 ) This study aimed to examine the development and implementation of the new industrial mathematics curriculum for Ethiopian Science and Technology universities. Ethiopia is in the process of transforming to industry led ...
• The impact of 8Ps learning model on the mathematical problem-solving performance of grade 12 learners in the concept of stationary points in differential calculus  Omoniyi, Adebayo Akinyinka ( 2022-11-11 ) Noting the centrality of problem solving to Mathematics and its capability to enhance learner performance in the subject, the study measured the impact of the use of 8Ps learning model on the mathematical problem-solving ...
• Enhancing mathematics teachers' professional development for creative teaching in Addis Ababa secondary schools  Anwar Seid Al-amin ( 2023-09-25 ) The Ethiopian education system is imported from the West, and the traditional method of instruction (pouring information) in the ICT revolution era wastes time and resources. Learners can get more information about mathematics ...
• Grade 9 mathematics teachers’ strategies to address mathematical proficiency in their teaching of linear equations : a case of selected schools in Gauteng North District  Mothudi, Teresa Ntsae ( 2022-12-10 ) The purpose of this research was to look into Grade 9 mathematics teachers’ strategies to address mathematical proficiency in their teaching of linear equations. The study was intrigued by the performance of learners in ...
• Grade 6 mathematics teachers’ development of learner mathematical proficiency in addition and subtraction of common fractions, in the Tshwane South District of Gauteng  Lendis, Ashley Pearl ( 2022-11-30 ) The study was motivated by the fact that learners are not performing well in the topic of fractions due to the lack of conceptual understanding of the concept. The purpose of this qualitative study based on an interpretive ...
• The performance and learning difficulties of Grade 10 learners in solving euclidean geometry problems in Tshwane West District  Olabode, Adedayo Abosede ( 2023-01-31 ) There is a growing trend of declining performance in the final year (Grade 12) mathematics examinations in the South African public school system. The study aimed to evaluate Grade 10 learners in the Tshwane West District ...
• A collaborative model for teaching and learning mathematics in secondary schools  Ngwenya, Vusani ( 2021-11 ) Mathematics pass rates in South African schools, as in many developing nations, continue to be a source of concern for educators and policymakers alike. Improving mathematics performance is non-negotiable if Africa is to ...
• Teachers improvement of mathematics achievements in rural schools of Mopani District : implications for professional development  Sambo, Sosa Isaac ( 2023-01-01 ) Throughout the globe, there is an outcry that learners’ performance in mathematics is below the expected standard. Hence, teachers need teacher development to improve their skills and knowledge in the subject. The purpose ...
• Grade 10 learners’ academic experiences of learning parabolic functions in schools of Vhembe district of Limpopo Province  Mudau, Takalani Lesley ( 2022-11-22 ) The objective of this study was to determine Grade 10 learners' academic performance in learning parabola functions. Furthermore, the study sought to unearth errors that learners make when learning parabola functions and ...
• An exploration of learning difficulties experienced by grade 12 learners in euclidean geometry : a case of Ngaka Modiri Molema district  Mudhefi, Fungirai ( 2022-08-20 ) The purpose of this study was to investigate the learning difficulties experienced by Grade 12 learners in Euclidean geometry. Despite the efforts exerted in terms of time, material and human resources in the teaching and ...
• The relationship between anxiety, working memory and achievement in mathematics in grade 5 learners : a case study of Tshepisong schools  Mnguni, Maria Tebogo ( 2022-01-21 ) This study investigated the relationship between anxiety, working memory and achievement in mathematics in grade 5 learners at Tshepisong schools. A sample of 300 grade 5 learners from Tshepisong schools was selected using ...
• The effects of using a graphic calculator as a cognitive tool in learning grade 10 data handling  Rambao, Mpho ( 2022-09 ) The integration of technology in the mathematics classroom is believed to have an influence on how the learners learn and change their perception of mathematics. Therefore, technology tools are developed to enhance the ...
• Integration of ethnomathematics in the teaching of probability in secondary school mathematics in Zimbabwe  Turugari, Munamato ( 2022-06 ) Underpinned by the social constructivism theory and the praxis of integrating ethnomathematics in the teaching of probability, this PAR developed a policy framework for facilitating the integration of ethno-mathematics in ...
• Evaluating Grade 10 learners’ change in understanding of similar triangles following a classroom intervention  Maweya, Amokelo Given ( 2022 ) Geometry, in particular Euclidean geometry, has been highlighted as a subject in mathematics that presents a variety of challenges to many secondary school learners. Many students struggle to gain appropriate knowledge of ...
• The impact of technology integration in teaching grade 11 Euclidean geometry based on van Hiele’s model  Bediako, Adjei ( 2021-10-10 ) This quantitative study reports on the impact of using GeoGebra software to teach Grade 11 geometry through van Hieles’ levels theory, merged with some elements of the Technological Pedagogical Content Knowledge framework. ...
• A framework towards improved instruction of probability to grade seven students : a case of South African schools in Mpumalanga province  Kodisang, Sophy Mamanyena ( 2022-02-03 ) This empirical phenomenological study explored teachers’ conceptual understanding of probability to gain insights into how this understanding enhanced their instructional classroom practice. The study was motivated by the ...
• The use and effect of Geogebra software in Calculus at Wachemo University, Ethiopia : an investigation  Tola Bekene Bedada ( 2021-08 ) With the rapid growth of technology in the 21st century, traditional teaching and learning methods are considered outdated and not suitable for the active learning processes of the constructivist learning approach. The ...

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• Undergraduate

Undergraduate Research Projects

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

• Undergraduate Research

Undergraduate Research programs are a great opportunity for undergraduates to build research experience, connect with faculty and researchers, and (sometimes) even earn some money. Undergraduate Research programs can take a variety of formats. Some are informal arrangements with a professor where you work independently on a problem but with guidance from the professor. Other programs are more formal, such as the numerous summer REU programs funded by the National Science Foundation.

These programs are typically an 8-10 week residential program with other students from various universities where you work together on a problem.

Summer REU programs typically involve paid travel expenses and a summer stipend and are very competitive to get admitted to. If you are interested in finding out more about Undergraduate Research opportunities at Purdue, or how to apply to summer REU programs, contact Jon Peterson at [email protected] .

Summer is traditionally a time to kick back and take a break from studies, but not so for several mathematics students who are in residence in the Mathematics Department during summers.

With support provided by Purdue alumni Andy Zoltners, Joel Spira, as well as the National Science Foundation and other funding, undergraduate math students engage in research projects under the guidance of mathematics faculty members.

• Purdue REU Opportunities

Summer REU Opportunities

Past Research Projects

• Faculty Research Areas
• Center for Computational & Applied Mathematics
• Funding Opportunities
• Research at Purdue

Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067

Phone: (765) 494-1901 - FAX: (765) 494-0548   Contact Us

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Undergraduate Research

Where to start:.

A good starting point is the Harvard College Undergraduate Research and Fellowships page. The Office of Undergraduate Research and Fellowships administers research programs for Harvard College undergraduates. Check out the website . Another resource is OCS , the Harvard Office of Career Services. It offers help on preparing a CV or cover letters and gives advice on how to network, interview, etc. Their website is here . Other Sources that can provide additional information on Scholarships, awards, and other grants:

• Committee on General Scholarships: more …
• Office of International Programs: more …
• Student Employment Office: more …

Independent study in Mathematics

Students who would like to do some independent study or a reading class please read the pamphlet page . about Math 91r.

THE ANNUAL OCS SUMMER OPPORTUNITIES FAIR

The Office of Career Services hosts summer programs to help you begin your summer search. Programs are both Harvard affiliated and public or private sector and include internships, public service, funding, travel, and research (URAF staff will be there to answer your questions!). Check out the website.

Harvard-Amgen Scholars program in Biotechnology

Check out the Harvard-Amgen Scholars Program Learn about Harvard’s Amgen 10-week intensive summer research program, one of ten Amgen U.S. programs that support research in biotechnology. The Harvard program includes faculty projects in FAS science departments, SEAS, the Wyss Institute for Biologically-inspired Engineering, and the School of Medicine, open to rising juniors and seniors in biotechnology-related fields.

PRIMO program

The Program for research in Markets and Organizations (PRIMO) is a 10-week program for Harvard undergraduates who wish to work closely with Harvard Business School faculty on research projects.

Harvard Undergraduate Research Events

• Wednesday, October 10, 12:00-1: 20 PM – Fall Undergraduate Research Spotlight. Come and meet Harvard undergraduate peers who will showcase their research projects and share their experiences conducting research at Harvard and abroad, followed by reception and deserts. Event program and list of presentations can be found here: here (pizza and desserts while supplies last). Free for Harvard students. Cabot Library 1st floor Discovery Bar.
• Wednesday, October 17, 12:00-1: 00 PM – Undergraduate Science Research Workshop. Workshop facilitators Dr. Margaret A. Lynch, (Assoc. Director of Science #Education) and Dr. Anna Babakhanyan, (Undergraduate Research Advisor) will help Harvard students learn about science research landscape at Harvard. You will learn about what kind of research (basic science vs. clinical, various research areas) is available at Harvard, where you can conduct research, the types of undergraduate research appointments, how to find a lab that fits, interviewing and more. In addition, the workshop will provide strategies for students to prepare for the Annual HUROS Fair, see below. No registration is required for this event (pizza while supplies last). Free for all Harvard students. Cabot Library first floor Discover Bar. More.

Outside Programs

Caltech always announces two summer research opportunities available to continuing undergraduate students. Examples: WAVE Student-Faculty Programs The WAVE Fellows program provides support for talented undergraduates intent on pursuing a Ph.D. to conduct a 10-week summer research project at Caltech. And then there is the AMGEN Scholars program. See the website for more details.

Johns Hopkins Summer 2018 Opportunities

The Johns Hopkins University Center for Talented Youth (CTY) is seeking instructors and teaching assistants for our summer programs. CTY offers challenging academic programs for highly talented elementary, middle, and high school students from across the country and around the world. Positions are available at residential and day sites at colleges, universities, and schools on the East and West coasts, as well as internationally in Hong Kong. Website

Math REU list from AMS

Mellon Mays opportunities awareness

The Mellon Mays Undergraduate Fellowship Program ( MMUF ) selects ten students in their sophomore year to join a tightly-knit research community during junior and senior years to conduct independent research in close collaboration with a faculty mentor. Join us at this information session to find out more about the program. MMUF exists to counter the under-representation of minority groups on college and university faculties nationwide through activities designed to encourage the pursuit of the Ph.D. in the humanities and core sciences.

MIT Amgen and UROP

You may be familiar with the Amgen Scholars Program, a summer research program in science and biotechnology. The Massachusetts Institute of Technology is a participant in the Amgen-UROP Scholars Program for a ninth year. UROP is MIT’s Undergraduate Research Opportunities Program. The mission of the Amgen-UROP Scholars Program is to provide students with a strong science research experience that may be pivotal in their undergraduate career, cultivate a passion for science, encourage the pursuit of graduate studies in the sciences, and stimulate interest in research and scientific careers. MIT is delighted to invite undergraduate students from other colleges and universities to join our research enterprise. We value the knowledge, experience, and enthusiasm these young scholars will bring to our campus and appreciate this opportunity to build a relationship with your faculty and campus.

More REU's, not only math

The National Science Foundation Research Experiences for Undergraduates (REU) NSF funds a large number of research opportunities for undergraduate students through its REU Sites program. An REU Site consists of a group of ten or so undergraduates who work in the research programs of the host institution. Each student is associated with a specific research project, where he/she works closely with the faculty and other researchers. Students are granted stipends and, in many cases, assistance with housing and travel. Undergraduate students supported with NSF funds must be citizens or permanent residents of the United States or its possessions. An REU Site may be at either the US or foreign location. By using the web page , search for an REU Site, you may examine opportunities in the subject areas supported by various NSF units. Also, you may search by keywords to identify sites in particular research areas or with certain features, such as a particular location. Students must contact the individual sites for information and application materials. NSF does not have application materials and does not select student participants. A contact person and contact information are listed for each site.

Here is a link with more information about summer programs for undergraduates at NSA: NSA The most math-related one is DSP, but those students who are more interested in computer science could also look at, say, CES SP. They are all paid with benefits and housing is covered. Note that application deadlines are pretty early (usually mid-October). The application process will involve usually a few interviews and a trip down to DC.

NSF Graduate Research Fellowships

US citizens and permanent residents who are planning to enter graduate school in the fall of 2019 are eligible (as are those in the first two years of such a graduate program, or who are returning to graduate school after being out for two or more years). The program solicitation contains full details. Information about the NSF Graduate Research Fellowship Program (GRFP) is here . The GRFP supports outstanding graduate students in NSF-supported science, technology, engineering, and mathematics disciplines who are pursuing research-based Masters and doctoral degrees at accredited United States institutions. The program provides up to three years of graduate education support, including an annual, 000 stipend. Applications for Mathematical Sciences topics are due October 26, 2018.

Pathway to Science

summer research listings from pathways to science.

Perimeter Institute

Applications are now being accepted for Perimeter Institute’s Undergraduate Theoretical Physics Summer Program. The program consists of two parts:

• Fully-Funded Two Week Summer School (May 27 to June 7, 2019) Students are immersed in Perimeter’s dynamic research environment — attending courses on cutting-edge topics in physics, learning new techniques to solve interesting problems, working on group research projects, and potentially even publishing their work. All meals, accommodation, and transportation provided
• Paid Research Internship (May 1 to August 30, 2019, negotiable) Students will work on projects alongside Perimeter researchers. Students will have the opportunity to develop their research skills and absorb the rich variety of talks, conferences, and events at the Perimeter Institute. Applicants can apply for the two-week summer school or for both the summer school and the research internship. Summer school and internship positions will be awarded by February 28, 2019. Selected interns will be contacted with the research projects topics. All research interns must complete the two-week summer school.

Apply online at perimeterinstitute.ca/undergrad

Stanford resident counselors

Stanford Pre-Collegiate Institutes is hiring Residential Counselors for the summer to work with the following courses:

• Cryptography (grades 9-10)
• Knot Theory (grades 10-11)
• Logic and Problem Solving (grades 8-9)
• Number Theory (grades 9-11)
• Excursions in Probability (grades 8-9)
• Discrete Mathematics (grades 9-10)
• The Mathematics of Symmetry (grades 10-11)
• Mathematical Puzzles and Games (grades 8-9)

Stanford Pre-Collegiate Institutes offers three-week sessions for academically talented high school students during June and July. Interested candidates can learn more about our positions and apply by visiting our employment website .

Summer Research 2019 at Nebraska

We are now accepting applications for the University of Nebraska’s 2019 Summer Research Program, and we’d like to encourage your students to apply. Details.

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181 Mathematics Research Topics From PhD Experts

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

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

Our Newest Research Topics in Math

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

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

Cool Math Topics to Research

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

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

Undergraduate Math Research Topics

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

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

Number Theory Topics to Research

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

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

Math Research Topics for High School

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

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

Complex Math Topics

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

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

Easy Math Research Paper Topics

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

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

Logic Topics to Research

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

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

Algebra Topics for a Research Paper

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

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

Math Education Research Topics

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

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

Computability Theory Topics to Research

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

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

Interesting Math Research Topics

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

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

Applied Math Research Topics

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

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

Geometry Topics for a Research Paper

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

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

Math Research Topics for Middle School

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

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

Math Research Topics for College Students

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

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

Calculus Topics for a Research Paper

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

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

Simple Math Research Paper Topics for High School

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

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

Business Math Topics

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

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

Probability and Statistics Topics for Research

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

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

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Collections of undergraduate research projects [closed]

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics:

• abstract algebra;
• linear algebra;
• number theory;
• mathematical physics;
• game theory;
• graph theory, combinatorics, probability, statistics;
• algorithms, computability, computational complexity, and other computer-science-related topics;
• et cetera...

To be precise, I'm searching for some references ( books , web-pages , online notes , etc.) that collect

• the statement of the problems (which are the most important thing);
• and preferably hints, guidance, or complete solutions ( if they have been found ) of the problems, and the actual complete projects.

I will start the list myself by mentioning the well-known book Student Research Projects in Calculus .

• reference-request
• soft-question
• $\begingroup$ If they have solutions, how can they be research problems? $\endgroup$ –  Matt Samuel Commented Dec 23, 2014 at 14:52
• 4 $\begingroup$ @MattSamuel: well, mathematicians do happen to solve problems sometimes :) $\endgroup$ –  Dal Commented Dec 23, 2014 at 15:00
• $\begingroup$ @MattSamuel Anyway, I've edited the question. $\endgroup$ –  Dal Commented Dec 23, 2014 at 15:00
• 2 $\begingroup$ Some departments have REU pages which list past/present projects and sometimes include writeups by past students. $\endgroup$ –  Antonio Vargas Commented Dec 25, 2014 at 19:48
• $\begingroup$ Please avoid the tags undergraduate-research and research . We are trying to remove them. $\endgroup$ –  Caleb Stanford Commented Jul 24, 2016 at 19:45

2 Answers 2

I looked into the question and found some projects that may be of interest:

Calculus and Differential Equations

Simple Differential Equations and the Growth and Decay of Ice Sheets :

In this project we will re-visit and expand upon a project of John Imbrie of the University of Virginia and his daughter Katherine matching periodic earth temperatures reflected in ice cores to when the earth axis tilt wobbled and the planets relative annual position to sun. We will investigate how key aspects of the ice-age record (such as shifts in dominant periodicities) follow from simple ordinary differential equations capturing the essential physics of the growth and decay of ice sheets.

The Gamma Function :

The theory of the gamma function was developed in connection with the problem of generalizing the factorial function of the natural numbers. The gamma function is defined as a definite, improper integral, and the notion of factorials is extended to complex and real arguments. This function crops up in many unexpected places in mathematical analysis, such as finding the volume of an n-dimensional “ball”. In this project we develop and explore the basic properties of this function.

Conic Sections via Cones :

In this project we will work our way through Conics of Apollonius of Perga ca. 262 BC – ca. 190 BC. In this work Apollonius develops simple and not so simple properties of conic sections, many of which we now only know through calculus. We will also attempt to illustrate the propositions in Conics using the powerful mathematics software Mathematica.

Linear Algebra

Affine Transformations and Homogeneous Coordinates :

In this project we will look at geometric transformations using homogeneous coordinates and matrices. Affine transformations include translation, rotation, reflection, shear, expansion/contraction, and similarity transformations. This will show the student the relationship between high school geometry, linear algebra, and group theory. We will also illustrate properties in geometry and linear algebra using the powerful mathematics software Mathematica or MATLAB.

Approximation of Functions with Simpler Functions :

In this project we will use functions with "nice" properties to approximate other functions. An example that might be familiar to students is using polynomials to approximate certain functions via Taylor/Maclaurin series. Eventually we will look at families of so-called “orthogonal functions” and how they are used to approximate other functions. We will use the powerful mathematics software Mathematica to illustrate the approximations.

Least is the Best :

A common concern in industry is optimization: minimizing the cost, maximizing the profit, optimizing resource utilization, and so on. Students learn basic optimization techniques in calculus courses. But to what is it applied? What if the objective function is non-differentiable? What if variables are discrete? In this project, students can choose their preferred "no-so-nice" application and explore heuristic approaches to estimate the optimum and the optimizer.

The kinematics of rolling. (Riemannian geometry/Non-holonomic mechanical systems) :

On a smooth stone, draw a curve beginning at a point p, and hold the stone over a flat table with p as the point of contact. Now roll the stone over the plane of the table so that at all times the point of contact lies on the curve, being careful not to allow the stone to slip or twist. We may equally well think that we are rolling the plane of the table over the surface of the stone along the given curve. Mechanical systems with this type of motion are said to have "non-holonomic" constraints, and are common fare in mechanics textbooks.

Now imagine a tangent vector to the plane at p. This rolling of the plane over the surface provides a way to transport v along the curve, keeping it tangent at all times. The resulting vector field over the curve is said to be a "parallel" vector field. Show that there is a unique way to carry out this parallel translation. (Find a differential equation that describes the parallel vector field and use some appropriate existence and uniqueness theorem.) Let c be a short path joining p and q, whose velocity vector field is parallel. Show that c is the shortest path contained in the surface that joins p and q.

Whether or not you fully succeed, this mechanical idea will give you a concrete way of thinking about ideas in differential geometry that might seem a bit abstract at first, such as Levi-Civita connection, parallel translation, geodesics, etc. Also look for an engineering text on Robotic manipulators and explain why such non-holonomic mechanical systems are important in that area of engineering.

I don't know of many places where these things are explained in a simple way. Perhaps Geometric Control Theory by Velimir Jurdjevic is a place to start. In the engineering literature, A Mathematical Introduction to Robotic Manipulation is a particularly good reference.

Geometry in very high dimensions. (Convex geometry)

Geometry in very high dimensions is full of surprises. Consider the following easy exercise as a warm-up. Let $B(n,r)$ represent the ball of radius $r$, centered at the origin, in Euclidian n-space. Show that for arbitrarily small positive numbers $a$ and $b$, there is a big enough $N$ such that $(100 - a)\%$ of the volume of $B(n,r)$ is contained in the shell $B(n,r) - B(n,r - b)$ for all $n > N$.

Here is a much more surprising fact that you might like to think about. Let $S(n-1)$ denote the sphere of radius 1 in dimension $n$. (It is the boundary of $B(n,1 )$.) Let f be a continuous function from $S(n-1)$ into the real line that does not increase distances, that is, $| f(p) - f(q) |$ is not bigger than $| p - q |$ for any two points $p$ and $q$ on the sphere. ($f$ is said to be a "1-Lipschitz" function.) Then there exists a number $M$ such that, for all positive $a$, no matter how small, the set of points $p$ in $S(n-1)$ such that $| f(p) - M |>a$ has volume smaller than $\exp(-na^2 / 2 )$. In words, this means that, taking away a set with very small volume (if the dimension is very large), $f$ is very nearly a constant function, equal to $M$.

This is much more than a geometric curiosity. In fact, such concentration of volume phenomenon is at the heart of statistics, for example. To make the point, consider the following. Let $S(n-1, n^{0.5})$ be the sphere in n-space whose radius is the $\sqrt{n}$. Let f denote the orthogonal projection from the sphere to one of the $n$ coordinate directions, which we agree to call the x-direction. Show that the part of the sphere that projects to an interval $a < x < b$ has volume very nearly (when $n$ is big) equal to the integral from $a$ to $b$ of the standard normal distribution. (This is easy to show if you use the central limit theorem).

For a nice introduction to this whole subject, see the article by Keith M. Ball in the volume Flavors of Geometry, Cambridge University Press, Ed.: S. Levy, 1997.

Hodge theory and Electromagnetism. (Algebraic topology/Physics)

Electromagnetic theory since the time of Maxwell has been an important source of new mathematics. This is particularly true for topology, specially for what is called "algebraic topology". One fundamental topic in algebraic topology with strong ties to electromagnetism is the so called "Hodge-de Rham theory". Although in its general form this is a difficult and technical topic, it is possible to go a long way into the subject with only Math 233. The article "Vector Calculus and the Topology of Domains in 3-Space", by Cantarella, DeTurck and Gluck (The American Mathematical Monthly, V. 109, N. 5, 409-442) is the ideal reference for a project in this area. (It has as well some inspiring pictures.)

Another direction to explore is the theory of direct current electric circuits (remember Kirkhoff's laws?). In fact, an electric circuit may be regarded as electric and magnetic field over a region in 3-space that is very nearly one dimensional, typically with very complicated topology (a graph). Solving circuit problems implicitly involve the kind of algebraic topology related to Hodge theory. (Hermann Weyl may have been the first to look into electric circuits from this point of view.) The simplification here is that the mathematics involved reduces to finite dimensional linear algebra. A nice reference for this is appendix B of The Geometry of Physics (T. Frankel), as well as A Course in Mathematics for Studentsof Physics vol. 2, by Bamberg and Sternberg.

Symmetries of differential equations. (Lie groups, Lie algebras/Differential equations)

Most of the time spent in courses on ODEs is devoted to linear differential equations, although a few examples of non-linear equations are also mentioned, only to be quickly dismissed as odd cases that cannot be approached by any general method for finding solutions. (One good and important example is the Riccati equation.) It turns out that there is a powerful general method to analyze nonlinear equations that sometimes allows you to obtain explicit solutions. The method is based on looking first for all the (infinitesimal) symmetries of the differential equation. (A symmetry of a differential equation is a transformation that sends solutions to solutions. An infinitesimal symmetry is a vector field that generates a flow of symmetries.) The key point is that finding infinitesimal symmetries amounts to solving linear differential equations and may be a much easier problem than to solve the equation we started with.

Use this idea to solve the Riccati equation. Choose your favorite non-linear differential equation and study its algebra of infinitesimal symmetries (a Lie algebra). What kind of information do they provide about the solutions of the equation? Since my description here is hopelessly vague, you might like to browse Symmetry Methods for Differential Equations - A Beginner's Guide by Peter Hydon, Cambridge University Press. It will give you a good idea of what this is all about.

Riemann surfaces and optical metric. (Riemannian geometry/Optics)

Light propagates in a transparent medium with velocity $c/n$, where c is a constant and n is the so called "refractive index" -- a quantity that can vary from point to point depending on the electric and magnetic properties of the medium. For a given curve in space, the time an imaginary particle would take to traverse its length, having at each point the same speed light would have there, is called the "optical length" of the curve. Therefore, the optical length is the line integral of n/c along the curve with respect to the arc-length parameter. According to Fermat's principle, the actual path taken by a light ray in space locally minimizes the "optical length". It is possible to use the optical length (for some given function n) to defined a new geometry whose geodesic curves are the paths taken by light rays. This is a particular type of Riemannian geometry, called "conformally" Euclidian. All this also makes sense in dimension 2.

One of the most famous paintings of Escher show a disc filled with little angels and demons crowding towards the boundary circle. What refractive index would produce the metric distortions shown in that picture?

A fundamental result about the geometry of surfaces states that, no matter what shape they have, you can always find a coordinate system in a neighborhood of any point that makes the surface conformally Euclidian. Why is this so? (This will require that you learn something about so called "isothermal coordinates".)

Failure of von Neumann's inequality.

Von Neumann proved that if A is a contractive matrix (has operator norm $\leq 1$) and $p(z)$ is a complex polynomial, then $p(A)$ has operator norm bounded by the supremum of $p$ on the unit circle. A two variable version of this result is true (Andô's inequality) but the three variable version is false. Counterexamples can be shown to exist either through probabilistic arguments (i.e. a random polynomial will fail the inequality) and there are also a few examples constructed through ad hoc methods. This project would involve trying to construct more interesting families of counterexamples to the three variable von Neumann inequality in order to understand "how badly" the inequality fails.

Multilinear Bohnenblust-Hille inequality

This is a different kind of inequality for polynomials. Multilinear polynomials satisfy an inequality bounding certain little $l^p$ norms of their coefficients by the supremum norm of the polynomial. This project would also involve looking for interesting examples to test the sharpness of known versions of this inequality.

algebra project 1

If $a_1,a_2,...a_n$ are integers with $gcd = 1$, then the Eulidean algorithm implies that there exists a $n \times n$-matrix $A$ with integer entries, with first row $= (a_1,a_2,...,a_n)$, and such that $\det(A) = 1$. A similar question was raised by J.P. Serre for polynomial rings over a field, with the a's being polynomials in several variables. This fundamental question generated an enormous amount of mathematics (giving birth to some new fields) and was finally settled almost simultaneously by D. Quillen and A. A. Suslin, independently. Now, there are fairly elementary proofs of this which require only some knowledge of polynomials and a good background in linear algebra. This could be an excellent project for someone who wants to learn some important and interesting mathematics. (These results seem to be of great interest to people working in control theory.)

algebra project 2

A basic question in number theory and theoretical computer science is to find a "nice" algorithm to decide whether a given number is prime or not. This has important applications in secure transmissions over the internet and techniques like RSA cryptosystems. Of course, the ancient method of Eratosthenes (sieve method) is one such algorithm, albeit a very inefficient one. All the methods availabe so far has been known to take exponential time. There are probabilistic methods to determine whether a number is prime, which take only polynomial time. The drawback is that there is a small chance of error in these methods. So, computer scientists have been trying for the last decade to find a deterministic algorithm which works in polynomial time. Recently, this has been achieved by three scientists from IIT, Kanpur, India. A copy of their article can be downloaded from www.cse.iitk.ac.in A nice project would be to understand their arguments (which are very elementary and uses only a little bit of algebra and number theory) and maybe to do a project on the history of the problem and its ramifications.

• VT past projects
• UMD past projects
• UC Berkeley past projects
• NYU project ideas
• NSF research experience for undergrads
• Washington State past projects
• Cornell REU
• U of Mary Washington opportunities, conferences, and past projects
• A few examples from Stanford and how to enroll if you are at Stanford
• BYU mentor section at the bottom has projects ideas

Not an answer, just one contribution.

The William's College SMALL summer program is an impressive model, at the high-end. "Around 500 students have participated in the project since its inception in 1988."

There are six areas this (2015) summer: Arithmetic Combinatorics (Leo Goldmakher), Combinatorial Geometry (Satyan Devadoss), Commutative Algebra (Susan Loepp), Geometry (Frank Morgan), Hyperbolic Knots (Colin Adams) and Number Theory & Harmonic Analysis (Steven Miller and Eyvi Palsson).

Here is a link for the project abstracts . Past projects have resulted in an impressive number of published papers .

• $\begingroup$ Thank you, Professor. As always, your contributions are precious. If you happen to find out other similar collections, please, do not hesitate to add them. $\endgroup$ –  Dal Commented Dec 31, 2014 at 10:48

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Research Topics

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Algebra and algebraic geometry.

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Discrete mathematics, group overview, harmonic analysis.

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UCI Mathematics

Math undergraduate research resources.

This page was designed to assist mathematics majors to find research on-campus opportunities with UCI faculty or off-campus opportunities with industrial partners, national labs, and other universities. It was last updated on 10/26/2019.

On-Campus Opportunities

1.1. Faculty research interests 1.2. MCBU 1.3. UROP/SURP 1.4. Math 199 1.5. Faculty grants

Off-Campus Opportunities

2.1. Internships 2.2. REUs 2.3. Other mathematics enhancement programs

FAQ’S about Undergraduate Mathematical Research

3.1. How to find faculty mentors at UCI 3.2. How to apply for external programs 3.3. Who is eligible

The UROP Office and their Services

1. On-Campus Opportunities

1.1 faculty research interests.

The faculty in the UCI mathematics department span a wide spectrum of interests and expertise. Active areas of research include: Applied and Computational Mathematics,  Ergodic Theory and Dynamical Systems, Differential Geomtry and Topology, Image Problems and Imaging, Logic and Foundations, Mathematical and Computational Biology, Mathematical Physics, Number Theory, Partial Differential Equations and Probability. 

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1.2 MCBU - Mathematical and Computational Biology for Undergraduates

MCBU is an NSF-funded program for training and research for UCI undergraduate students in mathematics and biology. The program provides an opportunity for undergraduate students to be trained to work at the interface of mathematics, biology and computation. Students have the opportunity to perform undergraduate research in paired teams of mathematics and biology students. Students will create laboratory data and utilize mathematical and computational tools to analyze their data to address a real world research problem.

Students interested in the MCBU summer research program should first enroll in Math 113A, offered every Spring quarter.

1.3 UROP/SURP

The UCI Undergraduate Research Opportunities Program provides funding for undergraduate research and creative projects mentored by UCI faculty through two separate annual Calls for Proposals , one in the Fall Quarter and one in the Spring Quarter. Once the Calls have been announced, students have approximately one month to submit their proposals. Proposals are evaluated based on the intellectual merit of the student’s research, a complete application, the level of support from the faculty mentor, the student’s transcript, and available funding. Samples of UROP proposals are available on the UROP website .

Students who receive a UROP grant must present their findings at the annual UCI Undergraduate Research Symposium held in May, and are invited to submit their research findings to the UCI Undergraduate Research Journal .

To be eligible for a UROP grant, students must be undergraduates in good academic standing. Students who will graduate within a quarter of submitting their proposal are not eligible to apply. Proposals that do not require funding or are already receiving adequate funding from departmental or other sources may be submitted for an Honorary Fellowship.

The UCI Summer Undergraduate Research Progra m (SURP) provides funding to support students’ research during the summer. Students are given the opportunity to become immersed in a research topic for a full-time ten-week period, or the equivalent of 400 hours, and receive a maximum stipend of $3,000.

To be eligible for a SURP grant, students must be undergraduates in good academic standing. Students must also have been involved in a faculty-mentored research project or creative activity for at least one quarter before the beginning of the Summer (Spring Quarter involvement is acceptable). Students who will graduate within a quarter of submitting their proposal are not eligible to apply. Evaluation criteria for SURP proposal are similar to the UROP’s ones. Moreover, just like for UROP, students who receive a SURP grant must present their findings at the annual UCI Undergraduate Research Symposium held in May are invited to submit their research findings to The UCI Undergraduate Research Journal.

1.4 Math 199

Math 199A-B-C (Special Studies in Mathematics) is a 4-units course designed for outstanding undergraduate mathematics majors who want to be engaged in supervised but independent reading of a mathematical topic or in research work. A student interested in Math 199 needs to register with one particular faculty member. Once the consent of the instructor is obtained, the student can enroll in the course without any additional authorization (following the standard procedures for signing up for a course).

1.5 Faculty grants

Some faculty occasionally have personal grants to support undergraduate student research.

2. Off-Campus Research Opportunities

Several off-campus research opportunities are available. They mainly fall into three categories: internships, REUs (summer programs focused on a single research topic under faculty guidance with a small group of great students) and other mathematics enhancement programs (e.g., programs aimed at prepare students for graduate school).

Most of these programs pay travel, room and board plus a student stipend.

2.1 Internships

Details to be updated soon.

Research Experiences for Undergraduates (or REUs) are among the most prestigious and most competitive summer research programs for undergraduates studying mathematics.

Individual REU sites typically consist of ten undergraduates working on a very specialized math program for 6 to 8 weeks, under direct supervision of some faculty members. Rom and board, along with a stipend for the student are generally provided.

As the program is funded by the NSF, undergraduates must be citizens or permanent residents of the US or its possessions. Applications are typically due between February and March. The length of the application ranges from a single letter of reference without supporting materials all the way up to something comparable to a college admissions application. The programs generally require between one and three letters of reference, a transcript, 0-2 essays, a letter of interest, a resume, a biographical form, or some combination thereof.

Directory of active REU sites  

2.3 Other mathematics enhancement programs

The following list of programs was updated on October 26, 2019

Semester Programs (Domestic And International)

• Budapest Semester in Math
• Math in Moscow
• Penn State’s MASS (Mathematics Advanced Study Semesters) Program

Summer Programs (Domestic And International)

• Park City Math Institute Summer Program for Undergraduates
• University of Nebraska Summer IMMERSE program
• The Summer Math Institute at Cornell University
• The Summer Applied Mathematics Institute at Carnegie Mellon
• UCLA Undergraduate Research center
• The Mathematical and Theoretical Biology Institute Summer Program
• NIH sponsored Summer Institutes for Training in Biostatistics
• At North Carolina State University www.stat.ncsu.edu/sibs/

Specifically for women

• EDGE Summer Program for Women

3. FAQ’S Questions About Undergraduate Mathematical Research

Below you will find answers to a number of frequently asked questions regarding undergraduate research in mathematics. For more information please attend the REU workshop which is organized every year by the department.

3.1 How to find faculty mentors at UCI

Here is some advice:

• Look at faculty research interests
• Take appropriate courses first
• Read math outside of class
• Identify your area of research interests
• Describe yourself- grades, coursework, goals, etc.
• Show you know something about their research area
• Explain what extent of research you desire
• Show your motivation, work ethic, independence, etc.
• For reading courses, explain why you are interested in that topic and how it will fit with your future goals
• Remember you are asking for a giant favor. Research and creative projects require dedication, planning, and a substantial time committment. 

3.2 How to apply for external programs

Basic steps:

• Carefully look at program descriptions
• Follow all application instructions
• Ask for letters of recommendation at least two weeks in advance
• Tailor it to the particular program
• Explain your interest in topic and preparation for it
• Emphasize your unique math experiences
• Deadlines around mid-February.

For tips about requesting a recommendation letter or writing a personal statement please check the Math Grad School Resources page on our website

3.3 Who is eligible?

• Most research opportunities are highly competitive and intended for the advanced math students going to graduate school
• Appropriate preparation is essential for a successful research experience
• Summer REU’s are mostly for students finishing their junior year (maybe sophomore)
• Math 199 course is mainly for seniors in honor’s program or considering grad school
• Some faculty just don’t work with undergrads, because it involves a big time commitment. Don’t take it personally!

4. The UROP Office and their Services

The Undergraduate Research Opportunities Program  (UROP) in the Division of Undergraduate Education encourages and facilitates research and creative activities by undergraduates from all schools and academic disciplines at UCI. On the UROP website you will find sample proposals and guidelines for the UROP/SURP grants, and a long list of  on-campus  and  off-campus research opportunities, including internships, research experiences and fellowships.

The UROP office sponsors a yearly  symposium  for undergraduate research and organizes a series of research-related workshops  for undergraduates throughout the year. Last but not least, UROP offers assistance to students and faculty through all phases of the research process, whether it is with proposal writing, developing research plans through project management skills, awarding grants to fund research projects, scholarly journal writing through  The UCI Undergraduate Research Journal , or presenting results of the research or creative project through the UCI Undergraduate Research Symposium. 

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

Algebra, Combinatorics, and Geometry

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

Analysis and Partial Differential Equations

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

Applied Analysis

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

Mathematical Biology

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

Mathematical Finance

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

Numerical Analysis and Scientific Computing

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

Topology and Differential Geometry

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

Undergraduate Research and Reading Programs

Undergraduate students at MIT Mathematics Department have several opportunities to participate in mathematical research and directed reading. Four programs dedicated to cultivating research with the guidance of graduate students and faculty are:

• SPUR - Summer Program in Undergraduate Research
• DRP - Directed Reading Program during IAP in January
• UROP - Undergraduate Research Opportunities Program
• UROP+ - Supervised UROP Summer Program
• MSRP - MIT Summer Research Program

Undergraduates also have an opportunity to lead recitations, grade, and tutor in mathematics. For more information visit Math Academic Services .

College of Arts and Sciences

Department of Mathematics

• Undergraduate Research

Are you interested in doing undergraduate research in mathematics, attend a conference, and publish your journal paper? If so, follow the links to the below for information about undergraduate research opportunities, conferences, and journals.

Research Opportunities

• Summer Science Institute (University of Mary Washington)
• Research Experiences for Undergraduates (National Science Foundation)
• REU Summer Programs (American Mathematical Society)
• Undergraduate Research (Mathematical Association of America)
• Summer Mathematics Program for Women Undergraduates (Carleton College)
• Summer Program for Women in Mathematics (George Washington University)
• Summer Undergraduate Mathematical Sciences Research Institute (Miami University)
• Mathematics Advanced Study Semesters Program (Penn State University)
• Mathematics Program in English for Undergraduates (University of Moscow)
• Budapest Semesters in Mathematics

Undergraduate research conferences

• Posters on the Hill 2020 (applications open 9/4 – 11/5, 2019)
• Network for Undergraduate Research in VA Conference (submission deadline is 11/1/2019)
• Rose-Hulman Undergraduate Mathematics Conference
• National Conferences on Undergraduate Research
• Illiana Undergraduate Mathematics Research Conference
• Nebraska Conference for Undergraduate Women in Mathematics
• Butler Undergraduate Research Conference

Journals of Undergraduate Research in Mathematics

• Minnesota Journal of Undergraduate Mathematics
• Rose-Hulman Undergraduate Mathematics Journal
• Involve: A Journal of Mathematics
• Missouri Journal of Mathematical Sciences
• SIAM Undergraduate Research Online
• Journal of Young Investigators
• Pi Mu Epsilon Journal
• Furman University Electronic Journal of Undergraduate Mathematics
• Mathematics Exchange
• PUMP Journal of Undergraduate Research
• Pursue: Undergraduate Research Journal

Below is a list of Mathematics faculty members at UMW involved with undergraduate research, with topics of interest.

Randall Helmstutler:

My undergraduate research interests lie primarily in generalizations of parts of abstract algebra and topology.  My true research interests are in an area called category theory, which is an attempt to give axiomatic approaches that unify algebra, topology, and other parts of purely theoretical mathematics.  I have had several students complete projects in category theory and would be interested in doing more undergraduate research in this area.  To see examples of my former students’ work, visit my student research page .

Debra L. Hydorn:

I am interested in working with students on projects in multivariate statistics (e.g., regression analysis, analysis of variance, variable reduction techniques).  Most projects will include the derivation of some theoretical results and simulations to verify distributional properties.  I am also interested in working with students in the use of spatial statistics, particularly in environmental applications.  Some course work in statistics is needed, preferably MATH 381, and some background in programming is helpful but not required.  To learn more about the kinds of projects I have mentored please visit my undergraduate research page .

Jangwoon “Leo” Lee:

I have various undergraduate research problems in applied mathematics. In a project, for instance, you would consider a mathematical model equation (e.g., differential equations or (stochastic) partial differential equations) of physical phenomenon. Then you would try to solve it by some mathematical methods (if possible) and by numerical methods using programming languages such as MATLAB (including built-in solvers in MATLAB) to make predictions about how the physical phenomenon will behave in different circumstances and/or evaluate the performance of numerical solvers of the model equation. In this project, you would learn how to analyze a real-life problem using the mathematical language that is used by many people in industry and government. This would be a great experience and also would give a great opportunity to you who may be interested in seeking employment at places like Dahlgren. My former students’ projects can be found in here .

Suzanne Sumner:

I have two projects in mind for undergraduate research.

1. Competing Species Models

Competing species forestry models use differential equations to examine the long-term population levels with two types of species of trees. These species are either pioneer (those trees that are deprivation intolerant) and climax (those trees that receive a benefit from having other trees nearby.) In particular, the primary concern is when the long-term tree population values level off or fluctuate. Hopf bifurcations mark a change in the dynamics from stable to unstable scenarios.

2. Honey Bee Biology Models

In recent years honey bees have been parasitized by two species of mites, the tracheal mite and the varroa mite. These mites have decimated numerous colonies, severely impacting the areas of agriculture that rely on honey bee pollination. Difference equation models treat the parasitism as a Susceptible-Infected-Removed (SIR) model as in earlier work by Dr. Wyatt A. Mangum. Some bees carry desirable traits that allow them to withstand mite infestation. Here the primary concern is determining the proportion of bees with these traits that a colony must have so that the overall percent infestation decreases. Bifurcation surfaces separate unstable scenarios where percent mite infestation increases from stable situations where percent infestation decreases.

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Megamenu Global

Megamenu featured, megamenu social, math/stats thesis and colloquium topics.

Updated: April 2024

Math/Stats Thesis and Colloquium Topics 2024- 2025

The degree with honors in Mathematics or Statistics is awarded to the student who has demonstrated outstanding intellectual achievement in a program of study which extends beyond the requirements of the major. The principal considerations for recommending a student for the degree with honors will be: Mastery of core material and skills, breadth and, particularly, depth of knowledge beyond the core material, ability to pursue independent study of mathematics or statistics, originality in methods of investigation, and, where appropriate, creativity in research.

An honors program normally consists of two semesters (MATH/STAT 493 and 494) and a winter study (WSP 031) of independent research, culminating in a thesis and a presentation. Under certain circumstances, the honors work can consist of coordinated study involving a one semester (MATH/STAT 493 or 494) and a winter study (WSP 030) of independent research, culminating in a “minithesis” and a presentation. At least one semester should be in addition to the major requirements, and thesis courses do not count as 400-level senior seminars.

An honors program in actuarial studies requires significant achievement on four appropriate examinations of the Society of Actuaries.

Highest honors will be reserved for the rare student who has displayed exceptional ability, achievement or originality. Such a student usually will have written a thesis, or pursued actuarial honors and written a mini-thesis. An outstanding student who writes a mini-thesis, or pursues actuarial honors and writes a paper, might also be considered. In all cases, the award of honors and highest honors is the decision of the Department.

Here is a list of possible colloquium topics that different faculty are willing and eager to advise. You can talk to several faculty about any colloquium topic, the sooner the better, at least a month or two before your talk. For various reasons faculty may or may not be willing or able to advise your colloquium, which is another reason to start early.

RESEARCH INTERESTS OF MATHEMATICS AND STATISTICS FACULTY

Here is a list of faculty interests and possible thesis topics.  You may use this list to select a thesis topic or you can use the list below to get a general idea of the mathematical interests of our faculty.

Colin Adams (On Leave 2024 – 2025)

Research interests:   Topology and tiling theory.  I work in low-dimensional topology.  Specifically, I work in the two fields of knot theory and hyperbolic 3-manifold theory and develop the connections between the two. Knot theory is the study of knotted circles in 3-space, and it has applications to chemistry, biology and physics.  I am also interested in tiling theory and have been working with students in this area as well.

Hyperbolic 3-manifold theory utilizes hyperbolic geometry to understand 3-manifolds, which can be thought of as possible models of the spatial universe.

Possible thesis topics:

• Investigate various aspects of virtual knots, a generalization of knots.
• Consider hyperbolicity of virtual knots, building on previous SMALL work. For which virtual knots can you prove hyperbolicity?
• Investigate why certain virtual knots have the same hyperbolic volume.
• Consider the minimal Turaev volume of virtual knots, building on previous SMALL work.
• Investigate which knots have totally geodesic Seifert surfaces. In particular, figure out how to interpret this question for virtual knots.
• Investigate n-crossing number of knots. An n-crossing is a crossing with n strands of the knot passing through it. Every knot can be drawn in a picture with only n-crossings in it. The least number of n-crossings is called the n-crossing number. Determine the n-crossing number for various n and various families of knots.
• An übercrossing projection of a knot is a projection with just one n-crossing. The übercrossing number of a knot is the least n for which there is such an übercrossing projection. Determine the übercrossing number for various knots, and see how it relates to other traditional knot invariants.
• A petal projection of a knot is a projection with just one n-crossing such that none of the loops coming out of the crossing are nested. In other words, the projection looks like a daisy. The petal number of a knot is the least n for such a projection. Determine petal number for various knots, and see how it relates to other traditional knot invariants.
• In a recent paper, we extended petal number to virtual knots. Show that the virtual petal number of a classical knot is equal to the classical petal number of the knot (This is a GOOD question!)
• Similarly, show that the virtual n-crossing number of a classical knot is equal to the classical n-crossing number. (This is known for n = 2.)
• Find tilings of the branched sphere by regular polygons. This would extend work of previous research students. There are lots of interesting open problems about something as simple as tilings of the sphere.
• Other related topics.

Possible colloquium topics : Particularly interested in topology, knot theory, graph theory, tiling theory and geometry but will consider other topics.

Christina Athanasouli

Research Interests:   Differential equations, dynamical systems (both smooth and non-smooth), mathematical modeling with applications in biological and mechanical systems

My research focuses on analyzing mathematical models that describe various phenomena in Mathematical Neuroscience and Engineering. In particular, I work on understanding 1) the underlying mechanisms of human sleep (e.g. how sleep patterns change with development or due to perturbations), and 2) potential design or physical factors that may influence the dynamics in vibro-impact mechanical systems for the purpose of harvesting energy. Mathematically, I use various techniques from dynamical systems and incorporate both numerical and analytical tools in my work. 

Possible colloquium topics:   Topics in applied mathematics, such as:

• Mathematical modeling of sleep-wake regulation
• Mathematical modeling vibro-impact systems
• Bifurcations/dynamics of mathematical models in Mathematical Neuroscience and Engineering
• Bifurcations in piecewise-smooth dynamical systems

Julie Blackwood

Research Interests:   Mathematical modeling, theoretical ecology, population biology, differential equations, dynamical systems.

My research uses mathematical models to uncover the complex mechanisms generating ecological dynamics, and when applicable emphasis is placed on evaluating intervention programs. My research is in various ecological areas including ( I ) invasive species management by using mathematical and economic models to evaluate the costs and benefits of control strategies, and ( II ) disease ecology by evaluating competing mathematical models of the transmission dynamics for both human and wildlife diseases.

• Mathematical modeling of invasive species
• Mathematical modeling of vector-borne or directly transmitted diseases
• Developing mathematical models to manage vector-borne diseases through vector control
• Other relevant topics of interest in mathematical biology

Each topic (1-3) can focus on a case study of a particular invasive species or disease, and/or can investigate the effects of ecological properties (spatial structure, resource availability, contact structure, etc.) of the system.

Possible colloquium topics:   Any topics in applied mathematics, such as:

Research Interest :  Statistical methodology and applications.  One of my research topics is variable selection for high-dimensional data.  I am interested in traditional and modern approaches for selecting variables from a large candidate set in different settings and studying the corresponding theoretical properties. The settings include linear model, partial linear model, survival analysis, dynamic networks, etc.  Another part of my research studies the mediation model, which examines the underlying mechanism of how variables relate to each other.  My research also involves applying existing methods and developing new procedures to model the correlated observations and capture the time-varying effect.  I am also interested in applications of data mining and statistical learning methods, e.g., their applications in analyzing the rhetorical styles in English text data.

• Variable selection uses modern techniques such as penalization and screening methods for several different parametric and semi-parametric models.
• Extension of the classic mediation models to settings with correlated, longitudinal, or high-dimensional mediators. We could also explore ways to reduce the dimensionality and simplify the structure of mediators to have a stable model that is also easier to interpret.
• We shall analyze the English text dataset processed by the Docuscope environment with tools for corpus-based rhetorical analysis. The data have a hierarchical structure and contain rich information about the rhetorical styles used. We could apply statistical models and statistical learning algorithms to reduce dimensions and gain a more insightful understanding of the text.

Possible colloquium topics:  I am open to any problems in statistical methodology and applications, not limited to my research interests and the possible thesis topics above.

Richard De Veaux 

Research interests: Statistics.

My research interests are in both statistical methodology and in statistical applications.  For the first, I look at different methods and try to understand why some methods work well in particular settings, or more creatively, to try to come up with new methods.  For the second, I work in collaboration with an investigator (e.g. scientist, doctor, marketing analyst) on a particular statistical application.  I have been especially interested in problems dealing with large data sets and the associated modeling tools that work for these problems.

• Human Performance and Aging.I have been working on models for assessing the effect of age on performance in running and swimming events. There is still much work to do. So far I’ve looked at masters’ freestyle swimming and running data and a handicapped race in California, but there are world records for each age group and other events in running and swimming that I’ve not incorporated. There are also many other types of events.
• Variable Selection.  How do we choose variables when we have dozens, hundreds or even thousands of potential predictors? Various model selection strategies exist, but there is still a lot of work to be done to find out which ones work under what assumptions and conditions.
• Problems at the interface.In this era of Big Data, not all methods of classical statistics can be applied in practice. What methods scale up well, and what advances in computer science give insights into the statistical methods that are best suited to large data sets?
• Applying statistical methods to problems in science or social science.In collaboration with a scientist or social scientist, find a problem for which statistical analysis plays a key role.

Possible colloquium topics:

• Almost any topic in statistics that extends things you’ve learned in courses —  specifically topics in Experimental design, regression techniques or machine learning
• Model selection problems

Thomas Garrity (On Leave 2024 – 2025)

Research interest:   Number Theory and Dynamics.

My area of research is officially called “multi-dimensional continued fraction algorithms,” an area that touches many different branches of mathematics (which is one reason it is both interesting and rich).  In recent years, students writing theses with me have used serious tools from geometry, dynamics, ergodic theory, functional analysis, linear algebra, differentiability conditions, and combinatorics.  (No single person has used all of these tools.)  It is an area to see how mathematics is truly interrelated, forming one coherent whole.

While my original interest in this area stemmed from trying to find interesting methods for expressing real numbers as sequences of integers (the Hermite problem), over the years this has led to me interacting with many different mathematicians, and to me learning a whole lot of math.  My theses students have had much the same experiences, including the emotional rush of discovery and the occasional despair of frustration.  The whole experience of writing a thesis should be intense, and ultimately rewarding.   Also, since this area of math has so many facets and has so many entrance points, I have had thesis students from wildly different mathematical backgrounds do wonderful work; hence all welcome.

• Generalizations of continued fractions.
• Using algebraic geometry to study real submanifolds of complex spaces.

Possible colloquium topics:   Any interesting topic in mathematics.

Leo Goldmakher

Research interests:   Number theory and arithmetic combinatorics.

I’m interested in quantifying structure and randomness within naturally occurring sets or sequences, such as the prime numbers, or the sequence of coefficients of a continued fraction, or a subset of a vector space. Doing so typically involves using ideas from analysis, probability, algebra, and combinatorics.

Possible thesis topics:  

Anything in number theory or arithmetic combinatorics.

Possible colloquium topics:   I’m happy to advise a colloquium in any area of math.

Susan Loepp

Research interests: Commutative Algebra.  I study algebraic structures called commutative rings.  Specifically, I have been investigating the relationship between local rings and their completion.  One defines the completion of a ring by first defining a metric on the ring and then completing the ring with respect to that metric.  I am interested in what kinds of algebraic properties a ring and its completion share.  This relationship has proven to be intricate and quite surprising.  I am also interested in the theory of tight closure, and Homological Algebra.

Topics in Commutative Algebra including:

• Using completions to construct Noetherian rings with unusual prime ideal structures.
• What prime ideals of C[[ x 1 ,…, x n ]] can be maximal in the generic formal fiber of a ring? More generally, characterize what sets of prime ideals of a complete local ring can occur in the generic formal fiber.
• Characterize what sets of prime ideals of a complete local ring can occur in formal fibers of ideals with height n where n ≥1.
• Characterize which complete local rings are the completion of an excellent unique factorization domain.
• Explore the relationship between the formal fibers of R and S where S is a flat extension of R .
• Determine which complete local rings are the completion of a catenary integral domain.
• Determine which complete local rings are the completion of a catenary unique factorization domain.

Possible colloquium topics:   Any topics in mathematics and especially commutative algebra/ring theory.

Steven Miller

For more information and references, see http://www.williams.edu/Mathematics/sjmiller/public_html/index.htm

Research interests :  Analytic number theory, random matrix theory, probability and statistics, graph theory.

My main research interest is in the distribution of zeros of L-functions.  The most studied of these is the Riemann zeta function, Sum_{n=1 to oo} 1/n^s.  The importance of this function becomes apparent when we notice that it can also be written as Prod_{p prime} 1 / (1 – 1/p^s); this function relates properties of the primes to those of the integers (and we know where the integers are!).  It turns out that the properties of zeros of L-functions are extremely useful in attacking questions in number theory.  Interestingly, a terrific model for these zeros is given by random matrix theory: choose a large matrix at random and study its eigenvalues.  This model also does a terrific job describing behavior ranging from heavy nuclei like Uranium to bus routes in Mexico!  I’m studying several problems in random matrix theory, which also have applications to graph theory (building efficient networks).  I am also working on several problems in probability and statistics, especially (but not limited to) sabermetrics (applying mathematical statistics to baseball) and Benford’s law of digit bias (which is often connected to fascinating questions about equidistribution).  Many data sets have a preponderance of first digits equal to 1 (look at the first million Fibonacci numbers, and you’ll see a leading digit of 1 about 30% of the time).  In addition to being of theoretical interest, applications range from the IRS (which uses it to detect tax fraud) to computer science (building more efficient computers).  I’m exploring the subject with several colleagues in fields ranging from accounting to engineering to the social sciences.

Possible thesis topics: 

• Theoretical models for zeros of elliptic curve L-functions (in the number field and function field cases).
• Studying lower order term behavior in zeros of L-functions.
• Studying the distribution of eigenvalues of sets of random matrices.
• Exploring Benford’s law of digit bias (both its theory and applications, such as image, voter and tax fraud).
• Propagation of viruses in networks (a graph theory / dynamical systems problem). Sabermetrics.
• Additive number theory (questions on sum and difference sets).

Possible colloquium topics: 

Plus anything you find interesting.  I’m also interested in applications, and have worked on subjects ranging from accounting to computer science to geology to marketing….

Ralph Morrison

Research interests:   I work in algebraic geometry, tropical geometry, graph theory (especially chip-firing games on graphs), and discrete geometry, as well as computer implementations that study these topics. Algebraic geometry is the study of solution sets to polynomial equations.  Such a solution set is called a variety.  Tropical geometry is a “skeletonized” version of algebraic geometry. We can take a classical variety and “tropicalize” it, giving us a tropical variety, which is a piecewise-linear subset of Euclidean space.  Tropical geometry combines combinatorics, discrete geometry, and graph theory with classical algebraic geometry, and allows for developing theory and computations that tell us about the classical varieties.  One flavor of this area of math is to study chip-firing games on graphs, which are motivated by (and applied to) questions about algebraic curves.

Possible thesis topics : Anything related to tropical geometry, algebraic geometry, chip-firing games (or other graph theory topics), and discrete geometry.  Here are a few specific topics/questions:

• Study the geometry of tropical plane curves, perhaps motivated by results from algebraic geometry.  For instance:  given 5 (algebraic) conics, there are 3264 conics that are tangent to all 5 of them.  What if we look at tropical conics–is there still a fixed number of tropical conics tangent to all of them?  If so, what is that number?  How does this tropical count relate to the algebraic count?
• What can tropical plane curves “look like”?  There are a few ways to make this question precise.  One common way is to look at the “skeleton” of a tropical curve, a graph that lives inside of the curve and contains most of the interesting data.  Which graphs can appear, and what can the lengths of its edges be?  I’ve done lots of work with students on these sorts of questions, but there are many open questions!
• What can tropical surfaces in three-dimensional space look like?  What is the version of a skeleton here?  (For instance, a tropical surface of degree 4 contains a distinguished polyhedron with at most 63 facets. Which polyhedra are possible?)
• Study the geometry of tropical curves obtained by intersecting two tropical surfaces.  For instance, if we intersect a tropical plane with a tropical surface of degree 4, we obtain a tropical curve whose skeleton has three loops.  How can those loops be arranged?  Or we could intersect degree 2 and degree 3 tropical surfaces, to get a tropical curve with 4 loops; which skeletons are possible there?
• One way to study tropical geometry is to replace the usual rules of arithmetic (plus and times) with new rules (min and plus).  How do topics like linear algebra work in these fields?  (It turns out they’re related to optimization, scheduling, and job assignment problems.)
• Chip-firing games on graphs model questions from algebraic geometry.  One of the most important comes in the “gonality” of a graph, which is the smallest number of chips on a graph that could eliminate (via a series of “chip-firing moves”) an added debt of -1 anywhere on the graph.  There are lots of open questions for studying the gonality of graphs; this include general questions, like “What are good lower bounds on gonality?” and specific ones, like “What’s the gonality of the n-dimensional hypercube graph?”
• We can also study versions of gonality where we place -r chips instead of just -1; this gives us the r^th gonality of a graph.  Together, the first, second, third, etc. gonalities form the “gonality sequence” of a graph.  What sequences of integers can be the gonality sequence of some graph?  Is there a graph whose gonality sequence starts 3, 5, 8?
• There are many computational and algorithmic questions to ask about chip-firing games.  It’s known that computing the gonality of a general graph is NP-hard; what if we restrict to planar graphs?  Or graphs that are 3-regular? And can we implement relatively efficient ways of computing these numbers, at least for small graphs?
• What if we changed our rules for chip-firing games, for instance by working with chips modulo N?  How can we “win” a chip-firing game in that context, since there’s no more notion of debt?
• Study a “graph throttling” version of gonality.  For instance, instead of minimizing the number of chips we place on the graph, maybe we can also try to decrease the number of chip-firing moves we need to eliminate debt.
• Chip-firing games lead to interesting questions on other topics in graph theory.  For instance, there’s a conjectured upper bound of (|E|-|V|+4)/2 on the gonality of a graph; and any graph is known to have gonality at least its tree-width.  Can we prove the (weaker) result that (|E|-|V|+4)/2 is an upper bound on tree-width?  (Such a result would be of interest to graph theorists, even the idea behind it comes from algebraic geometry!)
• Topics coming from discrete geometry.  For example:  suppose you want to make “string art”, where you have one shape inside of another with string weaving between the inside and the outside shapes.  For which pairs of shapes is this possible?

Possible Colloquium topics:   I’m happy to advise a talk in any area of math, but would be especially excited about talks related to algebra, geometry, graph theory, or discrete mathematics.

Shaoyang Ning (On Leave 2024 – 2025)

Research Interest :  Statistical methodologies and applications. My research focuses on the study and design of statistical methods for integrative data analysis, in particular, to address the challenges of increasing complexity and connectivity arising from “Big Data”. I’m interested in innovating statistical methods that efficiently integrate multi-source, multi-resolution information to solve real-life problems. Instances include tracking localized influenza with Google search data and predicting cancer-targeting drugs with high-throughput genetic profiling data. Other interests include Bayesian methods, copula modeling, and nonparametric methods.

• Digital (disease) tracking: Using Internet search data to track and predict influenza activities at different resolutions (nation, region, state, city); Integrating other sources of digital data (e.g. Twitter, Facebook) and/or extending to track other epidemics and social/economic events, such as dengue, presidential approval rates, employment rates, and etc.
• Predicting cancer drugs with multi-source profiling data: Developing new methods to aggregate genetic profiling data of different sources (e.g., mutations, expression levels, CRISPR knockouts, drug experiments) in cancer cell lines to identify potential cancer-targeting drugs, their modes of actions and genetic targets.
• Social media text mining: Developing new methods to analyze and extract information from social media data (e.g. Reddit, Twitter). What are the challenges in analyzing the large-volume but short-length social media data? Can classic methods still apply? How should we innovate to address these difficulties?
• Copula modeling: How do we model and estimate associations between different variables when they are beyond multivariate Normal? What if the data are heavily dependent in the tails of their distributions (commonly observed in stock prices)? What if dependence between data are non-symmetric and complex? When the size of data is limited but the dimension is large, can we still recover their correlation structures? Copula model enables to “link” the marginals of a multivariate random variable to its joint distribution with great flexibility and can just be the key to the questions above.
• Other cross-disciplinary, data-driven projects: Applying/developing statistical methodology to answer an interesting scientific question in collaboration with a scientist or social scientist.

Possible colloquium topics:   Any topics in statistical methodology and application, including but not limited to: topics in applied statistics, Bayesian methods, computational biology, statistical learning, “Big Data” mining, and other cross-disciplinary projects.

Anna Neufeld

Research interests:  My research is motivated by the gap between classical statistical tools and practical data analysis. Classic statistical tools are designed for testing a single hypothesis about a single, pre-specified model. However, modern data analysis is an adaptive process that involves exploring the data, fitting several models, evaluating these models, and then testing a potentially large number of hypotheses about one or more selected models. With this in mind, I am interested in topics such as (1) methods for model validation and selection, (2) methods for testing data-driven hypotheses (post-selection inference), and (3) methods for testing a large number of hypotheses. I am also interested in any applied project where I can help a scientist rigorously answer an important question using data. 

• Cross-validation for unsupervised learning. Cross-validation is one of the most widely-used tools for model validation, but, in its typical form, it cannot be used for unsupervised learning problems. Numerous ad-hoc proposals exist for validating unsupervised learning models, but there is a need to compare and contrast these proposals and work towards a unified approach.
• Identifying the number of cell types in single-cell genomics datasets. This is an application of the topic above, since the cell types are typically estimated via unsupervised learning.
• There is growing interest in “post-prediction inference”, which is the task of doing valid statistical inference when some inputs to your statistical model are the outputs of other statistical models (i.e. predictions). Frameworks have recently been proposed for post-prediction inference in the setting where you have access to a gold-standard dataset where the true inputs, rather than the predicted inputs, have been observed. A thesis could explore the possibility of post-prediction inference in the absence of this gold-standard dataset.
• Any other topic of student interest related to selective inference, multiple testing, or post-prediction inference.
• Any collaborative project in which we work with a scientist to identify an interesting question in need of non-standard statistics.
• I am open to advising colloquia in almost any area of statistical methodology or applications, including but not limited to: multiple testing, post-selection inference, post-prediction inference, model selection, model validation, statistical machine learning, unsupervised learning, or genomics.

Allison Pacelli

Research interests:   Math Education, Math & Politics, and Algebraic Number Theory.

Math Education.  Math education is the study of the practice of teaching and learning mathematics, at all levels. For example, do high school calculus students learn best from lecture or inquiry-based learning? What mathematical content knowledge is critical for elementary school math teachers? Is a flipped classroom more effective than a traditional learning format? Many fascinating questions remain, at all levels of education. We can talk further to narrow down project ideas.

Math & Politics.  The mathematics of voting and the mathematics of fair division are two fascinating topics in the field of mathematics and politics. Research questions look at types of voting systems, and the properties that we would want a voting system to satisfy, as well as the idea of fairness when splitting up a single object, like cake, or a collection of objects, such as after a divorce or a death.

Algebraic Number Theory.  The Fundamental Theorem of Arithmetic states that the ring of integers is a unique factorization domain, that is, every integer can be uniquely factored into a product of primes. In other rings, there are analogues of prime numbers, but factorization into primes is not necessarily unique!

In order to determine whether factorization into primes is unique in the ring of integers of a number field or function field, it is useful to study the associated class group – the group of equivalence classes of ideals. The class group is trivial if and only if the ring is a unique factorization domain. Although the study of class groups dates back to Gauss and played a key role in the history of Fermat’s Last Theorem, many basic questions remain open.

  Possible thesis topics:

• Topics in math education, including projects at the elementary school level all the way through college level.
• Topics in voting and fair division.
• Investigating the divisibility of class numbers or the structure of the class group of quadratic fields and higher degree extensions.
• Exploring polynomial analogues of theorems from number theory concerning sums of powers, primes, divisibility, and arithmetic functions.

Possible colloquium topics:   Anything in number theory, algebra, or math & politics.

Anna Plantinga

Research interests:   I am interested in both applied and methodological statistics. My research primarily involves problems related to statistical analysis within genetics, genomics, and in particular the human microbiome (the set of bacteria that live in and on a person).  Current areas of interest include longitudinal data, distance-based analysis methods such as kernel machine regression, high-dimensional data, and structured data.

• Impacts of microbiome volatility. Sometimes the variability of a microbial community is more indicative of an unhealthy community than the actual bacteria present. We have developed an approach to quantifying microbiome variability (“volatility”). This project will use extensive simulations to explore the impact of between-group differences in volatility on a variety of standard tests for association between the microbiome and a health outcome.
• Accounting for excess zeros (sparse feature matrices). Often in a data matrix with many zeros, some of the zeros are “true” or “structural” zeros, whereas others are simply there because we have fewer observations for some subjects. How we account for these zeros affects analysis results. Which methods to account for excess zeros perform best for different analyses?
• Longitudinal methods for compositional data. When we have longitudinal data, we assume the same variables are measured at every time point. For high-dimensional compositions, this may not be the case. We would generally assume that the missing component was absent at any time points for which it was not measured. This project will explore alternatives to making that assumption.
• Applied statistics research. In collaboration with a scientist or social scientist, use appropriate statistical methodology (or variations on existing methods) to answer an interesting scientific question.

Any topics in statistical application, education, or methodology, including but not restricted to:

• Topics in applied statistics.
• Methods for microbiome data analysis.
• Statistical genetics.
• Electronic health records.
• Variable selection and statistical learning.
• Longitudinal methods.

Cesar Silva

Research interests :  Ergodic theory and measurable dynamics; in particular mixing properties and rank one examples, and infinite measure-preserving and nonsingular transformations and group actions.  Measurable dynamics of transformations defined on the p-adic field.  Measurable sensitivity.  Fractals.  Fractal Geometry.

Possible thesis topics:    Ergodic Theory.   Ergodic theory studies the probabilistic behavior of abstract dynamical systems.  Dynamical systems are systems that change with time, such as the motion of the planets or of a pendulum.  Abstract dynamical systems represent the state of a dynamical system by a point in a mathematical space (phase space).  In many cases this space is assumed to be the unit interval [0,1) with Lebesgue measure.  One usually assumes that time is measured at discrete intervals and so the law of motion of the system is represented by a single map (or transformation) of the phase space [0,1).  In this case one studies various dynamical behaviors of these maps, such as ergodicity, weak mixing, and mixing.  I am also interested in studying the measurable dynamics of systems defined on the p-adics numbers.  The prerequisite is a first course in real analysis.  Topological Dynamics.  Dynamics on compact or locally compact spaces.

Topics in mathematics and in particular:

• Any topic in measure theory.  See for example any of the first few chapters in “Measure and Category” by J. Oxtoby. Possible topics include the Banach-Tarski paradox, the Banach-Mazur game, Liouville numbers and s-Hausdorff measure zero.
• Topics in applied linear algebra and functional analysis.
• Fractal sets, fractal generation, image compression, and fractal dimension.
• Dynamics on the p-adic numbers.
• Banach-Tarski paradox, space filling curves.

Mihai Stoiciu

Research interests: Mathematical Physics and Functional Analysis. I am interested in the study of the spectral properties of various operators arising from mathematical physics – especially the Schrodinger operator. In particular, I am investigating the distribution of the eigenvalues for special classes of self-adjoint and unitary random matrices.

Topics in mathematical physics, functional analysis and probability including:

• Investigate the spectrum of the Schrodinger operator. Possible research topics: Find good estimates for the number of bound states; Analyze the asymptotic growth of the number of bound states of the discrete Schrodinger operator at large coupling constants.
• Study particular classes of orthogonal polynomials on the unit circle.
• Investigate numerically the statistical distribution of the eigenvalues for various classes of random CMV matrices.
• Study the general theory of point processes and its applications to problems in mathematical physics.

Possible colloquium topics:  

Any topics in mathematics, mathematical physics, functional analysis, or probability, such as:

• The Schrodinger operator.
• Orthogonal polynomials on the unit circle.
• Statistical distribution of the eigenvalues of random matrices.
• The general theory of point processes and its applications to problems in mathematical physics.

Elizabeth Upton

Research Interests: My research interests center around network science, with a focus on regression methods for network-indexed data. Networks are used to capture the relationships between elements within a system. Examples include social networks, transportation networks, and biological networks. I also enjoy tackling problems with pragmatic applications and am therefore interested in applied interdisciplinary research.

• Regression models for network data: how can we incorporate network structure (and dependence) in our regression framework when modeling a vertex-indexed response?
• Identify effects shaping network structure. For example, in social networks, the phrase “birds of a feather flock together” is often used to describe homophily. That is, those who have similar interests are more likely to become friends. How can we capture or test this effect, and others, in a regression framework when modeling edge-indexed responses?
• Extending models for multilayer networks. Current methodologies combine edges from multiple networks in some sort of weighted averaging scheme. Could a penalized multivariate approach yield a more informative model?
• Developing algorithms to make inference on large networks more efficient.
• Any topic in linear or generalized linear modeling (including mixed-effects regression models, zero-inflated regressions, etc.).
• Applied statistics research. In collaboration with a scientist or social scientist, use appropriate statistical methodology to answer an interesting scientific question.
• Any applied statistics research project/paper
• Topics in linear or generalized linear modeling
• Network visualizations and statistics

Mathematics Education Project Topics PDF Materials for Students

Access 150 Mathematics Education Project Topics PDF Research Materials for Mathematics Education Students

Best Project Topics in Mathematics Education

Here is the List of 150 Best Mathematics Education Research Project Topics and Materials for (Final Year and Undergraduate) Mathematics Education Students in Nigeria & other English Speaking Countries:

Relative Academic Performances Of Secondary School Students In School Certificate Mathematics & English Language. A Case Study Of Enugu North L.G.A

Availability of laboratory facilities for effective teaching – learning of mathematics, integrated science and computer science in junior secondary schools. case study of enugu north l.g.a, effect of class size to the teaching and learning of mathematics. in enugu north local government area of enugu state, effect of student perception on teaching and learning mathematics. a case study of igboeze north local government area of enugu state, problem of teaching and learning of mathematics in senior secondary schools. case study of enugu north lga of enugu state, impact of laboratory practical on senior secondary school student academic achievement in biology, chemistry and mathematics. case study of ss2 in enugu north lga, impact of laboratory practical on senior secondary school student academic achievement in ss2 biology, chemistry and mathematics. a case study of enugu north lga of enugu state, problems militating against the effective teaching and learning of mathematics in junior secondary schools. (a case study of enugu west senatorial zone, enugu state), investigation into academic indiscipline and failure among secondary school students in (english language mathematics, igbo language, agricultural science, economics. a case study of nigeria, causes and effects of mass failure in junior secondary school certificate examination (jssce) mathematics. a case study of nsukka educatiopn zone of enugu state, influence of teachers qualification on students performance in mathematics., study on the collaborative methods of teaching mathematics in secondary schools. in lagos state, comparative analysis of male and female students performance in english and mathematics., effects of teachers’ mastery of subject matter and utilization of instructional materials on the academic achievement of senior secondary school students in mathematics., effect of problem-solving approach on learning on junior secondary school academic achievement in mathematics., comparison of junior and high school attitude towards mathematics., study habits and student achievement in mathematics., implications of smartphone on the academic performance of senior secondary school mathematics students. in katsina metropolis, impact of story book on academic performance of mathematics students in junior secondary school., practical based instructional strategy and students’ academic performance in mathematics. in ikot ekpene l. g. a., influence of counselling on the academic achievement in maths of junior secondary school students. in ibadan south west local government area, ibadan, oyo state, perception of students on the causes and effects of mathematics anxiety among students of tertiary institutions in nigeria., attitude and interest as correlates of students academics performance in mathematics., impact of teacher and qualification on students academic performance in mathematics., perception of students in teaching and learning of mathematics., influence of parental socio-economic status on career aspirations among secondary school mathematics student., survey to the factors responsible for the students’ poor performance in mathematics in selected secondary schools., effects of improvised instructional material and the achievement of student in mathematics education., difficulties encountered by students in solving algebraic problem in senior secondary school., effects of ict on teaching and learning of mathematics in secondary school., impact of mathematical anxiety on academic performance of students., effects of heuristic method of teaching and academic performance of mathematics students in senior secondary school. in mkpat enin local government area, teaching and learning of mathematics using computer assisted instruction (cai) in staff demonstration., perception of teachers on the causes and effects of mathematics anxiety among secondary school students. in bayelsa state, factors influencing the attitude of secondary school students towards the study of mathematics., assessing senior secondary school students’ motivation to learn mathematics as related to gender and performance in mathematics., assessment of the difficulties encountered by students in solving algebraic words problems in junior secondary school., evaluation of the implementation of the secondary school mathematics core-curriculum by teachers., effect of teacher’s perception of use of ict on mathematics students academic achievement of junior secondary school. in eket lga, impact of professional development on subject and pedagogical content knowledge of mathematics teachers., effect of mathematics laboratory on students performance in mathematics in secondary schools. in enugu state, investigative study on the causes of mass failure in mathematics in external examinations., secondary school student attitudes towards the study of mathematics and their academic achievement., instructional materials utilization on jss 3 students’ academic performance in mathematics. in ibesikpo azutan local government, small group discussions learning pattern and its impact on student academic performance in mathematics., evaluation of students academic achievement in mathematics in the junior secondary school., students teachers conception of mathematics., effect of the use of technology on the instruction of mathematics and english secondary schools., factors leading to poor performance in mathematics subject., influence of gender school and type on students achievement in senior secondary school mathematics..

Downloadable Mathematics Education Project Topics and PDF/DOC Materials END HERE. NOTE: Below are Mathematics Education Research Areas that students & researchers can develop independently .

• Innovative Pedagogical Approaches: Explore novel teaching methods and strategies in mathematics education, such as flipped classrooms, inquiry-based learning, and project-based learning, assessing their effectiveness in enhancing student engagement and understanding.
• Integration of Technology: Investigate the integration of technology, including computer-based learning tools, educational apps, and interactive simulations, to facilitate mathematical understanding and skill development among students.
• Assessment Methods: Examine various assessment methods in mathematics education, including formative assessment, summative assessment, and alternative assessment techniques, to evaluate student learning outcomes accurately.
• Cultural Relevance in Mathematics Education: Analyze the cultural factors that influence mathematics teaching and learning, exploring ways to make mathematical content more culturally relevant and accessible to diverse student populations.
• Gender and Mathematics: Investigate gender differences in mathematical achievement and attitudes towards mathematics, exploring strategies to address gender disparities and promote equity in mathematics education.
• Mathematics Anxiety: Explore the phenomenon of mathematics anxiety among students and its impact on learning outcomes, identifying effective strategies for alleviating anxiety and promoting positive attitudes towards mathematics.
• Teacher Professional Development: Examine professional development programs for mathematics teachers, focusing on strategies for enhancing pedagogical knowledge, content knowledge, and instructional practices.
• Mathematical Problem Solving: Investigate the development of mathematical problem-solving skills among students, exploring instructional approaches and learning environments that foster problem-solving abilities.
• Mathematics Curriculum Development: Analyze mathematics curriculum frameworks and standards, exploring approaches to curriculum design and implementation that promote coherence, rigor, and relevance.
• Mathematical Modeling: Explore the use of mathematical modeling in the classroom, investigating how modeling tasks can engage students in authentic mathematical inquiry and real-world problem-solving.
• Mathematics and Multilingual Learners: Investigate effective instructional strategies for teaching mathematics to multilingual learners, considering language barriers and cultural differences in mathematical understanding.
• Mathematics and Special Education: Explore strategies for teaching mathematics to students with diverse learning needs, including those with disabilities or learning difficulties, focusing on inclusive instructional practices.
• Mathematics and Gifted Education: Examine approaches to challenging and enriching the mathematical learning experiences of gifted students, considering differentiated instruction and enrichment programs.
• Mathematics and Social Justice: Investigate the intersection of mathematics education and social justice, exploring ways to address inequities in access to high-quality mathematics instruction and opportunities.
• Mathematics Teacher Identity: Explore the development of teacher identity in mathematics education, considering factors that shape teachers’ beliefs, attitudes, and practices in teaching mathematics.
• Mathematics Teacher Collaboration: Investigate collaborative practices among mathematics teachers, including professional learning communities, lesson study groups, and co-teaching arrangements, to promote teacher collaboration and collective efficacy.
• Parental Involvement in Mathematics Education: Examine the role of parents and families in supporting children’s mathematical learning, exploring strategies for enhancing parental involvement and communication between home and school.
• Cross-Curricular Connections: Explore interdisciplinary connections between mathematics and other subject areas, such as science, technology, engineering, and the arts, fostering integrated approaches to teaching and learning.
• History of Mathematics Education: Investigate the historical development of mathematics education, examining influential figures, movements, and reforms that have shaped the field over time.
• Ethics in Mathematics Education Research: Reflect on ethical considerations in mathematics education research, including issues related to participant consent, confidentiality, and potential harm to participants.
• Mathematics Teacher Beliefs: Explore the beliefs and attitudes of mathematics teachers towards teaching and learning, investigating the impact of these beliefs on instructional practices and student outcomes.
• Mathematics and Motivation: Investigate motivational factors that influence student engagement and achievement in mathematics, exploring strategies for fostering intrinsic motivation and a growth mindset.
• Mathematics and Socioeconomic Status: Examine the relationship between socioeconomic status and mathematics achievement, considering the impact of poverty, access to resources, and educational opportunities on student outcomes.
• Assessment for Learning: Explore the principles of assessment for learning in mathematics education, focusing on how formative assessment practices can enhance student understanding and inform instructional decisions.
• Mathematics and Neuroscience: Investigate insights from cognitive neuroscience that inform our understanding of mathematical learning processes, exploring implications for instructional design and intervention strategies.
• Mathematics Teacher Preparation: Examine pre-service and in-service teacher preparation programs in mathematics education, evaluating the effectiveness of different approaches in preparing teachers for the classroom.
• Mathematics and Gifted Education: Investigate approaches to identifying and serving gifted students in mathematics, considering issues related to assessment, curriculum differentiation, and talent development.
• Mathematics and Social Media: Explore the use of social media platforms for mathematics education, considering how online communities, resources, and collaborative tools can support teaching and learning.
• Mathematics and Environmental Education: Investigate connections between mathematics and environmental education, exploring ways to integrate mathematical concepts and skills into the study of environmental issues and sustainability.
• Mathematics and Critical Thinking: Examine the role of mathematics in promoting critical thinking skills, exploring instructional strategies that encourage students to analyze, evaluate, and solve complex problems.
• Mathematics and Cultural Diversity: Investigate cultural perspectives on mathematics teaching and learning, considering how cultural norms, values, and practices influence mathematical reasoning and problem-solving approaches.
• Mathematics and Entrepreneurship Education: Explore connections between mathematics education and entrepreneurship education, considering how mathematical thinking and problem-solving skills are essential for entrepreneurial success.
• Mathematics and Global Competence: Investigate the role of mathematics education in fostering global competence, including intercultural understanding, communication skills, and awareness of global issues.
• Mathematics and Environmental Justice: Explore the intersection of mathematics education and environmental justice, considering how mathematical modeling and data analysis can inform advocacy and decision-making.
• Mathematics and Early Childhood Education: Investigate effective approaches to teaching mathematics in early childhood settings, focusing on developmentally appropriate activities and instructional strategies.
• Mathematics and Indigenous Knowledge: Explore connections between mathematics education and indigenous knowledge systems, considering culturally relevant approaches to teaching and learning mathematics.
• Mathematics and Health Education: Investigate connections between mathematics education and health education, exploring how mathematical concepts and skills can be applied to understanding health-related data and making informed decisions.
• Mathematics and Social Emotional Learning: Explore the intersection of mathematics education and social-emotional learning, considering how mathematical tasks and collaborative activities can promote skills such as self-regulation, perseverance, and empathy.
• Mathematics and Career Readiness: Investigate the role of mathematics education in preparing students for future careers, considering the relevance of mathematical skills in various industries and professions.
• Mathematics and Sustainable Development Goals: Explore connections between mathematics education and the United Nations Sustainable Development Goals, considering how mathematical thinking and problem-solving can contribute to addressing global challenges such as poverty, inequality, and climate change.

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The transition from school to university in mathematics education research: new trends and ideas from a systematic literature review

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• Published: 11 November 2022
• Volume 113 , pages 7–34, ( 2023 )

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• Pietro Di Martino   ORCID: orcid.org/0000-0002-6524-866X 1 ,
• Francesca Gregorio   ORCID: orcid.org/0000-0001-7646-8898 2 , 3 &
• Paola Iannone   ORCID: orcid.org/0000-0001-7904-5380 4  

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Investigating the transition between educational levels is one of the main themes for the future of mathematics education. In particular, the transition from secondary school to STEM degrees is problematic for the widespread students’ difficulties and significant for the implications that it has on students’ futures. Knowing and understanding the past is key to imagine the future of a research field. For this reason, this paper reports a systematic review of the literature on the secondary-tertiary transition in Mathematics Education from 2008 to 2021. We constructed two corpuses: one from the proceedings of three international conferences in mathematics education (PME, ICME, and INDRUM) and the other from peer reviewed research papers and book chapters returned by the databases ERIC and Google Scholar. A clear evolution in perspectives since 2008 emerges from the analysis of the two corpuses: the research focus changed from a purely cognitive to a more holistic one, including socio-cultural and — to a lesser extent — affective issues. To this end, a variety of research methods were used, and specific theoretical models were developed in the considered papers. The analysis also highlights a worrisome trend of underrepresentation: very little research comes from large geographical areas such as South America or Africa. We argue that this gap in representation is problematic as research on secondary tertiary transition concerns also consideration of socio-cultural and contextual factors.

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1 Introduction

The study of the transition from secondary school to university mathematics — also referred to as the secondary-tertiary transition (STT) — has a long tradition in mathematics education research. Seminal studies were conducted by Tall since the 1980s, investigating significant cognitive discontinuities in STT (Tall, 1991 ; Tall & Vinner, 1981 ). Tall’s studies inspired the development of research specifically focused on students’ difficulties in STT. At the end of the past millennium, De Guzman et al. ( 1998 ) summarised the main findings of this field of research and identified three categories related to students’ difficulties in STT: epistemological-cognitive, sociological-cultural, and didactical.

A quarter of a century later, students’ difficulties in STT are still a significant issue for modern society. The high dropout of undergraduate students in STEM subjects reported in several countries (Rach & Heinze, 2017 ) is problematic for at least two reasons. The first relates to the need of advanced mathematics competencies for economies to flourish (Adkins & Noyes, 2016 ). The second is linked to equity, given the opportunities afforded by STEM degrees (science, technology, engineering, and mathematics) in terms of social mobility and future earnings (see Higher Education Statistics Agency, 2018 , for the UK context). Indeed, the European Mathematical Society recently conducted a survey amongst mathematicians to collect information to devise national and international actions that may help reduce students’ dropout during the STT (Koichu & Pinto, 2019 ). The authors underline that the STT is a complex and multi-faceted process, provoking frustration in first-year students, as well as in lecturers.

The difficulties that students encounter when moving from school mathematics to the mathematics in a STEM degree still represent a strange story, not fully explained, that involves and overwhelms even students considered excellent in mathematics during their school experience (Di Martino & Gregorio, 2019 ). Understanding this complex and apparently unexplained phenomenon requires many theoretical approaches and, as Schoenfeld ( 2000 ) notices, ‘findings are rarely definitive; they are usually suggestive. Evidence is not on the order of proof but is cumulative’ (p. 648). This cumulative nature of findings suggests the relevance of literature reviews on selected topics: they allow researchers to identify trends and provide insights for possible developments of the research field (Pan, 2016 ).

An influential review of research on STT was conducted by Gueudet in 2008. Gueudet highlights three types of transitions involving individual, social, and institutional factors: transition in ways of thinking, transition to proof and the technical language of mathematics, and institutional transition related to changes in the didactical contract. Gueudet’s review also confirms that up to that time, the tertiary transition in mathematics was mainly studied through cognitive and epistemological lenses, even though socio-cultural and affective issues had since assumed an important role in mathematics education (Lerman, 2000 ). Gueudet ( 2008 ) concludes that there is ‘the need for further research, and for teaching designs grounded in their findings’ (p. 252). Her call was heard by the mathematics education research community, and research on STT has since adopted new theoretical perspectives, discussed new results, and highlighted new lines of interest. Indeed, the transitions to higher education are one of the themes emerging from a recent international survey of researchers in mathematics education answering the question: ‘on what themes should research in mathematics education focus in the coming decade?’ (Bakker et al., 2021 , p. 2).

Recently, Bergsten and Jablonka ( 2019 ) traced the development of studies on STT in terms of the theoretical approaches taken. However, a systematic review of the research on STT in the last 15 years is still missing. To fill this gap, we analyse the research on STT in mathematics education since 2008.

We developed our systematic review to answer the following research questions related to the period 2008–2021:

RQ1: Which are the methods and the specific theoretical frameworks used to approach STT?

RQ2: What are the main themes of STT research emerging from an analysis of published research?

RQ3: What is the geographical distribution of STT research? Do we know enough about diverse cultural context coming from a wide range of countries?

2 Methodology

The development of a systematic review involves several aspects: in this section, we briefly describe the underlying choices of our review and their reasons, as well as its constraints and the possibility of reproducibility.

The development of a literature review is a systematic process (Pan, 2016 ), involving three steps needed to identify the corpus of interest (Green et al., 2006 ): selection of the topic, definition of the sources of information, and definition of the selection criteria employed (Table 1 ).

The relevance of the topic and the reasons for the review (Step 1) were discussed in the introduction of this paper.

The description of Steps 2 and 3 is of crucial importance for a systematic literature review: it allows other scholars to replicate the selection of the corpus by following the given criteria. Related to these two steps, we differentiate between the two searches we conducted: one focused on conference proceedings, the other on refereed journal papers and book chapters. For both searches, the starting year was 2008 and the ending was October 2021. However, the database and, in part, the automatic search criteria were different between the two searches.

According to our focus, we included in this review studies describing the STT to university mathematics of students enrolled in STEM degrees. These studies may involve pre-service secondary mathematics teachers in those countries where they are taught mathematics in the same lectures of students enrolled in a STEM degree. However, we consider the STT for pre-service teachers (including teachers for primary and lower secondary education) as a special case of transition to tertiary mathematics deserving, in our view, a separate discussion, not addressed in this paper.

We developed a specific search for conference proceedings since international conferences are likely to be a forum to share ideas which are then developed in journal papers or book chapters, and results emerging from conference proceedings often anticipate trends in the development of research themes.

The first choice we made was related to which conference’s proceedings to include in the review. We decided to include two of the most popular (by number of participants) international conferences in mathematics education: the International Congress on Mathematical Education (ICME) and the Conference of the International Group for the Psychology of Mathematics Education (PME). For the likely relevance to our focus, we also included the conferences organised by the International Network for Didactic Research in University Mathematics (INDRUM).

Since our third research question concerned the representation of geographical regions in mathematics education research, one of the main reasons for the selection of ICME, PME, and INDRUM was that the international program committees of these conferences are opened to members coming from all over the world. Using this criterion, we did not include in our search other potentially relevant conferences such as CERME (Congress of the European society for Research in Mathematics Education), RUME (the Research on Undergraduate Mathematics Education of the Mathematical Association of America), PME-NA (North American Chapter of the International Group for the Psychology of Mathematics Education), and other PME regional conferences.

The database for the construction of this first corpus therefore consisted of the published proceedings of ICME, PME, and INDRUM since 2008. We identified all the contributions including the key term ‘transition’ in the title, abstract, or text (automatic search criterion). We then considered only contributions where the occurrence was related to the secondary-tertiary transition and with STT as the significant focus of the paper (not mentioned as a secondary topic).

For the construction of the second corpus (which includes journal papers and book chapters), we first considered what search criteria to use, as this is a crucial step for every literature review. Gusenbauer and Haddaway ( 2020 ) recognised three main quality criteria for literature searches: completeness (identifying all the most significant resources about the topic), transparency, and reproducibility. Through the analysis of the systematic search qualities of 28 academic search engines, the authors suggested to conduct literature reviews using at least two different search engines, according to their quality with respect the three criteria mentioned above. Regarding Google Scholar, Gusenbauer and Haddaway ( 2020 ) observe that this search engine has several limitations in terms of transparency and reproducibility of searches; however, ‘it is considered a suitable supplementary source of evidence (including on grey literature)’ (p. 196) for systematic literature reviews.

Based on the above considerations, we used two databases: ERIC (Education Resources Information Center, https://eric.ed.gov ), the online database of education research promoted by the US Department of Education, and Google Scholar ( https://scholar.google.com ). The use of Google Scholar in our review appeared important to bring to the fore contributions by researchers who may not have access to the main journals and conferences of our community (for example due to the high cost of conference fees and travel). We also followed Haddaway et al.’s ( 2015 ) advice to focus on the first 200 to 300 results returned by Google Scholar for systematic reviews as supplementary source of evidence.

Since the databases were not specific to mathematics education, we used a strict automatic search criterion. We used ‘transition AND mathematics AND school AND university’ as search terms. We observed a posteriori that the inclusion of the word ‘university’ in the Google Scholar search was not useful: Google Scholar searches the entire document — unlike ERIC which searches only the title, the abstract, and the keywords — and therefore the word university featured in all documents where the names of authors and their institutions were included. At the end of this first phase, following the recommendation of Haddaway et al. ( 2015 ), we considered the corpus produced by the union of the 252 results produced from ERIC and the first 300 results by relevance from Google Scholar taking care to eliminate repetition of entries between the two sets of results since, as expected, there was a significant overlap.

The selection criteria used to include a study identified using the automatic search criteria were the same for both databases. Assuming the definition of STT given in the Encyclopaedia of Mathematics Education as ‘the process experienced by students leaving secondary school and entering different kinds of postsecondary institutions: universities, engineering schools, etc.’ (Gueudet & Thomas, 2020 , p. 762), we selected the contributions describing a transitional process , considering a ‘before’ and an ‘after’ this process. This means we excluded studies where the focus was solely on one of the stages of this transition (e.g., students’ experiences with proof at the start of a mathematics degree) and did not consider the ‘before’ and ‘after’ of the STT. Because of this choice, papers focusing on the design of transition to proof courses or university preparatory classes were not included in our review.

A margin of subjectivity in the application of the selection criteria clearly exists in this process. However, the first two authors developed an investigator triangulation to balance individual biases (Mok & Clarke, 2015 ). The two authors applied independently the selection criteria, sharing their outcome. They then discussed the cases of disagreement until a full agreement for the selection of the final corpuses was reached.

The result of this process was the two final corpuses: the first one including 59 reports presented in the three selected conferences and the second one including 55 papers from peer reviewed journals and books.

Once we selected the corpus, aligned with the research questions of the systematic review, each paper was categorised according to: year of publication, type of study (theoretical, empirical, didactical design) and — if empirical — methods (quantitative, qualitative, mixed), instruments, sample, context Footnote 1 (country where the research was developed, how many and which schools, university, or texts were involved in the research), theoretical framework(s), research question(s), and focus (cognitive, affective, socio-cultural). This classification allowed for multiple labels: for example, a paper could be classified with cognitive and socio-cultural foci if explicit references to both cognitive and socio-cultural issues were made in the theoretical framework or in the research question(s).

In addition to the described classification, we also noted the main findings of the study for each paper.

The recognition of the main research themes was based on a coding process related to the research questions aimed to recognise frequencies and patterns (Cohen et al., 2007 ). Initially, a very specific label was assigned to each research question of the paper considered, then — following Miles et al. ( 2003 ) — the labels evolved during the process to describe wider categories capable to bring together several similar focuses (an example of this evolution is summarised in Fig.  1 ).

Example of the coding process for the research questions

3 Results and discussion

3.1 stt in conference proceedings from 2008 to 2021.

As previously described, we considered in our review three international conferences: ICME, INDRUM, and PME.

3.1.1 ICME congresses

ICME is a large international congress held every 4 years involving researchers worldwide. Four ICME congresses took place since 2008: the proceedings of ICME-11 (Monterrey, Mexico 2008) and of ICME-14 (Shanghai, China, 2021, online conference) are not currently available, while those of ICME-12 (Seoul, Korea, 2012) and ICME-13 (Hamburg, Germany, 2016) are published as open access Springer books. According to the spirit and nature of the activities of the Topic Study Groups (TSG), the existing proceedings consist of abstracts of the discussions developed in the sessions rather than a collection of research reports. For this reason, the discussion which follows is short and mainly descriptive.

The evolution of the presence of STT in the discussion of the ICME community can be recognised by the analysis of the proceedings, scientific programs, and books related to developed activities. In what follows, we detail the evolution of research on STT within these four ICME congresses.

Although one of the TSGs of ICME-11 was about university mathematics, STT was not mentioned in the description of its scientific program.

The situation was different 4 years later at ICME-12. The transition ‘problem’ was explicitly mentioned in the call for papers and in the description of TSG 2: ‘Mathematics education at the tertiary level and access to tertiary level’. Within this TSG, we found five contributions on STT by authors from five different continents (Brazil, Canada, South Korea, South Africa, Sweden). Four out of these five contributions analysed students’ difficulties with specific mathematics topic (matrices, axiomatic method, calculus) with a predominantly cognitive approach. There was also a survey team dedicated to STT, and the outcomes of this work were findings of an international survey of 79 mathematics lecturers from 21 countries. Findings pointed to ‘a multi-faceted web of cognitive, curricular and pedagogical issues’ (Thomas et al., 2015 , p. 278) related to STT, also depending on the institutional context. Despite the evident differences between institutions, and therefore the emergence of socio-cultural aspects, the lens of this survey is still strongly cognitive, without specific reference to affective issue. However, the final mention of the emerging interest for how students experience their first encounters with advanced mathematical topics represented a first step towards the inclusion of affective issues in the discussion.

At ICME-13, STT was a recurrent topic: seven contributions presented in TSG2: ‘Mathematics education at the tertiary level’ were focused on STT. In addition, transition issues were the topic of a discussion group and of the plenary panel. The latter inspired an ICME monograph where the main results of the research and ‘the blind spots that remain unquestioned’ (Gueudet et al., 2016 , p. 19) were discussed. The summary of the results included considerations about institutional differences (for example in class size or equipment) and pedagogical and cognitive issues, but no mention of affect was made.

At ICME-14, four contributions in TSG2 were focused on STT. In particular, Pinto, Gamlieli, and Koichu discussed the results of an international survey conducted among 310 tertiary mathematics lecturers from 30 countries. This confirms the potential of ICME Congresses to foster discussion between researchers from different contexts. The wide geographical reach of the ICME Congresses is particularly relevant for research on STT considering that international comparisons are still rare elsewhere in our field.

To conclude, the diachronic analysis of the proceedings and scientific programmes of these four ICME editions shows an increasing interest for STT. However, while a lack of attention to affective issues clearly emerges, the discussion about socio-cultural issues is promoted through the involvement of researchers from a variety of countries, with a considerable participation of researchers from countries currently underrepresented in mathematics education.

3.1.2 INDRUM conferences

The growing interest for undergraduate mathematics education in our field is also evidenced by the foundation of the International Network for Didactic Research in University Mathematics (INDRUM) in 2015. Since 2016, this network holds a biannual conference with open access to online proceedings. To date, the following conferences have been held: INDRUM 2016 (Montpellier, France, Nardi et al., 2016 ), INDRUM 2018 (Kristiansand, Norway, Durand-Guerrier et al., 2018 ), and INDRUM 2020 (virtually held in Bizerte, Tunisia, Hausberger et al., 2020 ). The book Research and Development in University Mathematics Education (Durand-Guerrier et al., 2021 ) provides a detailed overview of the discussion on topics in the 2016 and 2018 INDRUM conferences. Focusing on STT and according to our criteria, we identified 18 contributions relevant to our review (Table 2 ).

We first notice a clear prevalence (78%) of contributions by European researchers, probably related to the origin of the INDRUM group. Within this European prevalence, there is a strong presence of French authors who account for 39% of the reports on STT. The French influence on the INDRUM community is not only evident in terms of participation: it is also evident from the theoretical frameworks used in the research reports included in the proceedings of these conferences.

Three out of the four theoretical papers on STT used the anthropological theory of the didactic (ADT) (Chevallard, 1992 ) for identifying praxeologies in STT. The fourth, the paper by Artigue ( 2016 ), discussed the challenges of the research in mathematics education at the tertiary level. As for the empirical papers, four reported and interpreted students’ difficulties in STT using the lens of the anthropological theory of didactic. The common hypothesis of these studies was that several phenomena in STT can be interpreted as institutional issues, determined by the strong discontinuity between school and university praxeologies. This approach was extensively described in Gueudet and Pepin ( 2017 ).

Using this framework, Winsløw ( 2008 ) introduced a model with repeated cycles of two transitions to university mathematical praxeologies: the first is related to the need for students to extend their praxeologies considering the role of theory in mathematics. The second transition takes place when the elements of the theory become objects; students need to work with autonomously: in this case, the emergence of new objects can require further transitions.

Other important theoretical frameworks developed by French researchers and adopted in papers included in the INDRUM proceedings for studying STT are the theory of didactical situations (Brousseau, 2002 ) — that Bloch and Gibel ( 2016 ) used for developing a tool for modelling students’ reasoning processes — and the Instrumental Approach (Rabardel, 2002 ), used by Gueudet and Pepin ( 2016 ) for theorising the use of technology in STT.

The INDRUM papers pay great attention to cognitive and epistemological issues in STT; several of the authors’ theoretical approaches fall ‘under the umbrella term advanced mathematical thinking’ (Hochmuth et al., 2021 , p. 195) and are often situated within APOS (action-process-object-schema) theory (Dubinsky, 1991 ). However, the use of the ATD also shows some consideration of institutional and social perspectives, but only one paper (Gueudet & Pepin, 2016 ) analysed and compared two case studies from different national contexts (France and UK).

Only two papers in our INDRUM sample have an affective focus. Quéré ( 2016 ) analysed the different forms of autonomy required in the move from secondary to tertiary education in mathematics, also discussing the role of theories of success and expectations in students’ difficulties in STT. Geisler and Rolka ( 2018b ) discussed the relationship between students’ success in mathematics in their first university year and some affective variables (self-concept, interest, view of mathematics, basic needs, self-efficacy).

3.1.3 PME conferences

PME conferences have annual frequency; therefore, we analysed 13 editions: from Mexico 2008 to the edition virtually held in Thailand in 2021. Footnote 2 PME includes several types of contributions and activities: research report, short oral, working group, discussion group, poster, and plenary. In the following, we will focus on the 26 selected research reports (Table 3 ).

Table 3 illustrates both a small but steadily growing presence of empirical research reports on STT since 2009, and the absence of theoretical reports. The latter may also result from the strict length limitation of the contributions. The analysis of representation by country returns a clear picture: 23 out of 26 reports (88%) were developed in a European country by European researchers (one of these papers discusses a comparison between France and Brazil, also involving a Brazilian author). These data highlight an issue of underrepresentation in the PME conferences concerning the discussion about STT.

Moreover, 16 (62%) out of 26 reports are by researchers from German universities. This regional predominance is also reflected in the research methodologies: the quantitative approach is prevalent in these empirical studies (62.5% versus 29% for the qualitative approach and 8% for the mixed one), and samples are usually large, notwithstanding the presence of three interesting case studies. Several reports focus on the identification of significant correlations between academic success/failure and other variables. This aim is related to a clear definition of the variables involved, to the development of instruments for measuring these variables and the students’ success or failure, and to the adoption of statistical models (the more frequently adopted being the Rasch model).

Considering the 7-year period 2008–2014, we found 9 research reports about STT measuring students’ preparedness in mathematics and students’ learning strategies at the beginning of their university experience. These studies aim to determine whether and to what extent mathematics dropout can be predicted by the analysis of cognitive performance at the beginning of university (Halverscheid & Pustelnik, 2013 ). In first period, two exceptions are represented by the paper by Di Martino and Maracci ( 2009 ), stressing the need to go beyond a purely cognitive approach also in research about STT, and the paper by Dias et al. ( 2010 ), that, within the ATD framework, developed a socio-cultural comparison between Brazil and France regarding STT, considering differences in educational systems and in educational cultures.

In the subsequent 7-year period (2015–2021), we found 17 research reports about STT. The clear difference in the number of reports between the period 2008–2014 and the period 2015–2021 highlights the growing interest for STT in the mathematics education community. The analysis of the frameworks and the research questions used in the research reports in the period 2015–2021 shows a greater consideration for affective and socio-cultural constructs in STT research. Ufer ( 2015 ) analysed the relationship between students’ motivations to choose a mathematical programme and their success, Jeschke et al. ( 2016 ) attempted to measure students’ ‘academic buoyancy’ and its role in early dropout, and Kouvela et al. ( 2017 ) studied the students’ identities as mathematics learners and the influence that messages given by their lecturers have on the development of these identities.

The 2018 edition of PME in Umeå deserves special attention because of its variety of approaches to STT. Bampili et al. ( 2018 ) analysed how social and institutional issues shape the development of a new identity for first year mathematics students by adopting the theoretical framework of Communities of Practice (Wenger, 1998 ). Meehan et al. ( 2018 ) studied how STT affects high-achieving students’ ‘sense of belonging to math’, that is related ‘to whether one feels a member of a mathematical community and feels valued and accepted by that community’ (p. 371). Di Martino and Gregorio ( 2018 ) introduced and analysed the so called ‘first-time phenomenon’, that is the cognitive and emotional reactions of (successful) students to the first experience of failure in mathematics. Geisler and Rolka ( 2018b ) discussed the relationship between affective variables (such as self-concept and beliefs about the nature of mathematics) and academic procrastination.

Therefore, the diachronic analysis of the research reports on STT in the 13 PME conferences shows a clear evolution toward a more holistic view of STT that increasingly includes socio-cultural and affective considerations.

3.2 STT in journal papers and book chapters

Following the selection criteria described in the method section, we obtained a corpus of 55 papers (Table 4 ). As hypothesised, we found several papers extending the ideas discussed by the authors in the international conferences presented in the previous section; therefore, the analysis of this corpus is particularly significant to gain a clear picture of the state of the art of STT research.

The data about the university affiliation of the authors and the geographical contexts where the studies were developed underlines again an issue of representation: the largest number of contributions comes from authors working in European universities reporting studies developed in Europe (63%). Research concerning other regions, such as South America, Asia, Africa, is almost completely absent in the identified corpus.

Concerning the analysis of the methods in the empirical studies, we found a balance between qualitative (43%) and quantitative (37%) approaches (20% used a mixed approach), as well as a variety of instruments and targets (summarised in Table 5 ).

In particular, the analysis of National School Standards or textbooks allows us to compare different contexts, leading to the awareness that the systems in school and university are culturally embedded (Frank & Thompson, 2021 ). To this aim, Vollstedt et al. ( 2014 ) elaborated a framework for analysing and comparing mathematics textbooks.

The analysis of the research methods highlights two related issues: most data are collected through online surveys (but the work by Geisler and Rolka ( 2021 ) represents a recent exception) and the sample is almost exclusively a sample of volunteers. This latter aspect, recurrent in the mathematics education research, appears to be particularly relevant in STT research since it involves adults, asking them to report an event often perceived as a personal failure.

3.3 Theoretical models

The 55 selected papers include a great variety of theoretical frameworks: several of these theoretical approaches are related to Tall’s ideas of advanced mathematical thinking and the three worlds of mathematics (e.g., Deeken et al., 2020 ; Hong et al., 2009 ), others are based on the Anthropological Theory of the Didactic (e.g., Hausberger, 2018 ), and others are within the theory of commognition by Sfard ( 2008 ). In the latter case, the focus is on the shift of mathematical discourses between secondary school and university (Thoma & Nardi, 2018 ).

Recently, more authors have adopted frameworks related to affective constructs not originally developed in the field of STT: Hernandez-Martinez et al. ( 2011 ) focused on mathematical identity adopting the perspective of Sfard and Prusak ( 2005 ); Ufer et al. ( 2017 ) conceptualised interest within the person-object theory of interest by Krapp ( 2002 ); Dibbs ( 2019 ) used engagement theory (Fredricks et al., 2004 ) to study students’ affective reactions to failure in a calculus course; Geisler ( 2021 ) analysed the role of attitude in the dropout from university mathematics within the three-dimensional model of attitude (Di Martino & Zan, 2010 ).

In 2008, Clark and Lovric observed that ‘perhaps the most notable feature of the existing body of research on transition is the absence of a theoretical model’ (p. 25). In a special issue of the Mathematics Education Research Journal dedicated to transitions, two specific (and influential) models were discussed: the three worlds of mathematics by Tall ( 2008 ) and the rite of passage by Clark and Lovric ( 2008 ).

The three worlds of Mathematics is a theory about the development of mathematical thinking that Tall presented at PME in Bergen (Tall, 2004 ). In this theory, the development of mathematical thinking is described as the development of perceptions of three different but interrelated worlds. In 2008, Tall used this theory to focus on the changes in thinking involved in the STT and on the individual development needed for this transition. According to this theoretical model, there are three fundamental mental structures that shape long-term learning and mathematical thinking: recognition of patterns, repetition of sequences of actions, and language. In this cognitive and epistemological perspective, Tall ( 2008 ) identified ‘three worlds of mathematics that develop in sophistication in quite different ways’ (p. 7): conceptual embodiment, proceptual symbolism (APOS theory is included in this world), and axiomatic formalism. Tall explained how, in his view, the blending of embodiment and symbolism gives a more accurate way of developing sophistication in mathematical thinking. In the final version of his theoretical model, Tall ( 2013 ) included an affective dimension recognising the role of emotions in the interpretation of previous experiences with mathematics, and therefore in the individual development of mathematical thinking.

Clark and Lovric ( 2008 ) adapted a well-established anthropological theory to the study of STT: that of rites of passage. This model recognised three stages in STT (Fig.  2 ).

The three stages of the rite of passage

The liminal stage includes the period from the last part of high school to the first part of university. It is characterised by an unavoidable crisis, known mathematical routines are challenged, and first year students need to find their place in a new mathematics community. This model appeared initially strongly influenced by the cognitive perspective in STT; Clark and Lovric ( 2008 ) discussed only the cognitive shock of the passage from informal to formal language and reasoning in mathematics. A year later, Clark and Lovric ( 2009 ) recognised that the rite of passage inevitably leads to the emergence of affective reactions: the initial reaction of euphoria, the feelings of inadequacy during the crisis, and the recovery after the resolution of the crisis. These affective reactions are particularly strong for those students who were successful in secondary school: this is the case of most of the first-year students in mathematics (Di Martino & Gregorio, 2019 ).

The rite of passage model has two main implications. First, dealing with a significant crisis is a necessary step for a successful passage: instead of avoiding the crisis — for example making the new context like the old one — the crisis needs to be understood to offer students support to overcome it. As Thomas and Klymchuk ( 2012 ) observed: ‘the existence of demanding aspects of transition that are difficult to control is not in itself a good enough reason to ignore those that can be managed to produce a better experience for students’ (p. 285). An unsuccessful rite of passage may result in a never completed incorporation stage, and students’ dropout can be interpreted in this perspective. Second, the rite of passage model stresses the fact that socio-cultural aspects cannot be disregarded in the analysis of STT: the old and new communities through which the rite of passage is accomplished are context specific.

3.4 Main themes of research

Through the coding process, we identified, inductively, four main research themes. We describe them in turn below.

The mathematical gap between secondary school and university. This gap is described in terms of students’ thinking (Godfrey & Thomas, 2008 ), approach and content (Brandell et al., 2008 ), acceptance criteria for justification (Selden, 2012 ; Sommerhoff & Ufer, 2019 ) and for legitimate mathematical activity (Jablonka et al., 2017 ), teaching style and assessment (Thomas & Klymchuk, 2012 ), didactic contract (Pepin, 2014b ) and messages students receive (Kouvela et al., 2018 ), identity (Jooganah & Williams, 2016 ), and high school calculus outcomes and university calculus requirements (Ghedamsi & Lecorre, 2021 ). These studies are mainly conducted within a socio-cultural perspective, adopting the three worlds of mathematics framework (Tall, 2008 ). In recent years, the approach to STT has become more holistic, including new viewpoints. For example, some studies consider the social and discursive perspective related to the analysis of commognitive conflicts (Thoma & Nardi, 2018 ); others describe the gap between school and university mathematical experiences through the description of the perception of the main actors involved: students (e.g., Hernandez-Martinez et al., 2011 ; O’Shea and Breen, 2021 ), schoolteachers (Hong et al., 2009 ), lecturers (Klymchuk et al., 2011 ; Deeken et al., 2020 ), or compare these different perceptions (Corriveau, 2017 ).

The potential of technology to facilitate the STT. Two issues emerge within this category: the effective use of CAS (Computer Algebra Systems) in STT (Hong & Thomas, 2015 ; Varsavsky, 2012 ) and the analysis of the potential of ICT (Information and Communication Technology) to facilitate the STT (Bardelle & Di Martino, 2012 ; Daza et al., 2013 ). It is interesting to observe that these kinds of studies are limited to the period 2012–2015: the recent events related to the pandemic could (should?) generate new interest towards this line of research in STT (Chan et al., 2021 ).

The factors correlated with academic success. The studies in this category consider a wide range of cognitive, social, and affective factors and are mostly quantitative, involving large samples. Their explicit goal is to highlight significant correlations between academic success and other factors, such as students’ attitudes (Geisler, 2021 ), standardised test results (Culpepper et al., 2010 ), attended secondary school (Adamuti-Trache et al., 2013 ), prior knowledge (Rach & Ufer, 2020 ), students’ learning prerequisites (Rach & Heinze, 2017 ), interest (Kosiol et al., 2019 ), and students’ beliefs (Geisler & Rolka, 2021 ).

Failure in STT. This category represents studies of failure to transition: they are mostly qualitative, based on the collection of narratives, often framed as case studies (Hernandez-Martinez, 2016 ). The shared assumption of these studies is that the difficulties in STT are inevitable — according to Clark and Lovric ( 2009 ) even essential — and there is a thin line between success and failure. The description and interpretation of the failure is considered a significant key to understand, prevent, and overcome the students’ difficulties in STT. The definition of failure in STT varies in these studies: there is a local meaning, i.e., negative results in some first-year university exam (Dibbs, 2019 ) and a global meaning, i.e., students who leave university studies (Di Martino & Gregorio, 2019 ).

These four themes of research are highly specific to mathematics. This is evident for the first two themes which deal with subject-related issues. However, it is also the case for the latter two themes. Regarding the third theme, the factors considered are mathematics-related: for example, students’ beliefs and attitudes reveal students’ perceptions about the nature of the mathematics they are encountering at university. The fourth theme, that of failure/success in STT, is related to the students’ vision of the nature of mathematics. A change in one’s mathematical theory of failure/success is a change in one’s vision of mathematics. The mathematics epistemology conveyed by the mathematical culture in the new university environment affects students’ mathematical theories of success through the introduction of new mathematical symbols, knowledge, customs, and requirements for success. Therefore, all the elements of difficulty, not only the cognitive ones, but also the affective and sociocultural ones, are highly specific to mathematics.

4 Conclusion and directions for research

First, a meta-reflection about literature searches based on fully automated search engines such as Google Scholar. Such searches not only present an issue of non-reproducibility (Gusenbauer & Haddaway, 2020 ) but also present an issue of control (or rather of poor control). These searches are affected by external and contextual factors. The list produced is based on an unknown ranking algorithm (Beel & Gipp, 2009 ) and the profiling process and geolocation of who develops the search also play a role. The introduction of external selection criteria not fully controlled by researchers is a significant dilemma for the scientific community.

Notwithstanding this significant issue, the academic search engines are an essential resource to develop systematic literature reviews. We believe that adopting the recommendations of Haddaway et al. ( 2015 ) — for example using a primary and a supplementary search engine, adopting a suitable search query, and considering a large sample of the papers — it is possible to obtain a representative picture of the state of the art in the field investigated, also highlighting the so called ‘grey literature’ (Haddaway et al., 2015 ).

Our review of the research literature on STT from 2008 to 2021 highlights the variety of theoretical frameworks in use, the growing awareness of the complexity of phenomena at play in STT and the consequent adoption of a more holistic approach to STT that goes beyond a purely cognitive interpretation, including socio-cultural and affective issues. The first studies focused on the cognitive and epistemological obstacles of the shift toward advanced mathematical thinking have been integrated and complemented by studies considering social, cultural, and affective issues in the last decade. On the other hand, Artigue ( 2021 ) recently underlined how ‘the socio-cultural turn... has not yet impacted research’ (p. 9) at university level and we argue that the ‘affective turn’ has also not been completely fulfilled in STT research.

The research results obtained considering social, cultural, and affective issues have already some clear implications for the teaching and learning of mathematics at secondary school and at university level. Since the seminal work of Tall ( 1991 ), we know much about the difference between tertiary mathematics and secondary school mathematics: mathematics is understood and presented in a different (advanced) way at the university level. The more recent research on the STT highlights how tertiary transition involves other changes than the purely cognitive and other actors. Many difficulties the learners experience in the passage from school mathematics to university mathematics appear to be related to a sudden change in their mathematics identity (Hernandez-Martinez et al., 2011 ). Many successful students develop a different view of mathematics in their passage to university, often perceiving that their ability in mathematics is suddenly reduced and, consequently, developing very strong negative emotions in their university experience (Geisler & Rolka, 2021 ). This affective phenomenon is however related to an epistemological aspect: the meaning of being a successful student in mathematics. The individuals’ theory of success in mathematics are rooted and consolidated during the school experience (Di Martino & Zan, 2010 ), going often into crisis during the tertiary transition. Whether the development of the students’ theories of success during STT is unavoidable and related to the epistemology of the advanced mathematics encountered at university is an open question at the boundary between epistemological and didactical issues.

Concerning the second research question we posed, we identified four main themes of the STT research: the mathematical gap between secondary school and university, the potential of technology to facilitate STT, and the factors correlated to academic success and to academic failure in STT. These strands of research produced significant outcomes. However, we believe that more research is needed to improve our understanding of the relationship between cognitive, affective, social, and cultural aspects of the STT. We encourage studies that consider the dynamic nature of STT — that consider STT as a process — and bring into play the cultural context in which this crisis takes place. In this perspective, comparative studies between cultural or institutional contexts are still rare: an interesting exception is represented by the study of Deeken et al. ( 2020 ) that, through the Delphi method, described which students’ mathematical abilities are considered minimal prerequisites by university mathematics lecturers. Studies involving and comparing several institutions or countries are rare (Di Martino et al., 2022 , is one such example), but much needed, since they can help to understand the role of contextual factors and the generalisability of studies conducted in a specific context. Further studies in this direction would surely represent a valuable addition to the current body of research.

From a methodological point of view, the analysis of existing research highlights on the one hand the use of several different instruments, approaches, and samples. On the other hand, it points to two significant issues. First, most data are collected through online surveys: this choice is ‘not neutral’. Psychologist have discussed the impact of computer versus paper–pencil survey in collecting self-reports (Bates & Cox, 2008 ) and, in mathematics education, recent publications address the impact in students’ performance in online versus pen and paper tasks (Lemmo, 2021 ).

Second, the participants of these studies are almost exclusively volunteers, and we need to consider and discuss the limitations related to this. The effects of a sample of volunteers appear to be particularly significant in the analysis of failure in STT, where students who drop out or are about to do so are often the focus of research. The volunteer sampling in the STT may create a bias towards higher achieving students (Vollstedt et al., 2014 ).

In our overview, we included papers such as Doukhan ( 2020 ), Griese ( 2017 ), Jablonka et al. ( 2017 ), and Hong and Thomas ( 2015 ). Those papers focus on STT concerning non-specialist students: i.e., students who study mathematics as part of their degree but are not enrolled in a degree course in mathematics. The study of the differences in the mathematical transition to the various STEM degrees — the differences in the mathematical transition between specialist and non-specialist students — appears to be a significant perspective for further research.

These differences can be related to the different mathematical identities of the first-year students, to the different degrees of discontinuity of mathematical contents (for example, calculus) as they are presented at school and as they are presented in the different degrees, as well as to the different mathematical practices and requests in the different degrees. In our view, this research should also involve epistemological, socio-cultural, and affective aspects.

Finally, the result of our systematic review confirms a need of our community: that of learning more about STT in the areas of the world that are not represented in our final corpus. This is not only an important issue of equity and participation, but — considering the role of cultural, affective, and social factors in the STT — our limited knowledge of many educational contexts signifies a limited understanding of the STT, an understanding too culturally bound. We firmly believe that the mathematics education community should make all possible efforts to support researchers from underrepresented educational contexts interested in developing research about STT.

Data availability

The datasets analysed during the current study are available from the correspondent author upon reasonable request. These datasets were derived from the following public domain resources: Google Scholar and ERIC.

For the theoretical studies, we reported and considered the author’s nationality.

The PME conference was not held in 2020, due to the Covid-19 pandemic.

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Di Martino, P., Gregorio, F. & Iannone, P. The transition from school to university in mathematics education research: new trends and ideas from a systematic literature review. Educ Stud Math 113 , 7–34 (2023). https://doi.org/10.1007/s10649-022-10194-w

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