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Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2023 2023.

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

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

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

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

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

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

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

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

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

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

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

Symmetric Generation , Ana Gonzalez

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

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

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

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

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

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

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

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

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

Permutation and Monomial Progenitors , Crystal Diaz

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

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

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

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

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

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

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

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

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

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

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

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

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

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

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

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

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

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

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

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

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

LIFE EXPECTANCY , Ali R. Hassanzadah

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

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

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

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

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

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

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

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

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

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

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

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

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

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

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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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Dissertations

Here is the complete list of all doctoral dissertations granted by the School of Math, which dates back to 1965. Included below are also all masters theses produced by our students since 2002. A combined listing of all dissertations and theses , going back to 1934, is available at Georgia Tech's library archive. For the post PhD employment of our graduates see our  Alumni Page .

Doctoral Dissertations

Masters dissertations.

dissertation on mathematics

Dissertations and Placements 2010-Present

Kimoi Kemboi Thesis: Full exceptional collections of vector bundles on linear GIT quotients Advisor: Daniel Halpern-Leistner First Position: Postdoc at the Institution for Advanced Study and Princeton

Max Lipton Thesis: Dynamical Systems in Pure Mathematics Advisor: Steven Strogatz First Position: NSF Mathematical Sciences Postdoctoral Fellow at Massachusetts Institute of Technology

Elise McMahon Thesis: A simplicial set approach to computing the group homology of some orthogonal subgroups of the discrete group  Advisor: Inna Zakharevich First Position: Senior Research Scientist at Two Six Technologies

Peter Uttenthal Thesis: Density of Selmer Ranks in Families of Even Galois Representations Advisor: Ravi Kumar Ramakrishna First Position: Visiting Assistant Professor at Cornell University

Liu Yun Thesis: Towers of Borel Fibrations and Generalized Quasi-Invariants Advisor: Yuri Berest First Position: Postdoc at Indiana University Bloomington

Romin Abdolahzadi Thesis: Anabelian model theory Advisor: Anil Nerode First Position: Quantitative Analyst, A.R.T. Advisors, LLC

Hannah Cairns Thesis: Abelian processes, and how they go to sleep Advisor: Lionel Levine First Position: Visiting Assistant Professor, Cornell University

Shiping Cao Thesis: Topics in scaling limits on some Sierpinski carpet type fractals Advisor: Robert Strichartz (Laurent Saloff-Coste in last semester) First Position: Postdoctoral Scholar, University of Washington

Andres Fernandes Herrero Thesis: On the boundedness of the moduli of logarithmic connections Advisor: Nicolas Templier First Position: Ritt Assistant Professor, Columbia University

Max Hallgren Thesis: Ricci Flow with a Lower Bound on Ricci Curvature Advisor: Xiaodong Cao First Position: NSF Postdoctoral Research Fellow, Rutgers University

Gautam Krishnan Thesis: Degenerate series representations for symplectic groups Advisor: Dan Barbasch First Position: Hill Assistant Professor, Rutgers University

Feng Liang Thesis: Mixing time and limit shapes of Abelian networks Advisor: Lionel Levine

David Mehrle Thesis: Commutative and Homological Algebra of Incomplete Tambara Functors Advisor: Inna Zakharevich First Position: Postdoctoral Scholar, University of Kentucky

Itamar Sales de Oliveira Thesis: A new approach to the Fourier extension problem for the paraboloid Advisor: Camil Muscalu First Position: Postdoctoral Researcher, Nantes Université

Brandon Shapiro Thesis: Shape Independent Category Theory Advisor: Inna Zakharevich First Position: Postdoctoral Fellow, Topos Institute

Ayah Almousa Thesis: Combinatorial characterizations of polarizations of powers of the graded maximal ideal Advisor: Irena Peeva First position: RTG Postdoctoral Fellow, University of Minnesota

Jose Bastidas Thesis: Species and hyperplane arrangements Advisor: Marcelo Aguiar First position: Postdoctoral Fellow, Université du Québec à Montréal

Zaoli Chen Thesis: Clustered Behaviors of Extreme Values Advisor: Gennady Samorodnitsky First Position: Postdoctoral Researcher, Department of and Statistics, University of Ottawa

Ivan Geffner Thesis: Implementing Mediators with Cheap Talk Advisor: Joe Halpern First Position: Postdoctoral Researcher, Technion – Israel Institute of Technology

Ryan McDermott Thesis: Phase Transitions and Near-Critical Phenomena in the Abelian Sandpile Model Advisor: Lionel Levine

Aleksandra Niepla Thesis:  Iterated Fractional Integrals and Applications to Fourier Integrals with Rational Symbol Advisor: Camil Muscalu First Position: Visiting Assistant Professor, College of the Holy Cross

Dylan Peifer Thesis: Reinforcement Learning in Buchberger's Algorithm Advisor: Michael Stillman First Position: Quantitative Researcher, Susquehanna International Group

Rakvi Thesis: A Classification of Genus 0 Modular Curves with Rational Points Advisor: David Zywina First Position: Hans Rademacher Instructor, University of Pennsylvania

Ana Smaranda Sandu Thesis: Knowledge of counterfactuals Advisor: Anil Nerode First Position: Instructor in Science Laboratory, Computer Science Department, Wellesley College

Maru Sarazola Thesis: Constructing K-theory spectra from algebraic structures with a class of acyclic objects Advisor: Inna Zakharevich First Position: J.J. Sylvester Assistant Professor, Johns Hopkins University

Abigail Turner Thesis: L2 Minimal Algorithms Advisor: Steven Strogatz

Yuwen Wang Thesis: Long-jump random walks on finite groups Advisor: Laurent Saloff-Coste First Position: Postdoc, University of Innsbruck, Austria

Beihui Yuan Thesis:  Applications of commutative algebra to spline theory and string theory Advisor: Michael Stillman First Position: Research Fellow, Swansea University

Elliot Cartee Thesis: Topics in Optimal Control and Game Theory Advisor: Alexander Vladimirsky First Position: L.E. Dickson Instructor, Department of , University of Chicago

Frederik de Keersmaeker Thesis: Displaceability in Symplectic Geometry Advisor: Tara Holm First Position: Consultant, Addestino Innovation Management

Lila Greco Thesis: Locally Markov Walks and Branching Processes Advisor: Lionel Levine First Position: Actuarial Assistant, Berkshire Hathaway Specialty Insurance

Benjamin Hoffman Thesis: Polytopes And Hamiltonian Geometry: Stacks, Toric Degenerations, And Partial Advisor: Reyer Sjamaar First Position: Teaching Associate, Department of , Cornell University

Daoji Huang Thesis: A Bruhat Atlas on the Wonderful Compactification of PS O(2 n )/ SO (2 n -1) and A Kazhdan-Lusztig Atlas on G/P Advisor: Allen Knutson First Position: Postdoctoral Associate, University of Minnesota

Pak-Hin Li Thesis: A Hopf Algebra from Preprojective Modules Advisor: Allen Knutson First position: Associate, Goldman Sachs

Anwesh Ray Thesis: Lifting Reducible Galois Representations Advisor: Ravi Ramakrishna First Position: Postdoctoral Fellowship, University of British Columbia

Avery St. Dizier Thesis: Combinatorics of Schubert Polynomials Advisor: Karola Meszaros First Position: Postdoctoral Fellowship, Department of , University of Illinois at Urbana-Champaign

Shihao Xiong Thesis: Forcing Axioms For Sigma-Closed Posets And Their Consequences Advisor: Justin Moore First Position: Algorithm Developer, Hudson River Trading

Swee Hong Chan Thesis: Nonhalting abelian networks Advisor: Lionel Levine First Position: Hedrick Adjunct Assistant Professor, UCLA

Joseph Gallagher Thesis: On conjectures related to character varieties of knots and Jones polynomials Advisor: Yuri Berest First Position: Data Scientist, Capital One

Jun Le Goh Thesis: Measuring the Relative Complexity of Mathematical Constructions and Theorems Advisor: Richard Shore First Position: Van Vleck Assistant Professor, University of Wisconsin-Madison

Qi Hou Thesis: Rough Hypoellipticity for Local Weak Solutions to the Heat Equation in Dirichlet Spaces Advisor: Laurent Saloff-Coste First Position: Visiting Assistant Professor, Department of , Cornell University

Jingbo Liu Thesis: Heat kernel estimate of the Schrodinger operator in uniform domains Advisor: Laurent Saloff-Coste First Position: Data Scientist, Jet.com

Ian Pendleton Thesis:  The Fundamental Group, Homology, and Cohomology of Toric Origami 4-Manifolds Advisor: Tara Holm

Amin Saied Thesis: Stable representation theory of categories and applications to families of (bi)modules over symmetric groups Advisor: Martin Kassabov First Position: Data Scientist, Microsoft

Yujia Zhai Thesis:  Study of bi-parameter flag paraproducts and bi-parameter stopping-time algorithms Advisor: Camil Muscalu First Position: Postdoctoral Associate, Université de Nantes 

Tair Akhmejanov Thesis: Growth Diagrams from Polygons in the Affine Grassmannian Advisor: Allen Knutson First position: Arthur J. Krener Assistant Professor, University of California, Davis

James Barnes Thesis:  The Theory of the Hyperarithmetic Degrees Advisor: Richard Shore First position: Visiting Lecturer, Wellesley College

Jeffrey Bergfalk Thesis:  Dimensions of ordinals: set theory, homology theory, and the first omega alephs Advisor: Justin Moore Postdoctoral Associate, UNAM - National Autonomous University of Mexico

TaoRan Chen Thesis: The Inverse Deformation Problem Advisor: Ravi Ramakrishna

Sergio Da Silva Thesis: On the Gorensteinization of Schubert varieties via boundary divisors Advisor: Allen Knutson First position: Pacific Institute for the Mathematical Sciences (PIMS) postdoctoral fellowship, University of Manitoba

Eduard Einstein Thesis:  Hierarchies for relatively hyperbolic compact special cube complexes Advisor: Jason Manning First position: Research Assistant Professor (Postdoc), University of Illinois, Chicago (UIC)

Balázs Elek Thesis:  Toric surfaces with Kazhdan-Lusztig atlases Advisor: Allen Knutson First position: Postdoctoral Fellow, University of Toronto

Kelsey Houston-Edwards Thesis:  Discrete Heat Kernel Estimates in Inner Uniform Domains Advisor: Laurent Saloff-Coste First position: Professor of Math and Science Communication, Olin College

My Huynh Thesis:  The Gromov Width of Symplectic Cuts of Symplectic Manifolds. Advisor: Tara Holm First position: Applied Mathematician, Applied Research Associates Inc., Raleigh NC.

Hossein Lamei Ramandi Thesis: On the minimality of non-σ-scattered orders Advisor: Justin Moore First position:  Postdoctoral Associate at UFT (University Toronto)

Christine McMeekin Thesis: A density of ramified primes Advisor: Ravi Ramakrishna First position: Researcher at Max Planck Institute

Aliaksandr Patotski Thesis:  Derived characters of Lie representations and Chern-Simons forms Advisor: Yuri Berest First position: Data Scientist, Microsoft

Ahmad Rafiqi Thesis:  On dilatations of surface automorphisms Advisor: John Hubbard First position: Postdoctoral Associate, Sao Palo, Brazil

Ying-Ying Tran Thesis:  Computably enumerable boolean algebras Advisor: Anil Nerode First position: Quantitative Researcher

Drew Zemke Thesis:  Surfaces in Three- and Four-Dimensional Topology Advisor: Jason Manning First position: Preceptor in , Harvard University

Heung Shan Theodore Hui Thesis: A Radical Characterization of Abelian Varieties  Advisor: David Zywina First position: Quantitative Researcher, Eastmore Group

Daniel Miller Thesis: Counterexamples related to the Sato–Tate conjecture Advisor: Ravi Ramakrishna First position: Data Scientist, Microsoft

Lihai Qian Thesis: Rigidity on Einstein manifolds and shrinking Ricci solitons in high dimensions Advisor: Xiaodong Cao First position: Quantitative Associate, Wells Fargo

Valente Ramirez Garcia Luna Thesis: Quadratic vector fields on the complex plane: rigidity and analytic invariants Advisor: Yulij Ilyashenko First position: Lebesgue Post-doc Fellow, Institut de Recherche Mathématique de Rennes

Iian Smythe Thesis: Set theory in infinite-dimensional vector spaces Advisor: Justin Moore First position: Hill Assistant Professor at Rutgers, the State University of New Jersey

Zhexiu Tu Thesis: Topological representations of matroids and the cd-index Advisor: Edward Swartz First position: Visiting Professor - Centre College, Kentucky

Wai-kit Yeung Thesis: Representation homology and knot contact homology Advisor: Yuri Berest First position: Zorn postdoctoral fellow, Indiana University

Lucien Clavier Thesis: Non-affine horocycle orbit closures on strata of translation surfaces: new examples Advisor: John Smillie First position: Consultant in Capital Markets, Financial Risk at Deloitte Luxembourg

Voula Collins Thesis: Crystal branching for non-Levi subgroups and a puzzle formula for the equivariant cohomology of the cotangent bundle on projective space Advisor: Allen Knutson FIrst position: Postdoctoral Associate, University of Connecticut

Pok Wai Fong Thesis: Smoothness Properties of symbols, Calderón Commutators and Generalizations Advisor: Camil Muscalu First position: Quantitative researcher, Two Sigma

Tom Kern Thesis: Nonstandard models of the weak second order theory of one successor Advisor: Anil Nerode First position: Visiting Assistant Professor, Cornell University

Robert Kesler Thesis: Unbounded multilinear multipliers adapted to large subspaces and estimates for degenerate simplex operators Advisor: Camil Muscalu First position: Postdoctoral Associate, Georgia Institute of Technology

Yao Liu Thesis: Riesz Distributions Assiciated to Dunkl Operators Advisor: Yuri Berest First position: Visiting Assistant Professor, Cornell University

Scott Messick Thesis: Continuous atomata, compactness, and Young measures Advisor: Anil Nerode First position: Start-up

Aaron Palmer Thesis: Incompressibility and Global Injectivity in Second-Gradient Non-Linear Elasticity Advisor: Timothy J. Healey First position: Postdoctoral fellow, University of British Columbia 

Kristen Pueschel Thesis: On Residual Properties of Groups and Dehn Functions for Mapping Tori of Right Angled Artin Groups Advisor: Timothy Riley First position: Postdoctoral Associate, University of Arkansas

Chenxi Wu Thesis: Translation surfaces: saddle connections, triangles, and covering constructions. Advisor: John Smillie First position: Postdoctoral Associate, Max Planck Institute of

David Belanger Thesis: Sets, Models, And Proofs: Topics In The Theory Of Recursive Functions Advisor: Richard A. Shore First position: Research Fellow, National University of Singapore

Cristina Benea Thesis: Vector-Valued Extensions for Singular Bilinear Operators and Applications Advisor: Camil Muscalu First position: University of Nantes, France

Kai Fong Ernest Chong Thesis: Face Vectors and Hilbert Functions Advisor: Edward Swartz First position: Research Scientist, Agency for Science, Technology and Research, Singapore

Laura Escobar Vega Thesis: Brick Varieties and Toric Matrix Schubert Varieties Advisor: Allen Knutson First position: J. L. Doob Research Assistant Professor at UIUC

Joeun Jung Thesis: Iterated trilinear fourier integrals with arbitrary symbols Advisor: Camil Muscalu First position: Researcher, PARC (PDE and Functional Analysis Research Center) of Seoul National University

Yasemin Kara Thesis: The laplacian on hyperbolic Riemann surfaces and Maass forms Advisor: John H. Hubbard Part Time Instructor, Faculty of Engineering and Natural Sciences, Bahcesehir University

Chor Hang Lam Thesis: Homological Stability Of Diffeomorphism Groups Of 3-Manifolds Advisor: Allen Hatcher

Yash Lodha Thesis: Finiteness Properties And Piecewise Projective Homeomorphisms Advisor: Justin Moore and Timothy Riley First position: Postdoctoral fellow at Ecole Polytechnique Federale de Lausanne in Switzerland

Radoslav Zlatev Thesis: Examples of Implicitization of Hypersurfaces through Syzygies Advisor: Michael E. Stillman First position: Associate, Credit Strats, Goldman Sachs

Margarita Amchislavska Thesis: The geometry of generalized Lamplighter groups Advisor: Timothy Riley First position: Department of Defense

Hyungryul Baik Thesis: Laminations on the circle and hyperbolic geometry Advisor: John H. Hubbard First position: Postdoctoral Associate, Bonn University

Adam Bjorndahl Thesis: Language-based games Advisor: Anil Nerode and Joseph Halpern First position: Tenure Track Professor, Carnegie Mellon University Department of Philosophy

Youssef El Fassy Fihry Thesis: Graded Cherednik Algebra And Quasi-Invariant Differential Forms Advisor: Yuri Berest First position: Software Developer, Microsoft

Chikwong Fok Thesis: The Real K-theory of compact Lie groups Advisor: Reyer Sjamaar First position: Postdoctoral fellow in the National Center for Theoretical Sciences, Taiwan

Kathryn Lindsey Thesis: Families Of Dynamical Systems Associated To Translation Surfaces Advisor: John Smillie First position: Postdoctoral Associate, University of Chicago

Andrew Marshall Thesis: On configuration spaces of graphs Advisor: Allan Hatcher First position: Visiting Assistant Professor, Cornell University

Robyn Miller Thesis: Symbolic Dynamics Of Billiard Flow In Isosceles Triangles Advisor: John Smillie First position: Postdoctoral Researcher at Mind Research Network

Diana Ojeda Aristizabal Thesis: Ramsey theory and the geometry of Banach spaces Advisor: Justin Moore First position: Postdoctoral Fellow, University of Toronto

Hung Tran Thesis: Aspects of the Ricci flow Advisor: Xiaodong Cao First position: Visiting Assistant Professor, University of California at Irvine

Baris Ugurcan Thesis: LPLP-Estimates And Polyharmonic Boundary Value Problems On The Sierpinski Gasket And Gaussian Free Fields On High Dimensional Sierpinski Carpet Graphs Advisor: Robert S. Strichartz First position: Postdoctoral Fellowship, University of Western Ontario

Anna Bertiger Thesis: The Combinatorics and Geometry of the Orbits of the Symplectic Group on Flags in Complex Affine Space Advisor: Allen Knutson First position: University of Waterloo, Postdoctoral Fellow

Mariya Bessonov Thesis: Probabilistic Models for Population Dynamics Advisor: Richard Durrett First position: CUNY City Tech, Assistant Professor, Tenure Track

Igors Gorbovickis Thesis: Some Problems from Complex Dynamical Systems and Combinatorial Geometry Advisor: Yulij Ilyashenko First position: Postdoctoral Fellow, University of Toronto

Marisa Hughes Thesis: Quotients of Spheres by Linear Actions of Abelian Groups Advisor: Edward Swartz First position: Visiting Professor, Hamilton College

Kristine Jones Thesis: Generic Initial Ideals of Locally Cohen-Macaulay Space Curves Advisor: Michael E. Stillman First position: Software Developer, Microsoft

Shisen Luo Thesis: Hard Lefschetz Property of Hamiltonian GKM Manifolds Advisor: Tara Holm First position: Associate, Goldman Sachs

Peter Luthy Thesis: Bi-parameter Maximal Multilinear Operators Advisor: Camil Muscalu First position: Chauvenet Postdoctoral Lecturer at Washington University in St. Louis 

Remus Radu Thesis: Topological models for hyperbolic and semi-parabolic complex Hénon maps Advisor: John H. Hubbard First position: Milnor Lecturer, Institute for Mathematical Sciences, Stony Brook University

Jenna Rajchgot Thesis: Compatibly Split Subvarieties of the Hilbert Scheme of Points in the Plane Advisor: Allen Knutson First position: Research member at the Mathematical Sciences Research Institute (fall 2012); postdoc at the University of Michigan

Raluca Tanase Thesis: Hénon maps, discrete groups and continuity of Julia sets Advisor: John H. Hubbard First position: Milnor Lecturer, Institute for Mathematical Sciences, Stony Brook University

Ka Yue Wong Thesis: Dixmier Algebras on Complex Classical Nilpotent Orbits and their Representation Theories Advisor: Dan M. Barbasch First position: Postdoctoral fellow at Hong Kong University of Science and Technology

Tianyi Zheng Thesis: Random walks on some classes of solvable groups Advisor: Laurent Saloff-Coste First position: Postdoctoral Associate, Stanford University

Juan Alonso Thesis: Graphs of Free Groups and their Measure Equivalence Advisor: Karen Vogtmann First position: Postdoc at Uruguay University

Jason Anema Thesis: Counting Spanning Trees on Fractal Graphs Advisor: Robert S. Strichartz First position: Visiting assistant professor of mathematics at Cornell University

Saúl Blanco Rodríguez Thesis: Shortest Path Poset of Bruhat Intervals and the Completecd-Index Advisor: Louis Billera First position: Visiting assistant professor of mathematics at DePaul University

Fatima Mahmood Thesis: Jacobi Structures and Differential Forms on Contact Quotients Advisor: Reyer Sjamaar First position: Visiting assistant professor at University of Rochester

Philipp Meerkamp Thesis: Singular Hopf Bifurcation Advisor: John M. Guckenheimer First position: Financial software engineer at Bloomberg LP

Milena Pabiniak Thesis: Hamiltonian Torus Actions in Equivariant Cohomology and Symplectic Topology Advisor: Tara Holm First position: Postdoctoral associate at the University of Toronto

Peter Samuelson Thesis: Kauffman Bracket Skein Modules and the Quantum Torus Advisor: Yuri Berest First position: Postdoctoral associate at the University of Toronto

Mihai Bailesteanu  Thesis: The Heat Equation under the Ricci Flow Advisor: Xiaodong Cao First position: Visiting assistant professor at the University of Rochester

Owen Baker  Thesis:  The Jacobian Map on Outer Space Advisor: Karen Vogtmann First position: Postdoctoral fellow at McMaster University

Jennifer Biermann  Thesis: Free Resolutions of Monomial Ideals Advisor: Irena Peeva First position: Postdoctoral fellow at Lakehead University

Mingzhong Cai  Thesis: Elements of Classical Recursion Theory: Degree-Theoretic Properties and Combinatorial Properties Advisor: Richard A. Shore First position: Van Vleck visiting assistant professor at the University of Wisconsin at Madison

Ri-Xiang Chen  Thesis: Hilbert Functions and Free Resolutions Advisor: Irena Peeva First position: Instructor at Shantou University in Guangdong, China

Denise Dawson  Thesis: Complete Reducibility in Euclidean Twin Buildings Advisor: Kenneth S. Brown First position: Assistant professor of mathematics at Charleston Southern University

George Khachatryan Thesis: Derived Representation Schemes and Non-commutative Geometry Advisor: Yuri Berest First position: Reasoning Mind

Samuel Kolins  Thesis: Face Vectors of Subdivision of Balls Advisor: Edward Swartz First position: Assistant professor at Lebanon Valley College

Victor Kostyuk Thesis: Outer Space for Two-Dimensional RAAGs and Fixed Point Sets of Finite Subgroups Advisor: Karen Vogtmann First position: Knowledge engineering at Reasoning Mind

Ho Hon Leung  Thesis: K-Theory of Weight Varieties and Divided Difference Operators in Equivariant KK-Theory Advisor: Reyer Sjamaar First position: Assistant professor at the Canadian University of Dubai

Benjamin Lundell  Thesis: Selmer Groups and Ranks of Hecke Rings Advisor: Ravi Ramakrishna First position: Acting assistant professor at the University of Washington

Eyvindur Ari Palsson  Thesis: Lp Estimates for a Singular Integral Operator Motivated by Calderón’s Second Commutator Advisor: Camil Muscalu First position: Visiting assistant professor at the University of Rochester

Paul Shafer  Thesis: On the Complexity of Mathematical Problems: Medvedev Degrees and Reverse Advisor: Richard A. Shore First position: Lecturer at Appalachian State University

Michelle Snider  Thesis: Affine Patches on Positroid Varieties and Affine Pipe Dreams Advisor: Allen Knutson First position: Government consulting job in Maryland

Santi Tasena Thesis: Heat Kernel Analysis on Weighted Dirichlet Spaces Advisor: Laurent Saloff-Coste First position: Lecturer professor at Chiang Mai University, Thailand

Russ Thompson  Thesis: Random Walks and Subgroup Geometry Advisor: Laurent Saloff-Coste First position: Postdoctoral fellow at the Mathematical Sciences Research Institute

Gwyneth Whieldon Thesis: Betti Numbers of Stanley-Reisner Ideals Advisor: Michael E. Stillman First position: Assistant professor of mathematics at Hood College

Andrew Cameron Thesis: Estimates for Solutions of Elliptic Partial Differential Equations with Explicit Constants and Aspects of the Finite Element Method for Second-Order Equations Advisor: Alfred H. Schatz First position: Adjunct instructor of mathematics at Tompkins Cortland Community College

Timothy Goldberg Thesis: Hamiltonian Actions in Integral Kähler and Generalized Complex Geometry Advisor: Reyer Sjamaar First position: Visiting assistant professor of mathematics at Lenoir-Rhyne University

Gregory Muller Thesis: The Projective Geometry of Differential Operators Advisor: Yuri Berest First position: Assistant professor at Louisiana State University 

Matthew Noonan Thesis: Geometric Backlund transofrmation in homogeneous spaces Advisor: John H. Hubbard

Sergio Pulido Niño Thesis: Financial Markets with Short Sales Prohibition Advisor: Philip E. Protter First position: Postdoctoral associate in applied probability and finance at Carnegie Mellon University

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This collection contains a selection of the latest doctoral theses completed at the School of Mathematics. Please note this is not a comprehensive record.

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Recent Submissions

Statistical and machine learning approaches to genomic medicine , using markov chain monte carlo in vector generalized linear mixed models: with an application to integral projection models in ecology , symmetries of riemann surfaces and magnetic monopoles , kan extensions in probability theory , regression analysis for extreme value responses and covariates , categorical torelli theorems for fano threefolds , laplacians for structure recovery on directed and higher-order graphs , efficient interior point algorithms for large scale convex optimization problems , solving sampling and optimization problems via tamed langevin mcmc algorithms in the presence of super-linearities , algebraic combinatorial structures for singular stochastic dynamics , stochastic modelling and inference of ocean transport , convergence problems for singular stochastic dynamics , classification of supersymmetric black holes in ads₅ , bps cohomology for 2-calabi—yau categories , quantitative finance informed machine learning , efficient model fitting approaches for estimating abundance and demographic rates for marked and unmarked populations , path-based splitting methods for sdes and machine learning for battery lifetime prognostics , certain geometric maximal functions in harmonic analysis , examining the effects of magnetic fields in neutron star mergers through numerical simulations , on slope stability of tangent sheaves on smooth toric fano varieties .

dissertation on mathematics

Senior Thesis

This page is for Undergraduate Senior Theses.  For Ph.D. Theses, see here .

So that Math Department senior theses can more easily benefit other undergraduate, we would like to exhibit more senior theses online (while all theses are available through Harvard University Archives , it would be more convenient to have them online). It is absolutely voluntary, but if you decide to give us your permission, please send an electronic version of your thesis to cindy@math. The format can be in order of preference: DVI, PS, PDF. In the case of submitting a DVI format, make sure to include all EPS figures. You can also submit Latex or MS word source files.

If you are looking for information and advice from students and faculty about writing a senior thesis, look at this document . It was compiled from comments of students and faculty in preparation for, and during, an information session. Let Wes Cain ([email protected]) know if you have any questions not addressed in the document.

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Digital Commons @ USF > College of Arts and Sciences > Mathematics and Statistics > Theses and Dissertations

Mathematics and Statistics Theses and Dissertations

Theses/dissertations from 2023 2023.

Classification of Finite Topological Quandles and Shelves via Posets , Hitakshi Lahrani

Applied Analysis for Learning Architectures , Himanshu Singh

Rational Functions of Degree Five That Permute the Projective Line Over a Finite Field , Christopher Sze

Theses/Dissertations from 2022 2022

New Developments in Statistical Optimal Designs for Physical and Computer Experiments , Damola M. Akinlana

Advances and Applications of Optimal Polynomial Approximants , Raymond Centner

Data-Driven Analytical Predictive Modeling for Pancreatic Cancer, Financial & Social Systems , Aditya Chakraborty

On Simultaneous Similarity of d-tuples of Commuting Square Matrices , Corey Connelly

Symbolic Computation of Lump Solutions to a Combined (2+1)-dimensional Nonlinear Evolution Equation , Jingwei He

Boundary behavior of analytic functions and Approximation Theory , Spyros Pasias

Stability Analysis of Delay-Driven Coupled Cantilevers Using the Lambert W-Function , Daniel Siebel-Cortopassi

A Functional Optimization Approach to Stochastic Process Sampling , Ryan Matthew Thurman

Theses/Dissertations from 2021 2021

Riemann-Hilbert Problems for Nonlocal Reverse-Time Nonlinear Second-order and Fourth-order AKNS Systems of Multiple Components and Exact Soliton Solutions , Alle Adjiri

Zeros of Harmonic Polynomials and Related Applications , Azizah Alrajhi

Combination of Time Series Analysis and Sentiment Analysis for Stock Market Forecasting , Hsiao-Chuan Chou

Uncertainty Quantification in Deep and Statistical Learning with applications in Bio-Medical Image Analysis , K. Ruwani M. Fernando

Data-Driven Analytical Modeling of Multiple Myeloma Cancer, U.S. Crop Production and Monitoring Process , Lohuwa Mamudu

Long-time Asymptotics for mKdV Type Reduced Equations of the AKNS Hierarchy in Weighted L 2 Sobolev Spaces , Fudong Wang

Online and Adjusted Human Activities Recognition with Statistical Learning , Yanjia Zhang

Theses/Dissertations from 2020 2020

Bayesian Reliability Analysis of The Power Law Process and Statistical Modeling of Computer and Network Vulnerabilities with Cybersecurity Application , Freeh N. Alenezi

Discrete Models and Algorithms for Analyzing DNA Rearrangements , Jasper Braun

Bayesian Reliability Analysis for Optical Media Using Accelerated Degradation Test Data , Kun Bu

On the p(x)-Laplace equation in Carnot groups , Robert D. Freeman

Clustering methods for gene expression data of Oxytricha trifallax , Kyle Houfek

Gradient Boosting for Survival Analysis with Applications in Oncology , Nam Phuong Nguyen

Global and Stochastic Dynamics of Diffusive Hindmarsh-Rose Equations in Neurodynamics , Chi Phan

Restricted Isometric Projections for Differentiable Manifolds and Applications , Vasile Pop

On Some Problems on Polynomial Interpolation in Several Variables , Brian Jon Tuesink

Numerical Study of Gap Distributions in Determinantal Point Process on Low Dimensional Spheres: L -Ensemble of O ( n ) Model Type for n = 2 and n = 3 , Xiankui Yang

Non-Associative Algebraic Structures in Knot Theory , Emanuele Zappala

Theses/Dissertations from 2019 2019

Field Quantization for Radiative Decay of Plasmons in Finite and Infinite Geometries , Maryam Bagherian

Probabilistic Modeling of Democracy, Corruption, Hemophilia A and Prediabetes Data , A. K. M. Raquibul Bashar

Generalized Derivations of Ternary Lie Algebras and n-BiHom-Lie Algebras , Amine Ben Abdeljelil

Fractional Random Weighted Bootstrapping for Classification on Imbalanced Data with Ensemble Decision Tree Methods , Sean Charles Carter

Hierarchical Self-Assembly and Substitution Rules , Daniel Alejandro Cruz

Statistical Learning of Biomedical Non-Stationary Signals and Quality of Life Modeling , Mahdi Goudarzi

Probabilistic and Statistical Prediction Models for Alzheimer’s Disease and Statistical Analysis of Global Warming , Maryam Ibrahim Habadi

Essays on Time Series and Machine Learning Techniques for Risk Management , Michael Kotarinos

The Systems of Post and Post Algebras: A Demonstration of an Obvious Fact , Daviel Leyva

Reconstruction of Radar Images by Using Spherical Mean and Regular Radon Transforms , Ozan Pirbudak

Analyses of Unorthodox Overlapping Gene Segments in Oxytricha Trifallax , Shannon Stich

An Optimal Medium-Strength Regularity Algorithm for 3-uniform Hypergraphs , John Theado

Power Graphs of Quasigroups , DayVon L. Walker

Theses/Dissertations from 2018 2018

Groups Generated by Automata Arising from Transformations of the Boundaries of Rooted Trees , Elsayed Ahmed

Non-equilibrium Phase Transitions in Interacting Diffusions , Wael Al-Sawai

A Hybrid Dynamic Modeling of Time-to-event Processes and Applications , Emmanuel A. Appiah

Lump Solutions and Riemann-Hilbert Approach to Soliton Equations , Sumayah A. Batwa

Developing a Model to Predict Prevalence of Compulsive Behavior in Individuals with OCD , Lindsay D. Fields

Generalizations of Quandles and their cohomologies , Matthew J. Green

Hamiltonian structures and Riemann-Hilbert problems of integrable systems , Xiang Gu

Optimal Latin Hypercube Designs for Computer Experiments Based on Multiple Objectives , Ruizhe Hou

Human Activity Recognition Based on Transfer Learning , Jinyong Pang

Signal Detection of Adverse Drug Reaction using the Adverse Event Reporting System: Literature Review and Novel Methods , Minh H. Pham

Statistical Analysis and Modeling of Cyber Security and Health Sciences , Nawa Raj Pokhrel

Machine Learning Methods for Network Intrusion Detection and Intrusion Prevention Systems , Zheni Svetoslavova Stefanova

Orthogonal Polynomials With Respect to the Measure Supported Over the Whole Complex Plane , Meng Yang

Theses/Dissertations from 2017 2017

Modeling in Finance and Insurance With Levy-It'o Driven Dynamic Processes under Semi Markov-type Switching Regimes and Time Domains , Patrick Armand Assonken Tonfack

Prevalence of Typical Images in High School Geometry Textbooks , Megan N. Cannon

On Extending Hansel's Theorem to Hypergraphs , Gregory Sutton Churchill

Contributions to Quandle Theory: A Study of f-Quandles, Extensions, and Cohomology , Indu Rasika U. Churchill

Linear Extremal Problems in the Hardy Space H p for 0 p , Robert Christopher Connelly

Statistical Analysis and Modeling of Ovarian and Breast Cancer , Muditha V. Devamitta Perera

Statistical Analysis and Modeling of Stomach Cancer Data , Chao Gao

Structural Analysis of Poloidal and Toroidal Plasmons and Fields of Multilayer Nanorings , Kumar Vijay Garapati

Dynamics of Multicultural Social Networks , Kristina B. Hilton

Cybersecurity: Stochastic Analysis and Modelling of Vulnerabilities to Determine the Network Security and Attackers Behavior , Pubudu Kalpani Kaluarachchi

Generalized D-Kaup-Newell integrable systems and their integrable couplings and Darboux transformations , Morgan Ashley McAnally

Patterns in Words Related to DNA Rearrangements , Lukas Nabergall

Time Series Online Empirical Bayesian Kernel Density Segmentation: Applications in Real Time Activity Recognition Using Smartphone Accelerometer , Shuang Na

Schreier Graphs of Thompson's Group T , Allen Pennington

Cybersecurity: Probabilistic Behavior of Vulnerability and Life Cycle , Sasith Maduranga Rajasooriya

Bayesian Artificial Neural Networks in Health and Cybersecurity , Hansapani Sarasepa Rodrigo

Real-time Classification of Biomedical Signals, Parkinson’s Analytical Model , Abolfazl Saghafi

Lump, complexiton and algebro-geometric solutions to soliton equations , Yuan Zhou

Theses/Dissertations from 2016 2016

A Statistical Analysis of Hurricanes in the Atlantic Basin and Sinkholes in Florida , Joy Marie D'andrea

Statistical Analysis of a Risk Factor in Finance and Environmental Models for Belize , Sherlene Enriquez-Savery

Putnam's Inequality and Analytic Content in the Bergman Space , Matthew Fleeman

On the Number of Colors in Quandle Knot Colorings , Jeremy William Kerr

Statistical Modeling of Carbon Dioxide and Cluster Analysis of Time Dependent Information: Lag Target Time Series Clustering, Multi-Factor Time Series Clustering, and Multi-Level Time Series Clustering , Doo Young Kim

Some Results Concerning Permutation Polynomials over Finite Fields , Stephen Lappano

Hamiltonian Formulations and Symmetry Constraints of Soliton Hierarchies of (1+1)-Dimensional Nonlinear Evolution Equations , Solomon Manukure

Modeling and Survival Analysis of Breast Cancer: A Statistical, Artificial Neural Network, and Decision Tree Approach , Venkateswara Rao Mudunuru

Generalized Phase Retrieval: Isometries in Vector Spaces , Josiah Park

Leonard Systems and their Friends , Jonathan Spiewak

Resonant Solutions to (3+1)-dimensional Bilinear Differential Equations , Yue Sun

Statistical Analysis and Modeling Health Data: A Longitudinal Study , Bhikhari Prasad Tharu

Global Attractors and Random Attractors of Reaction-Diffusion Systems , Junyi Tu

Time Dependent Kernel Density Estimation: A New Parameter Estimation Algorithm, Applications in Time Series Classification and Clustering , Xing Wang

On Spectral Properties of Single Layer Potentials , Seyed Zoalroshd

Theses/Dissertations from 2015 2015

Analysis of Rheumatoid Arthritis Data using Logistic Regression and Penalized Approach , Wei Chen

Active Tile Self-assembly and Simulations of Computational Systems , Daria Karpenko

Nearest Neighbor Foreign Exchange Rate Forecasting with Mahalanobis Distance , Vindya Kumari Pathirana

Statistical Learning with Artificial Neural Network Applied to Health and Environmental Data , Taysseer Sharaf

Radial Versus Othogonal and Minimal Projections onto Hyperplanes in l_4^3 , Richard Alan Warner

Ensemble Learning Method on Machine Maintenance Data , Xiaochuang Zhao

Theses/Dissertations from 2014 2014

Properties of Graphs Used to Model DNA Recombination , Ryan Arredondo

Recursive Methods in Number Theory, Combinatorial Graph Theory, and Probability , Jonathan Burns

On the Classification of Groups Generated by Automata with 4 States over a 2-Letter Alphabet , Louis Caponi

Statistical Analysis, Modeling, and Algorithms for Pharmaceutical and Cancer Systems , Bong-Jin Choi

Topological Data Analysis of Properties of Four-Regular Rigid Vertex Graphs , Grant Mcneil Conine

Trend Analysis and Modeling of Health and Environmental Data: Joinpoint and Functional Approach , Ram C. Kafle

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Mathematics MSc dissertations

The Department of Mathematics and Statistics was host until 2014 to the MSc course in the Mathematics of Scientific and Industrial Computation (previously known as Numerical Solution of Differential Equations) and the MSc course in Mathematical and Numerical Modelling of the Atmosphere and Oceans. A selection of dissertation titles are listed below, some of which are available online:

2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

2014: Mathematics of Scientific and Industrial Computation

Amanda Hynes - Slow and superfast diffusion of contaminant species through porous media

2014: Applicable and Numerical Mathematics

Emine Akkus - Estimating forecast error covariance matrices with ensembles

Rabindra Gurung - Numerical solution of an ODE system arising in photosynthesis

2013: Mathematics of Scientific and Industrial Computation

Zeinab Zargar - Modelling of Hot Water Flooding as an Enhanced Oil Recovery Method

Siti Mazulianawati Haji Majid - Numerical Approximation of Similarity in Nonlinear Diffusion Equations

2013: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Yu Chau Lam - Drag and Momentum Fluxes Produced by Mountain Waves

Josie Dodd - A Moving Mesh Approach to Modelling the Grounding Line in Glaciology

2012: Mathematics of Scientific and Industrial Computation

Chris Louder - Mathematical Techniques of Image Processing

Jonathan Muir - Flux Modelling of Polynyas

Naomi Withey - Computer Simulations of Dipolar Fluids Using Ewald Summations

2012: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Jean-Francois Vuillaume - Numerical prediction of flood plains using a Lagrangian approach

2011: Mathematics of Scientific and Industrial Computation

Tudor Ciochina - The Closest Point Method

Theodora Eleftheriou - Moving Mesh Methods Using Monitor Functions for the Porous Medium Equation

Melios Michael - Self-Consistent Field Calculations on a Variable Resolution Grid

2011: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Peter Barnet - Rain Drop Growth by Collision and Coalescence

Matthew Edgington - Moving Mesh Methods for Semi-Linear Problems

Samuel Groth - Light Scattering by Penetrable Convex Polygons

Charlotte Kong - Comparison of Approximate Riemann Solvers

Amy Jackson - Estimation of Parameters in Traffic Flow Models Using Data Assimilation

Bruce Main - Solving Richards' Equation Using Fixed and Moving Mesh Schemes

Justin Prince - Fast Diffusion in Porous Media

Carl Svoboda - Reynolds Averaged Radiative Transfer Model

2010: Mathematics of Scientific and Industrial Computation

Tahnia Appasawmy - Wave Reflection and Trapping in a Two Dimensional Duct

Nicholas Bird - Univariate Aspects of Covariance Modelling within Operational Atmospheric Data Assimilation

Michael Conland - Numerical Approximation of a Quenching Problem

Katy Shearer - Mathematical Modelling of the regulation and uptake of dietary fats

Peter Westwood - A Moving Mesh Finite Element Approach for the Cahn-Hilliard Equation

Kam Wong - Accuracy of a Moving Mesh Numerical Method applied to the Self-similar Solution of Nonlinear PDEs

2010: Mathematical and Numerical Modelling of the Atmosphere and Oceans

James Barlow - Computation and Analysis of Baroclinic Rossby Wave Rays in the Atlantic and Pacific Oceans

Martin Conway - Heat Transfer in a Buried Pipe

Simon Driscoll - The Earth's Atmospheric Angular Momentum Budget and its Representation in Reanalysis Observation Datasets and Climate Models

George Fitton - A Comparative Study of Computational Methods in Cosmic Gas Dynamics Continued

Fay Luxford - Skewness of Atmospheric Flow Associated with a Wobbling Jetstream

Jesse Norris - A Semi-Analytic Approach to Baroclinic Instability on the African Easterly Jet

Robert J. Smith - Minimising Time-Stepping Errors in Numerical Models of the Atmosphere and Ocean

Amandeep Virdi - The Influence of the Agulhas Leakage on the Overturning Circulation from Momentum Balances

2009: Mathematics of Scientific and Industrial Computation

Charlotta Howarth - Integral Equation Formulations for Scattering Problems

David Fairbairn - Comparison of the Ensemble Transform Kalman Filter with the Ensemble Transform Kalman Smoother

Mark Payne - Mathematical Modelling of Platelet Signalling Pathways Mesh Generation and its application to Finite Element Methods

Mary Pham - Mesh Generation and its application to Finite Element Methods

Sarah Cole - Blow-up in a Chemotaxis Model Using a Moving Mesh Method

2009: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Danila Volpi - Estimation of parameters in traffic flow models using data assimilation

Dale Partridge - Analysis and Computation of a Simple Glacier Model using Moving Grids

David MacLeod - Evaluation of precipitation over the Middle East and Mediterranean in high resolution climate models

Joanne Pocock - Ensemble Data Assimilation: How Many Members Do We Need?

Neeral Shah - Impact and implications of climate variability and change on glacier mass balance in Kenya

Tomos Roberts - Non-oscillatory interpolation for the Semi-Lagrangian scheme

Zak Kipling - Error growth in medium-range forecasting models

Zoe Gumm - Bragg Resonance by Ripple Beds

2008: Mathematics of Scientific and Industrial Computation

Muhammad Akram - Linear and Quadratic Finite Elements for a Moving Mesh Method

Andrew Ash - Examination of non-Time Harmonic Radio Waves Incident on Plasmas

Cassandra Moran - Harbour modelling and resonances

Elena Panti - Boundary Element Method for Heat Transfer in a Buried Pipe

Juri Parrinello - Modelling water uptake in rice using moving meshes

Ashley Twigger - Blow-up in the Nonlinear Schrodinger Equation Using an Adaptive Mesh Method

Chloe Ward - Numerical Evaluation of Oscillatory Integrals

Christopher Warner - Forward and Inverse Water-Wave Scattering by Topography

2008: Mathematical and Numerical Modelling of the Atmosphere and Oceans

Fawzi Al Busaidi - Fawzi Albusaidi

Christopher Bowden - A First Step Towards the Calculation of a Connectivity Matrix for the Great Barrier Reef

Evangelia-Maria Giannakopoulou - Flood Prediction and Uncertainty

Victoria Heighton - 'Every snowflake is different'

Thomas Jordan - Does Self-Organised Criticality Occur in the Tropical Convective System?

Gillian Morrison - Numerical Modelling of Tidal Bores using a Moving Mesh

Rachel Pritchard - Evaluation of Fractional Dispersion Models

2007: Numerical solution of differential equations

Tamsin Lee - New methods for approximating acoustic wave transmission through ducts (PDF 2.5MB)

Lee Morgan - Anomalous diffusion (PDF-1.5MB)

Keith Pham - Finite element modelling of multi-asset barrier options (PDF-3MB)

Alastair Radcliffe - Finite element modelling of the atmosphere using the shallow water equations (PDF-2.5MB)

Sanita Vetra - The computation of spectral representations for evolution PDE (PDF-3.2MB)

2007: Mathematical and numerical modelling of the atmosphere and oceans

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Promoting students’ interest and achievement in mathematics through “King and Queen of Mathematics” initiative

Journal of Research in Innovative Teaching & Learning

ISSN : 2397-7604

Article publication date: 29 August 2022

Issue publication date: 30 March 2023

The study explored the impact of the King and Queen of Mathematics Initiative (KQMI) in promoting students’ interest in learning mathematics and improving their achievement. The specific objectives of the study focused on the impact of the initiative in promoting interest in mathematics, assessing the contribution of the initiative to students’ achievements and investigating challenges encountered by the initiative.

Design/methodology/approach

The study used a case study design with a mixed-method approach. One ward secondary school was involved. The sample size was N  = 79, where 77 were grade three students in a science class and two teachers. Data collection involved documentary review, observation and interviews. Data analysis employed both content analysis and a dependent t -test to determine the effect size of the initiative.

The findings revealed that KQMI had a significant impact on improving performance in mathematics among students ( t (71) = −7.917, p  < 0.05). The study also showed that male students improved their performance more than their counterparts throughout the KQMI. The mathematics teacher revealed that students still need assistance to solve mathematical questions with different techniques to develop the expected competencies.

Research limitations/implications

The initiative was conducted only in one school, limiting the findings’ generalization. Also, the innovation faced different challenges, such as accessing adequate resources and students with little knowledge of mathematics, which the initiative aimed to address.

Practical implications

Pedagogical innovations enhance the promotion of students’ interest in learning mathematics and hence improve their performance. Also, through pedagogical innovations, teachers improve their teaching skills and practices from students’ feedback.

Originality/value

The KQMI is a new pedagogical innovation modified from the existing innovations such as game-based method, task design, mobile learning and mathematics island.

  • Mathematics
  • Students’ achievement
  • Students’ interest
  • Pedagogical innovation

Kihwele, J.E. and Mkomwa, J. (2023), "Promoting students’ interest and achievement in mathematics through “King and Queen of Mathematics” initiative", Journal of Research in Innovative Teaching & Learning , Vol. 16 No. 1, pp. 115-133. https://doi.org/10.1108/JRIT-12-2021-0083

Emerald Publishing Limited

Copyright © 2022, Jimmy Ezekiel Kihwele and Jamila Mkomwa

Published in Journal of Research in Innovative Teaching & Learning . Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode .

Introduction

Mathematics is an abstract subject; hence, it causes many students to lose interest, thus resulting in low achievement ( Yeh et al. , 2019 ). Apart from an abstraction of mathematics causing low interest in students, experiencing anxiety about learning mathematics also has contributed to disliking the subject ( Summer, 2020 ). This paper explores the contribution of pedagogical innovation in promoting students’ interest in learning mathematics hence improving achievement.

Lack of interest in learning mathematics results in low achievement. Interest is one of the attitudinal and influential variables that are predictors of students’ achievement in learning or avoidance of learning mathematics ( Singh et al. , 2002 ). Studies have shown the trend of poor performance in mathematics in many parts of the world ( Mazana et al. , 2020 ; Mbugua et al. , 2012 ; Ndume et al. , 2020 ; Sa'ad et al. , 2014 ). The trend of poor performance is associated with students’ low interest in studying mathematics. Students feel the subject is boring. Factors such as teachers’ lack of innovative pedagogies, the subject’s broad content and students’ inadequate practices amplify students’ low interest in learning ( Shoaib and Saeed, 2016 ). Pedagogical innovations in facilitating learning play a central role in addressing the challenges of students’ low interest and achievement in the subject.

Peteros et al. (2020) assert that recognizing and awarding students for their improved performance helps them boost their confidence and interest in the subject. An enjoyable learning environment significantly impacts students’ interest in studying mathematics and improves their performance ( Mazana et al. , 2019 ). Despite the expected positive results from implementing the innovation to promote interest to learn mathematics and improve performance, Maass et al. (2019) warn that implementing innovation in the classroom is a challenging and demanding activity that requires teacher’s commitment and motivation. The King and Queen of Mathematics Initiative (KQMI) adopted the awarding and recognition of students with improved achievement. The subject teacher crowned students the title of King or Queen to kindle students’ interest and make mathematics learning more enjoyable.

The status of students’ achievement in mathematics

Globally, students’ performance in mathematics has been a challenging issue given its importance in this era of science and technology. In Nigeria, Sa'ad et al. (2014) report that students perform poorly in mathematics, citing students’ negative attitudes and lack of innovative teaching methods as the cause. In Kenya, Mbugua et al. (2012 ) report similar factors for low achievement in mathematics. Peteros et al. (2020) report that, in the Philippines, the level of performance in mathematics in 2020 was low as the majority of students (53.01%) performed below the average. The implication is that many teachers fail to make the mathematics learning process enjoyable for students. Mazana et al. (2019) report that developing a positive attitude among students is when they enjoy the subject through various innovative and engaging methods. This positive attitude has a significant impact on improving achievement in the subject. Studies have revealed that apart from students struggling with mathematics achievements in Pakistan, female students have performed better than male students ( Khan et al. , 2018 ). In the light of the above, it necessitates emphasizing pedagogical innovations to eliminate the challenges and enhance students’ achievement in mathematics.

In Tanzania, the state of performance in mathematics subject is low. Ndume et al. (2020) show that the pass rate of mathematics in form four national examination is 16%. The trend of failure in the subject is high as research shows that in 2012 alone, 69% of form four students failed in the subject ( Mazana et al. , 2020 ). Mazana et al. (2019) reported factors associated with low achievement in mathematics as (1) students’ attitude towards mathematics, (2) the perception that the subject is complex, (3) low level of self-confidence, (4) bad grades attained in the classroom tests discourage students, (5) poor background and (6) irrelevance of the content to real-life situations.

As Mazana et al. (2019) posit, the persistence of the factors results in a small number of passing in form four national examination results. It implies that few students will continue in science subjects in advanced secondary education and at the university level. Despite increasing students from 240,160 in 2014 to 435,345 in 2020, the pass rate has remained low, as shown in Table 1 . The pass rate had progressively decreased from 2.82% in 2015 to 0.09% in 2020. This performance decrease manifests the necessity of applying pedagogical innovations to promote interest and improve students’ achievement in learning mathematics.

Due to poor performance in mathematics and the associated factors, teachers have been innovating and experimenting with various initiatives to promote students’ interest in learning the subject to improve achievement. Some of the pedagogical innovations include mobile learning in mathematics ( Ndume et al. , 2020 ), mathematics island ( Yeh et al. , 2019 ) and task design ( Coles and Brown, 2016 ).

Therefore, this study explores the KQMI in promoting students’ interest and achievement in mathematics. The initiative intends to create an enjoyable learning environment while focusing on recognizing and awarding students to promote interest and improve achievements in mathematics.

The genesis of KQMI

The KQMI is a pedagogical innovation of mathematics teachers at the school. The KQMI primarily was adopted from a nearby primary school called Tuishime, where higher achievers were crowned the titles of King for a male and Queen for a female student at the end of the academic year, i.e. November each year. The event took place on Parents’ Day at the school to inspire students to work hard and improve their achievements.

The KQMI started in a new academic year in January 2020 at Lemara Secondary School, where this study was conducted. A mathematics teacher called Jamila adopted the initiative, and she was committed to seeing students’ mathematics achievement improve in the school where she taught. Mostly, ward secondary schools (secondary schools built in each ward in Tanzania are classified by a generic name as ward secondary school) are regarded as low-quality schools because they are newly established, under-resourced and located in catchment areas, i.e. the villages in the vicinity, affecting classroom attendance due to dropouts. Notably, the enrolled students always have an average or low performance in Primary School Leaving Examination. For example, in many ward secondary schools, the pass rate in mathematics is between 5 and 8% lower than that of well-established public and private schools. As a result, students will achieve low in mathematics in national examinations.

Interestingly, most students in many ward secondary schools fail in mathematics while passing other science subjects such as biology, chemistry and physics. According to Mazana et al. (2019) , mathematics is a compulsory subject at the lower secondary level; hence, students who aspire to continue with any science or business combinations at the upper secondary level must have a pass in mathematics. This requirement hinders their dream as they cannot join science subjects in advanced levels of learning as they did not pass mathematics.

The teacher at Lemara Secondary School was motivated to adopt and implement the KQMI to address the challenge of massive failure in mathematics and the need to help these students to reach their dreams of continuing in higher levels of education in mathematics, science and business subjects. The implementation of KQMI involved the third-grade science class where the teacher was assigned to teach. Since the class expected to sit for the grade four national examination the following year, the initiative gave the teachers and students more time to learn.

The KQMI involved a weekly mathematics test competition where it crowned one male and female student who emerged with the highest score as King and Queen of Mathematics for one week. Every Friday, the teacher administered a test and could ask for help from other teachers in administering and invigilating the test. After every three months, students sat for comprehensive tests. The mathematics teacher who implemented the KQMI dedicated her time to helping students as she had to work extra hours to make the initiative realistic, attainable and sustainable. The extra work included marking students’ tests during the weekend and spending hours for remedial classes on the weekdays. In some cases, the teacher had to incur costs for printing tests which is rare for many teachers. The King and Queen received a special badge to identify them and recognize their efforts and achievement, and they had to strive hard to retain their titles.

Problem statement

Studies have shown that students’ performance in mathematics is poor, and the data have confirmed this poor performance trend ( Mazana et al. , 2019 , 2020 ). The crucial factors include the perception that mathematics is a complex subject and the lack of self-confidence among students due to their low grades. Despite students’ poor background in mathematics, the mentioned factors have resulted in a negative attitude towards the subject, which has affected their interest, thus, leading to poor performance in the subject. The introduction of KQMI focused on promoting students’ interest in the subject to attract self-confidence and improve mathematics achievement. The study then focused on exploring whether the KQMI has a significant impact on promoting students’ interest in learning mathematics and improving their performance.

Research questions

How has KQMI promoted students’ interest in learning mathematics?

To what extent the KQMI has succeeded in improving students’ achievement in mathematics?

What challenges affect the implementation of the KQMI?

Research hypotheses

The KQMI has a significant impact on improving students’ achievement in mathematics.

After implementing the KQMI, female students will achieve better mathematics than male students.

Literature review

Focusing on an educational paradigm rooted in critical pedagogy, the Socratic method, futures studies, and peace education, this essay takes the position that classrooms of the future should be transformed into safe harbours where students are afforded the opportunity to explore, deconstruct and share knowledge of themselves, their experiences, and the world in which they live. ( 2016 , p. 1)

Based on the above call for pedagogical innovations to enhance learning, promoting students’ interest to learn mathematics and improving their achievement in the subject is the central focus. The following section presents a review of these aspects.

Harackiewicz et al. (2016) define interest as “an individual’s momentary experience of being captivated by an object and more lasting feelings that the object is enjoyable and worth further exploration”. In the context of learning, Wong and Wong (2019) define interest as the state of engaging students in learning mathematics while enjoying the learning process. This study considers interest as the state of students being confident and free in interacting with teachers and colleagues in learning mathematics while showing they like and enjoy the learning process. Emefa et al. (2020) define interest as a psychological state occurring during the interaction between a person and a specific subject or activity, including the process of willingness, attention, concentration and positive feeling towards that particular subject or activity.

Interest is a construct of motivation and other constructs like perceived control, collaboration involvement and efficacy ( Ahmed, 2016 ). Also, Järvelä and Renninger (2014) concur that the concept of motivation is broader than interest, implying that interest, together with other factors, results in motivation. Knoll (2000) added that interest is a significant initiator of motivated behaviour; hence, before a student is motivated in learning, one must be interested first. Järvelä and Renninger further assert that interest is a cognitive and affective motivational variable that advances through four phases: (1) triggering of interest, (2) sustained, (3) emerging interest and (4) extending to a more well-developed individual interest.

In their study, Toli and Kallery (2021) provided the characteristics of interest that include increased attention, efforts, effects and experience. They used a situational interest development model to enhance students’ interest in learning science. The findings revealed a significant positive correlation between students’ learning outcomes and interest in the subject.

The study focused on interest as a single construct without relating to other aspects of motivation. It explored students’ psychological state towards their willingness to participate, learn attentively and concentrate on the subject happily. As Harackiewicz et al. (2016) presented, all the four interest-enhancing initiatives, attention-getting settings, contexts evoking prior individual interest, problem-based learning and enhancing utility value, were considered in the KQMI. Again, Singh et al. (2002) reveal that motivation and interests serve the goal of enhancing students’ achievement in mathematics.

Pedagogical innovations to promote interest in learning mathematics

Improved students’ performance in mathematics begins with students’ interest in liking the subject. Students’ interest is an internal aspect that develops in a given environmental setting ( Azmidar et al. , 2017 ). These traits manifest students’ interest in learning mathematics. In their research, Wong and Wong (2019) found no significant correlation between students’ interest in the subject and performance. Their study further revealed that being interested in learning mathematics includes liking the subject, answering questions in mathematics class, desire to learn more about the subject and anxiety to know all about how to do mathematics problems. However, Frenzel et al. (2010) found that promoting students’ interest in learning mathematics was more beneficial to low achievers as they improve their performances over time. This finding contradicts Wong and Wong’s (2019) reports, however, they gave the factors concerned with the insignificance of interest and achievement.

Motivation has also been considered an essential factor in promoting students’ interest in learning mathematics. Yeh et al. (2019) assert that a low level of motivation results in low interest in learning mathematics and hence low achievement. This assertion indicates a correlation between students’ interest and their academic achievements. They further argue that game-based teaching methods engage learners, encourage critical thinking and construct motivation. Although Otoo et al. (2018) opine that motivation has no significant impact on promoting interest, such assertion has received little support from scholars.

Again, studies have uncovered various aspects of improving students’ interest in mathematics. According to Yeh et al. (2019) , three aspects indicate students’ interest in the subject: attitude, initiative and confidence. They further describe that students’ liking of the subject significantly influences their attitudes. The initiative is from participating voluntarily in mathematics activities even beyond class hours. Confidence is the ability to ask questions or request the teacher to re-explain concepts during the lesson. Hackett and Betz (1989) also confirm that confidence is central to enhancing students’ interest in learning and improving the subject’s achievement. Self-confidence enhances students’ interest, whereby self-confidence depends on the perceived usefulness of the content, background knowledge and the level of anxiety among students ( Otoo et al. , 2018 ). In this regard, promoting students’ interest in learning mathematics depends on students’ internal factors.

Teachers use motivation strategies such as rewards, recognition, encouragement and praise to boost students’ interest in learning mathematics ( Kashefi et al. , 2017 ). Another pedagogical innovation used was the Concrete-Pictorial-Abstract approach to raise students’ interest in studying mathematics ( Azmidar et al. , 2017 ). The approach starts with concrete objects to perform mathematical operations, followed by pictorial and the last move to abstraction. This process implies that interest also depends on external factors from students learning environment.

Therefore, developing students’ interest in learning mathematics depends on internal and external factors. Counselling, consultations and assessment results identify students with challenges and take time to understand them to help identify internal factors and use them in assisting. The external factors may include rewards, recognitions, remedial classes and praise. Teachers need to be aware of this and design pedagogical approaches that consider both factors. Combining these strategies bears a solid contribution to promoting students’ studying interests.

Pedagogical innovation in improving students’ achievement in mathematics

Without suitable teaching methods and effective use of time allocated for teaching, many students will fail to improve their academic achievements ( Mosha, 2018 ). Students have struggled to develop mathematical skills, which probably implies that the teaching methods used were less effective and impactful. The struggle has led to various pedagogical innovations to promote students’ achievement in the subject.

Innovative teaching methods significantly improve students’ achievement in mathematics ( Abd-Algani, 2019 ). Such teaching methods include evaluation for learning, digital tools and applications, constructive learning principles and differential teaching. Yeh et al. (2019) developed a game-based method to enhance students’ learning environment. Their innovation found that the teaching method increased students’ mathematics achievement ( Abd-Algani, 2019 ). Task design is also considered a pedagogical innovation that intends to enhance students’ learning, understanding and achievement in tests ( Coles and Brown, 2016 ). Coles and Brown further mention the principles of implementing task design: (1) lesson delivery beginning with contrasting examples to spark curiosity among students, (2) students showing similarities and differences and (3) students naming the differences are directly linked and results in learning.

Despite the efforts to implement pedagogical innovations to improve students’ achievement, teachers and students encounter some challenges. Teachers fail to implement innovative approaches in schools due to limited material and time resources and huge workloads ( Abd-Algani, 2019 ; Kashefi et al. , 2017 ). Apart from the challenges that teachers face, on the side of students, the readiness to learn, the level of motivation and background issues act as challenges ( Wang et al. , 2018 ). Wang et al. further reveal that ability of teachers to apply pedagogical innovations in classroom settings depends on the methodological resources they have at their disposal. The resources are necessary to support and ensure the effectiveness of innovative pedagogies used in teaching and learning.

However, the innovations implemented might enhance students’ interest and hence achievement in the subjects, but several other factors also significantly contribute to students’ learning and achievement. Students’ ability, attitudes and perceptions, socio-economic variables, parent and peer influences, school-related variables, family and home environment, motivational variables and instructional time affect students’ achievement ( Singh et al. , 2002 ). This study also compared students’ performance before, during and after the KQMI. The purpose of the comparison focused on understanding the consistent influence of other factors apart from the KQMI on students’ achievements in mathematics.

Theoretical framework

Several studies have shown the application of the Interest-Driven Creator (IDC) theory in promoting students’ interest in learning mathematics ( Wong et al. , 2020 ; Wong and Wong, 2019 ). The theory shows the issues that contribute to creating and sustaining students’ interest in learning. From the IDC theory, Wong et al. (2020) came up with a model that involves three stages and focuses on developing and maintaining interest in learning. The model stages are Triggering, Immersing and Extending. According to Harackiewicz et al. (2016) , triggering implies catching students’ interest through attention-catching situations or environmental stimuli that ignite a reaction or response, while immersing means a maintained response to engage in learning activities/tasks. Harackiewicz et al. further reveal that extending means internalized behaviour of re-engaging in particular learning activities and tasks as the outcome of the former two stages (see Figure 1 ).

Methodology

Design of the study.

A case study design was adopted to understand the effectiveness of the innovative initiative that aimed at promoting students’ interest in learning mathematics and improving their performance. A case study was appropriate because the design involved intensive analysis of individual units within a case. A researcher focuses on the process of tracing and allows multiple ways of collecting information ( Creswell, 2014 ; Denzin and Lincoln, 2018 ). In this case, the unit was a specific class, grade three, in a school. The design was flexible enough to allow multiple data collection methods, i.e. interviews, observation and documentary review.

Lemara Secondary School was the area of study. It is one of the ward secondary schools within Arusha Municipality in Arusha region, located in the northern part of Tanzania. Historically, ward secondary schools were introduced in 2004 when Tanzania implemented the Secondary Education Development Programme phase two (SEDP II). SEDP II aimed to expand the enrolment rate in secondary schools since many students failed to proceed with education after completing the primary level. Ward schools mushroomed quickly, and they started operating while under-resourced with both teaching and learning materials and the number of teachers. These challenges persisted for a long time – the poor performance in form four national examinations among ward secondary schools confirms this (see Table 1 ).

Lemara Secondary School, established in 2005, is a co-education school. Currently, the school has grade one up to grade form four students. Mathematics is one of the compulsory subjects for all students, whether they specialize in science, business or arts subjects.

Participants and the KQMI context

The study involved form third grade (form three) science class in 2020. The class fits in the study because science class requires a good command of mathematical skills; hence, promoting their interest and performance in mathematics could significantly impact their science subjects. The participants in the study were 77 students (40 females and 37 males) and two teachers (a mathematics teacher and the head of the school).

Despite the KQMI involving weekly tests to find the King and Queen of another week, it involved teachers’ use of participatory teaching methods and remedial sessions to help the low achievers who were willing to be assisted. The study did not focus on the weekly scores but on the examinations stipulated on the school calendar; midterm, terminal and annual examinations. These tests gave a clear understanding of the performance trend during the implementation of the KQMI. The winners each week were crowned and given a special badge to wear for the whole week while exempted from all school activities outside the classroom. Wearing the special badge and the exemption from activities meant recognizing their weekly achievement, thus attracting many students to compete for such respectful recognition.

Data collection methods

Data collection methods involved documentary review, classroom observation and interviews. The data collection process considered teaching, learning and assessment practices conducted from January 2020 to November 2020. These data collection methods allowed researchers to interact with practitioners involved in action research within their contexts. Through interacting with the practitioners, the researcher obtained adequate and rich information concerning the implementation of KQMI. The methods ensured that appropriate data were collected to provide evidence for evaluating the implementation of KQMI. Qualitative data analysis employed content and narrative analysis. Quantitative data analysis employed a t -test calculation to find whether the initiative had a significant impact on improving students’ mathematics performance.

Interviews: The mathematics teacher who implemented KQMI, the head of the school, and six selected students participated in the interview. The mathematics teacher was purposively selected because she was the one implementing the KQMI at school. The head of the school has vital information concerning supporting and monitoring the initiative. Further, the head of the school occasionally observed the teaching and learning process in mathematics class to improve students’ interest in learning the subject. Students were selected from each test, the highest and lowest achiever, making six students from four tests administered. Two students won the crowns twice, making six students participate in the interview instead of eight.

Documentary review: The researcher reviewed several documents, such as students’ score records, to gather relevant information. The score involved students from science class (KQMI class) and other students (non-KQMI class). The KQMI class scores were taken before, during and after the initiative. In the non-KQMI class, the researchers took the scores from examinations before and after the initiative. Also, the researcher reviewed the mathematics teacher’s lesson plan to understand how the teacher planned the lessons and the kind of recommendations she gave for improvement.

Observation: Researchers conducted classroom observation to understand the noticeable changes in students’ behaviours like participating in discussions, attendance and asking questions.

Data analysis plan

Data analysis involved statistical analysis and coding data into categories and themes based on the data type obtained. The scores obtained from the documentary review were analyzed using a t -test to determine whether the initiative significantly impacted students’ performance. Again, the analysis involved Cohen’s D statistical calculation to determine the effect size of the KQMI on students’ performance. The study had two hypotheses: (1) showing that the initiative has a significant impact on students’ performance and (2) showing that after implementing the initiative, female students would perform better than male students. Data from observation and interviews were coded and developed into themes – direct quotations from respondents supported the findings.

Dependability, trustworthiness and credibility

Multiple procedures ranging from the data collection to analysis ensured the research’s dependability, trustworthiness and credibility ( Creswell, 2012 ). The study employed a triangulation method involving multiple data collection methods such as interviews, observation and documentary review (see Table 2 ).

Ethical consideration

The researcher adhered to all research ethics. The researcher handled data confidentially while maintaining anonymity after obtaining participants’ consent to participate in the study ( Auerbach and Silverstein, 2003 ; Creswell, 2007 ). Respondents were informed about the purpose of the research. Participants granted their consent, and the researcher protected all respondents from physical, psychological and political harm or risk. The information collected and presented did not disclose participants’ identities to maintain anonymity. Ensuring anonymity, the researcher used pseudo names during the data presentation.

The study intended to explore the impact of KQMI in promoting students’ interest in learning mathematics and improving their achievement. Further, the study intended to uncover the challenges teachers and students faced during the initiative’s implementation. The findings have revealed that students revived their learning interests as they engaged more in learning activities. Achievement gradually improved as the average increased from 17.6 to 29.8 in the first year of implementing the KQMI. Despite the promising results of the initiative, teachers’ commitment and material and financial support emerged as threats to the KQMI’s sustainability. The following sections present these findings in detail.

The KQMI in promoting interest in learning mathematics

In teaching, I mainly use demonstration and activity-based methods to show them how to solve various mathematical problems. Later, I give them questions that they must solve, as I had demonstrated. Also, I conduct remedial classes in the evening for those who wish to come and share their difficult areas. I also adopted a mathematics clinic strategy from one of my friends, though it was for all classes, not only the science class that I implemented the KQMI. I did this because students had varying levels of confidence as some could not speak in front of the class, but when they came alone, they shared the challenging part of their learning. (Mathematics teacher, 2021)

The findings again have revealed that students’ classroom behaviours have changed positively as they actively participated in the learning activities. The findings revealed that students demonstrated passive learning behaviour before KQMI, but after the initiative, they showed interest in the subject as they actively engaged in learning activities. Now students asked questions, responded to the teacher’s questions and assignments and participated in discussions, particularly trying to link concepts they have learned in the classroom. They tried to link what they have learned with its application in real-life situations. Further, students request the teacher repeat what they did not understand well, as shown in Table 3 . The act of students asking questions to the teacher, participating in discussion and requesting to reteach concepts they did not understand well implies that they have improved their confidence hence understanding the subject. The KQMI enhanced students’ academic engagement since it increased the number of students attending remedial classes, unlike before. The academic engagement proved that students previously were afraid or disliked the subject due to low interest.

Students started spending more time studying mathematics than in other subjects, which improved performance in mathematics and not in the other subjects. Students living in school hostels were found in the class around 10 pm, solving mathematics questions in groups. One of the teachers on duty observed this situation while walking around the school premises. (Mathematics teacher, 2021)
Previously [when] you enter the class knowing there will be no questions, so even the preparations were not intensive enough. Nevertheless, as they started asking many questions and asking to repeat or clarify using simple language, I started having intensive preparations for lessons so that I may not seem less prepared or fail to respond to some questions. Also, I started taking a variety of books in classrooms. But most importantly, I used the feedback to identify and provide assistance to low achievers. (Mathematics teacher, 2021)

The trend of students’ achievement in mathematics during the KQMI

In the second research question, the focus was to determine whether the initiative improved students’ achievement and to what extent. The findings have revealed improvement in students’ achievement in mathematics. The mean in test 1 was 17.6, increasing to 29.8 in test 4. The teacher administered test 1 in March 2020, where students learned and covered a few topics compared to test 4 at the end of November after covering all the topics required in form three class.

The study compared students’ achievement through mathematics scores before and after implementing the KQMI. The scores before KQMI were taken from the form two standardized examination, while after the initiative, the scores were from the national form four examinations. The findings revealed that the performance in form two examination before the initiative was 31.2% of the KQMI class achieved the pass grades while only 6.6% of the non-KQMI class achieved the pass grades. After the initiative, 35.1% of KQMI class achieved a passing grade, while only 1% of the non-KQMI class achieved a passing grade. The grading is classified in a range of scores as A  = 75–100% (Excellent), B  = 65–74% (Very good), C  = 45–64% (Good), D  = 30–44% (Satisfactory) and F  = 0–29% (Fail). A grade from A–D is a pass and a grade of F is a fail.

The comparison of mathematics achievement, as shown in Table 4 , reveals that, despite the increment in complexity and quality of the content covered, the KQMI class had a low doping rate compared to that of the non-KQMI class. Before the initiative, 46.75% of students in a KQMI class obtained a passing grade, while a non-KQMI class had 6.6%. After implementing the initiative, 35.1% of students in a KQMI class obtained a passing grade compared to 1% of students in a non-KQMI class. Although the achievement had dropped for both classes, the KQMI class did not have a sharp drop.

The findings have shown that the KQMI has improved students’ achievement in mathematics. The study had two hypotheses for testing the significant impact of the initiative on students’ performance. The first hypothesis stated that the KQMI has a substantial impact on promoting students’ achievement in mathematics. The dependent t -test was employed to understand the effect of the KQMI in improving students’ performance. The study found that the initiative was statistically significant as the p -value was 0.000 in all pairs tested; hence, it rejected the null hypothesis and supported the research hypothesis.

The second research hypothesis stated that after implementing the KQMI, female students would perform better than male students. Despite the mathematics teacher and the head of the school being females, they have not inspired female students to improve their achievements. The study expected female students to have more self-confidence because of a female mathematics teacher. The findings contradict Khan et al. ’s (2018) study that female students outperform male students when a female teacher instructs a subject. In test 1, as the initiative started, there was no significant difference in achievement between male and female students, t (71) = 1.351, p  > 0.05. However, in test 4, it was found that there was a significant difference in achievement as males performed better than female students, t (74) = 2.951, p  < 0.05. In this view, the research hypothesis, “After implementing the KQMI, female students will have higher achievement in mathematics than male students”, was rejected, and the null hypothesis was accepted. Again, the descriptive statistics in Figure 2 reveal that in all tests except test 3, males had a higher average than female students. Therefore, the findings confirm that the initiative was less effective for female students than male students. This finding contradicts Khan et al. ’s (2018) report that females perform better than males in mathematics (see Figure 3 ).

The researchers calculated the pair of tests to determine the effect size of the training programme. The findings revealed that the T1*T2, T1*T3 and T1*T4 pairs had Cohen’s D greater than 0.8 hence implying the effect size of the training is as large as shown in Table 4 . The effect size of tests (see Cohen’s D in Table 4 ) revealed that as the number of topics increased and tests became comprehensive, the initiative’s effect reduced; for example, T1*T2 Cohen’s D is 1.55, wherein T1*T4 Cohen’s D is 0.99. However, as calculated in the t -test, the later pairs found that there was no significant improvement from test 2 to test 3 and test 3 to test 4 ( p  > 0.05) (see Table 5 ).

Challenges encountered during the implementation of KQMI

The study found that the teacher spent her weekend marking the tests or assignments and recording the scores to announce winners every Monday morning. Further, the teacher designed special badges for winners as a sign of recognition to the entire school. Hence, the initiative required intrinsic motivation of individual teachers and commitment regardless of little assistance from school management. In this view, other mathematics teachers were hesitant to join the initiative citing it as it adds more responsibilities to the workload they already had. As a new initiative without any reference for its success, it received little assistance at the school level. The school did not provide the required resources and facilities. The teacher used her resources like money and time to manage the initiative. The head of the school confirmed the findings as she said, “ We do not have enough resources to support every new initiative. But some creativity means added responsibility, so many teachers are against it. It should come from within to be successful .”

The teacher reported some discouragement from fellow teachers as they did not assist in administering tests or marking. In some cases, the mathematics teacher requested some students from higher classes to assist in administering and invigilating the test. The teacher wrote the test on the board, so it was tedious somehow and forced the teacher to have few questions than her prior expectations.

On the side of challenges-facing students, the study revealed that they were inspired to work hard and win the title, hence placing more effort in learning mathematics than other subjects. During the interview, the teacher revealed, “ Some teachers complained that students are focusing on only one subject. This situation, to some extent, lowered their performance in other science subjects. However, I asked them to motivate them using different strategies. I could not change my initiative because I wanted my students to improve their achievement .”

Apart from the King and Queen, I prepared badges for the most improved girls and most improved boys, who moved to one or more grades higher than the previous one. The challenge I faced was that some students scored very low and believed they would never improve, so they never put effort to improve no matter how the teacher assisted and motivated them. Also, I learned and changed how to recognize them due to some students’ discouragement as they believed they would never win or be recognized. (Mathematics teacher, 2021)

Discussions

Pedagogical innovations have proved to effectively promote students’ interests in learning mathematics ( Mazana et al. , 2019 ). The innovations help students to discard their long-rooted beliefs that mathematics is complicated and they cannot perform well. The KQMI, as pedagogical innovation, has significantly improved students’ interest in learning mathematics and improved performance through the designed teaching methods such as task-based ( Coles and Brown, 2016 ) and mathematics clinic. These task-based teaching and mathematics clinics are in the immersing stage in the theoretical framework where students engage in learning activities that develop interests in learning ( Wong and Wong, 2019 ). The initiative’s outcome saw students change their classroom behaviours where they became active in interacting and showing interest in the subject.

Through innovations, students activate their interests to participate in classroom activities and better use their private time to learn and solve mathematical problems. The findings have proved the improved achievement after the KQMI, as the first hypothesis has confirmed. However, the hypothesis predicted that female students would outperform male students because the mathematics teacher was female, but female students achieved lower than their counterparts. This finding led to the rejection of the second research hypothesis and accepted the null hypothesis. The findings resonate with Hackett and Betz (1989) and Chouinard et al. (2007) , who found that sex difference in mathematics self-efficacy correlates with sex variation in mathematics achievement.

In a sustained context, Wong et al. (2020) term self-directed learning as extending interest as students make meaningful internalization of the learning behaviour. Sa'ad et al. (2014) support the finding as they reveal that a lack of pedagogical innovations harms students’ academic achievement. Recognizing the achievement boosts students’ self-confidence; hence, it makes them free to make trials in solving problems, asking questions and urging the teacher to reteach some concepts they have not well mastered. This finding resonates with Peteros et al. (2020) that recognition boosts students’ confidence and interest in learning mathematics. Students with a high level of confidence are likelier to have high achievements in the subject ( Hackett and Betz, 1989 ).

Wong et al. (2020) affirm that triggering interest involves facilitating an activity that elicits initial interest. In this study, award-winning and recognition triggered students’ interests in participating actively in learning tasks and seeking assistance for improvement. The recognition, awards and good scores triggered students to engage in various activities. Students who developed an interest in learning mathematics have significantly improved the subject’s achievement. This finding contradicts the findings of Wong and Wong (2019) that there is no significant correlation between students’ interest and their performance. Since they spend more time learning and practising, it makes them more confident and internalizes the taught skills, making it easy to apply the learned skills even in tests and examinations ( Azmidar et al. , 2017 ; Frenzel et al. , 2010 ).

Implementing pedagogical innovations such as KQMI requires teachers’ self-commitment and intrinsic motivation ( Maass et al. , 2019 ). There are a few obstacles that jeopardize the sustainability of the initiative. The teacher experienced a lack of recognition at the school level, and assistance from fellow teachers threatened pedagogical innovations’ prosperity. The lack of cooperation is a challenge for mathematics teachers and the other teachers who may be motivated to try their innovative strategies in teaching their subjects. School management should motivate teachers and students to use pedagogical innovations by providing resources and facilities. Using personal resources among teachers and students demotivates them and obstructs the innovation to deliver the expected outcome.

Conclusions

The study intended to explore the impact of KQMI in promoting students’ interest in learning mathematics and improving their achievement. The initiative has promoted interest as students actively participated in learning activities. Comparing the achievement before and after the initiative and with other non-KQMI classes, the KQMI has significantly improved students’ achievement in mathematics. Pedagogical innovations such as KQMI have effectively promoted students’ interest in learning mathematics at Lemara Secondary School which saw their interest revised and achievement improved. Apart from the promising results of the initiative, teachers’ commitment and material and financial support emerged as threats to the sustainability of the pedagogical innovation. Supporting these pedagogical innovations is vital for sustainability and achieving a maximum outcome in improving general performance among the students.

It is crucial to pilot the initiative in other schools to determine its contribution to promoting interest and achievement in mathematics. Teachers should be provided with motivation and capacity-building training to adopt and implement pedagogical innovations such as the KQMI. Teachers and students should get the necessary support to improve mathematics performance, especially in under-resourced ward schools that lag in national examination results. Future studies should first focus on implementing the initiative in more schools and assess its impact on promoting students’ interests in learning mathematics and improving performance. Secondly, studies should aim at strategies to inspire more teachers to engage in pedagogical innovations and foster cooperation. The pedagogical innovation and collaboration will enhance teachers’ continuous professional development to see them transform their classroom teaching practices.

dissertation on mathematics

Interest development model modified from Wong et al. (2020)

dissertation on mathematics

Mean comparison in four administered tests

dissertation on mathematics

Mean comparison between male and female students in four tests

Form four national examination mathematics pass rate in seven consecutive years

Changes in students’ behaviour during the implementation of KQMI

Funding : The authors declare that this research received no funding from any organization or agency.

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Research on early childhood mathematics teaching and learning

  • Survey Paper
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  • Published: 23 June 2020
  • Volume 52 , pages 607–619, ( 2020 )

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  • Camilla Björklund   ORCID: orcid.org/0000-0001-5436-537X 1 ,
  • Marja van den Heuvel-Panhuizen 2 , 3 &
  • Angelika Kullberg 1  

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This paper reports an overview of contemporary research on early childhood mathematics teaching and learning presented at recent mathematics education research conferences and papers included in the special issue (2020–4) of ZDM Mathematics Education . The research covers the broad spectrum of educational research focusing on different content and methods in teaching and learning mathematics among the youngest children in the educational systems. Particular focus in this paper is directed to what lessons can be drawn from teaching interventions in early childhood, what facilitates children’s mathematical learning and development, and what mathematical key concepts can be observed in children. Together, these themes offer a coherent view of the complexity of researching mathematical teaching and learning in early childhood, but the research also brings this field forward by adding new knowledge that extends our understanding of aspects of mathematics education and research in this area, in the dynamic context of early childhood. This knowledge is important for future research and for the development of educational practices.

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1 Introduction

Early childhood mathematics education is a rich field of study and practice that includes the provision of stimulating activities and learning environments, organized and orchestrated by teachers, care-takers and other professionals with the aim of offering young children experiences that extend their knowledge and development of mathematical concepts and skills. Generally, early childhood mathematics education involves children aged 3–6 years, but in many countries even the youngest toddlers go to early childhood centres. Therefore, contemporary research on early mathematics education focuses on children from birth until they enter formal schooling in the first grade. To develop this field of research, a strong foundation of theory and methodology is necessary, along with consideration of the practical settings of young children’s learning as well as the societal needs and relevant educational policy frameworks. Moreover, from a didactical perspective, it also requires consideration of the essence of the mathematics to be taught to young children.

High-quality research grounded in theory is necessary for all areas of mathematics education, in order to move forward and contribute to the generation of new knowledge from which the educational practice can benefit. Since there is much evidence that later development in mathematics is laid in the early years (e.g., Duncan et al. 2007 ; Krajewski and Schneider 2009 ; Levine et al. 2010 ), such high-quality research is especially critical for early childhood mathematics education. Research involving young children entails certain challenges that cannot simply be solved by adopting research designs that are used with older students. The aim of gaining deep knowledge of how young children’s mathematical understanding can be fostered places high demands on research methods. As early as 40 years ago, Donaldson ( 1978 ) stated that children act differently in their everyday situations than they do in experiment situations, and this has been confirmed by many others since then. Thus, gaining knowledge about teaching and learning mathematics in the early years requires research that is conducted in various learning environments and that acknowledges that these learning environments are complex, multifaceted, and dynamic.

Research in mathematics education is a relatively recent scientific discipline beginning in the last century (Kilpatrick 2014 ). Investigating young children’s mathematical learning and teaching became part of this discipline much later. Early childhood mathematics has long been the research field of developmental psychology and cognitive sciences. From the studies of mental abilities and thinking in mathematical problem-solving carried out in these disciplines, we have gained knowledge about the influence of working memory and attention span (e.g., Ashcraft et al. 1992 ; Passolunghi and Costa 2016 ; Stipek and Valentino 2015 ), as well as about the role of innate abilities of numerical awareness in children’s mathematical performance (e.g., Butterworth 2005 ; Wynn 1998 ). Yet, these studies lack a deeper investigation of the mathematics that is performed and how it is developed by children. Neither do such investigations address why certain mathematical competencies are important or why some activities stimulate their development and others do not. Contrary to psychological research, mathematics education research has a didactic perspective, which means that it is linked to the perspective of the learning child, the teaching teacher, and the environment offering learning opportunities in which the teaching and learning take place. Above all, didactic research distinguishes itself from psychological research because it deals explicitly with the question of what the mathematics is in early childhood activities, both within and outside formal education.

2 A brief overview of the current field of early mathematics education research

As shown by the many publications on teaching and learning of mathematics in early childhood that have been released in the past few years, this area of mathematics education research has increasingly become a mature discipline. The same is reflected by the special interest groups, working groups, and research fora dedicated to mathematics education in the early years. No self-respecting conference today can afford not to pay attention to the area of early mathematics, and there are now also communities and conferences that focus exclusively on early childhood mathematics education. All these communities and conferences are the epicentres where the latest developments in this field are brought together. To set the scene for research on early childhood mathematics teaching and learning, without it being complete, we first provide a brief overview of recently presented and discussed early mathematics education research. As an orientation point for this overview, we used what has recently been presented by researchers at three international meetings.

2.1 CERME 11 thematic working group (TWG) on early years mathematics

A conference that already has a considerable track record for including early childhood mathematics as a fixed part of its programme is the biennial conference of the European Society for Research in Mathematics Education (ERME). This conference started in 2009 with a Thematic Working Group (TWG) on Early Years Mathematics. Since then, the number of participants in this group has grown consistently. In 2019, this TWG (that is, TWG13) consisted not only of European researchers but also attracted participants from Canada, Japan, and Malawi. The most dominant theme presented there involved studies of children’s emerging number knowledge. Many of these presentations were traditional in design, including giving children tasks that had to be solved both individually when the children were interviewed and when they worked in groups in a classroom setting. Based on these studies, researchers formulated descriptions of the children’s knowledge. Sometimes, learning trajectories could be generated from these empirical observations. However, within this TWG several examples of studies with more innovative designs and research settings were also presented, including different modes of exploring and expressing numbers, which can extend our knowledge of early childhood mathematics education. An example of such research is Bjørnebye’s ( 2019 ) study, in which a dice game including elements of multiple representations and embodiment of counting strategies opened up the possibility of observing how children’s actions and responses reflect their understanding. Other studies investigated how affordances of manipulatives and applications encouraged children to develop new ways of thinking about numbers either by working in a digital environment (Bakos and Sinclair 2019 ) or by using their fingers to represent numbers (Lüken 2019 ; Björklund and Runesson Kempe 2019 ).

A characteristic of the research community gathered at CERME11 TWG13 is that the participants generally had in common an interest in better understanding the mathematical thinking of the child. Therefore, it was considered crucial that research establish clues for how to recognize mathematical thinking in the early years. For this purpose, Sprenger and Benz ( 2019 ) used eye-tracking data, as this platform was considered to contribute to the analysis of children’s perception of structure in the process of determining quantities. Yet, what Sprenger and Benz discovered is that data from technological devices still need to be interpreted, and that other expressions of children’s perceptions and reasoning are necessary assets for drawing valid conclusions.

A further important issue that was present at CERME11 TWG13 was related to teaching practice. Specifically, several presentations addressed the questions of how mathematics education should be orchestrated in early childhood education and what opportunities to learn should be offered to children. For example, Breive ( 2019 ) investigated the link between inquiry-based education and open-ended problem-solving, and the role of the teacher in orchestrating such conditions for mathematical exploration. In her paper, Breive described the teachers’ behaviour in terms of the degrees of freedom offered to the children with respect to their actions related to the mathematical content and context. Based on the data she collected, Breive concluded that teachers’ ways of acting, and the accompanying learning opportunities, should be given more attention within early mathematics education research. Similarly, Vogler ( 2019 ), who observed teacher–child group interactions, concluded that so-called indirect learning (which can be found as a common approach in many preschool settings) may induce an obstacle to learning mathematics embedded in activities if there is not a mutual understanding of what learning content is the aim of the activity. In line with these two studies, other researchers who focused on teachers’ interactions with children also highlighted critical issues for educational practice and supported further research inquiries.

Another source for learning about the latest developments in early childhood mathematics education research is the POEM conferences (Mathematics education perspective on early mathematics learning between the poles of instruction and construction). The latest conference, POEM4, was held in 2018. The presentations published in the conference proceedings (Carlsen et al. 2020 ) all, in one way or another, reflect the question “In what way—and how much—should children be ‘educated’ in mathematics before entering primary school?” This was also the recurring question in the discussions between the participating scholars. Among the contributions, three themes stood out: children’s mathematical reasoning, early mathematics teaching, and parents’ role in children’s mathematical development. There was a strong interest in children’s reasoning abilities and strategies in problem-solving. For example, Tsamir et al. ( 2020 ) investigated how children express their understanding of patterning. For this purpose, the researchers provided preschoolers with patterns to be copied and compared, while observing their strategies. Children’s strategy use was also observed in relation to play situations. Bjørnebye and Sigurjonsson ( 2020 ) observed them in teacher-led outdoor games, while Lossius and Lundhaug ( 2020 ) observed child-initiated play activities. Some researchers used their observations of children’s encounters with mathematical content for theoretical discussions on how to understand children’s meaning-making, for example by taking the semiotic mediation perspective (e.g., Bartolini Bussi 2020 ) or through the lens of attentional processes (Verschaffel et al. 2020 ).

With respect to early mathematics teaching, at POEM4 it was discussed that teachers’ educational work largely concerns how to empower children in the learning process, assuming that children have agency in their learning (Radford 2020 ). Some of the presented studies (e.g., Palmér and Björklund 2020 ) specifically chose children's perspectives and problematized how seriation was made a content for learning in a children’s story. They showed how different manipulatives and tools used in teaching have different implications for what is made possible for the children to learn. A critical but essential notion was expressed by Tzekaki ( 2020 ), who underlined that whether children act and think mathematically and learn mathematical concepts depends on what is defined to be mathematical thinking and acting. In line with this perspective, Keuch and Brandt ( 2020 ) and Bruns et al. ( 2020 ) also raised the issue that teachers’ and student teachers’ knowledge of mathematics in early childhood education affects their readiness to exploit the content in ways that facilitate children’s mathematical learning.

The issue of the knowledge of mathematics in early childhood was also addressed in papers on the role of parents in children’s learning of mathematics. Parents are recognized as young children’s first educators, contributing to their mathematical understanding and skills. One example of this research focus is Lembrér’s ( 2020 ) study. In order to know what experiences children bring with them into preschool education and thus might inform their encounter with mathematics, she investigated what parents value in the mathematics activities in which their children are engaged at home.

2.3 ICME-13 monograph “Contemporary research and perspectives on early childhood mathematics education”

The ICME-13 Monograph “Contemporary research and perspectives on early childhood mathematics education” (Elia et al. 2018 ) is the third source for becoming informed about the state of the art in the field of teaching and learning mathematics in early childhood. This book, which has its foundations in the ICME-13 (International Congress on Mathematical Education) Topic Study Group 1 (TSG1) “Early childhood mathematics education” held in 2016, contains chapters on a broad range of topics grouped into five key themes: pattern and structure, number sense, embodied action and context, technology, and early childhood educators’ professional issues and education.

Within these themes, the domain-overarching theme of pattern and structure played a prominent role. As Mulligan and Mitchelmore ( 2018 ) showed in a series of studies, children’s awareness of mathematical structures turned out to be crucial for acquiring mathematical competence. Particularly children’s structuring skills were found to be critical to developing coherent mathematical concepts and relationships. These findings are in line with Lüken and Kampmann’s ( 2018 ) intervention study with first graders, in which 5 months of explicit teaching of pattern and structure during regular mathematics lessons resulted in significant differences between pre- and post-test arithmetic achievement scores in the intervention group. Moreover, the intervention was most beneficial to the low-achieving children.

The research within the theme number sense examined a large variety of different aspects of number development. For example, there was a study about children’s enumeration skills when making lists for designating and representing collections of objects (Dorier and Coutat 2016 ). Also, attention was paid to the use of numerical finger gestures and other bodily-based communication in order to facilitate the learning process (Rinvold 2016 ), children’s spontaneous focusing on numerosity (SFON) (Rathé et al. 2018 ; Bojorque et al. 2018 ), and the link between writing skills and number development (Adenegan 2016 ). Furthermore, an exploration of children’s ability to operate with numbers revealed that 5-year-olds were able to solve multiplication and division problems when they were presented in familiar contexts (Young-Loveridge and Bicknell 2016 ).

In the theme embodied action and context , Karsli’s ( 2016 ) video-ethnographic research in a pre-kindergarten classroom showed that young children’s hand and body movements hold rich potential for engaging them in mathematics, and underlined the importance of early childhood teachers’ attention to the embodied ways in which children engage with mathematics, with potential for creating teachable moments. Other studies investigated children’s engagement in the context of play. In Henschen’s ( 2016 ) study free play was examined, while Nakken et al. ( 2016 ) compared free with guided play, of which the latter resulted in the children exhibiting deeper mathematical thinking, and engagement with more specific mathematical concepts. Anderson and Anderson ( 2018 ) broadened the scope by investigating children’s learning of mathematics in their home environment. Thom’s ( 2018 ) and Elia’s ( 2018 ) research on geometrical and spatial thinking in early childhood offered further insights into the crucial role of the body and other semiotic resources (language, drawings, and artefacts) by which young children develop and communicate spatial-geometrical thinking. A general conclusion within this theme was that the limited ways in which young children are invited to engage with geometrical, spatial, and measurement concepts undervalue the embodied, gestural, in-context nature of their mathematical thinking.

The theme technology specifically addressed the integration of technology into early childhood mathematics teaching and learning both at school and at home. The focus was mostly on touch-screen tablet-based applications. Because this new technology significantly differs from the traditional physical aid materials, professional development is needed to help educators identify and implement effective uses of these applications. To learn more about the role of the educator (teacher or parent) in the child’s interaction with the software, Baccaglini-Frank ( 2018 ) carried out an analysis of student-software-teacher relations, revealing how the teacher’s goal of helping the children experience success actually limited their development of numerical abilities. The use of technology also opened a window to a new perspective in early childhood mathematics, namely by exposing young children to advanced mathematics such as understanding symmetric transformation (Fletcher and Ginsburg 2016 ) and dealing with large numbers (also in symbolic form) and ordinality (Sinclair 2018 ).

In the theme early childhood educators’ professional issues and education , Cooke and Bruns ( 2018 ) provided a comprehensive overview of the various contributions in TSG1, for which they proposed to distinguish conditions at three levels that influence opportunities for young children to develop mathematical understanding and skills. At the macro level, curricula provide a framework (aims, content to learn, and activities) for mathematics teaching and learning in early childhood, with varying views. Several papers mentioned the tensions regarding new curricula and frameworks that may impose mathematical content rather than allowing the child to develop understanding of mathematical concepts through play. At the meso level, with focus on the teachers’ competence, all involved papers agreed as to the importance that the teacher possess a fundamental understanding of mathematics as the basis for high-quality early mathematics education. However, different studies used different conceptualizations and instruments to measure teachers’ mathematical competence. The micro level refers to the mathematics educational programmes and materials, as well as to the required training for teachers to develop their ability to effectively select and implement such programmes that address children’s mathematical needs (Fritz-Stratmann et al. 2016 ).

In sum, the common themes that stand out from the three international meetings are children’s learning through play, and concerns regarding how to apply content-focused teaching, with or without technology. We found that a great deal of the research is on children’s mathematical thinking and learning, including two main areas concerning children’s emerging number knowledge and children’s learning of patterns. It is noteworthy that in both areas, how children perceive structure or how they manifest structuring abilities were analysed in several of the studies. There were also a number of studies that focused on how finger patterns, gestures, or bodily-based communication may facilitate children’s learning of numbers.

Children’s learning through free or guided play is also a main issue that was discussed. Teachers’ guiding interaction with children in play was shown to contribute more to deeper mathematical thinking and engagement with specific mathematical content. How teaching affects children’s learning opportunities in preschool was furthermore of great concern in several of the studies. A conclusion drawn from this research is that teachers’ ways of acting and the learning opportunities created for children should be given more attention. In what way, and how much, children should be educated before entering primary school remains a central issue.

3 The contributions of this special issue

In this special issue of ZDM Mathematics Education (Issue 2020–4), contemporary research on early childhood mathematics teaching and learning is discussed by researchers from all over the world. The initiative emanated from the 42nd PME conference in Umeå, Sweden (July 2018), where we had the opportunity to organize a Research Forum in which researchers involved in the field of early childhood mathematics education gathered to present and discuss theoretical and methodological challenges and outcomes of studies on learning and teaching arithmetic skills in early years (Björklund et al. 2018 ; Van den Heuvel-Panhuizen 2018 ). The conclusion of the Research Forum was that early childhood mathematics education research is key, but that more efforts are needed to bring together the state of the art within this field as a foundation for moving early childhood mathematics education research forward. This special issue again provides a window into the contemporary field of research on early childhood mathematics teaching and learning. To discuss what this special issue adds to this field and reflect on the challenges that lie ahead for research on early childhood mathematics education, in the next section we synthesize the themes that emerge from the 15 papers included in this special issue. Each theme highlights the papers’ shared knowledge and contributions to research methods. Many papers are related to several themes, but for our discussion we chose those papers that predominantly belong to a particular theme. In total, we identified three recurring themes: the early interventions and their effects, the facilitating factors for learning and development, and the mathematical key concepts that can be observed in children. Together, these themes bring to the fore aspects that are essential for understanding the learning and teaching of mathematics in the early years.

3.1 What lessons can be drawn from interventions?

Research shows that children’s development of mathematical skills and knowledge is often influenced by socio-economic and curricular factors, and by social interaction in both short- and long-term perspectives (Pruden et al. 2011 ). Thus, there is a raised awareness of the impact early childhood education may have on reducing differences in conditions for learning and on increasing and securing equal opportunities for a good foundation in learning for all children. Based on their meta study of early mathematics education research, Duncan et al. ( 2007 ) stated that early intervention counts and numerous references to the same study indicate that this is an important standpoint in research. Why else indulge in the challenging task of researching learning among the youngest in our education systems, if one does not believe that efforts made through teaching are significant for children’s wellbeing and lifelong learning path?

Research on teaching and learning mathematics often shares a common research design in which interventions are implemented (designed, conducted, and the outcomes assessed) with the aim of finding ways to improve teaching practice for the benefit of the learning child, and often to reduce socio-economic inequality. Intervention studies can be objects of research in different ways, focusing on the children’s learning outcomes or the teaching practices. Nevertheless, the goal is to enhance learning through improved teaching. In the papers in this special issue we find efforts to implement well-designed interventions, explicitly focusing on how to teach. Some implement and analyse fine-grained differences in (teaching) actions and the effects on children’s attention to certain content (Paliwal and Baroody 2020 ; Mulligan et al. 2020 ), while others study the effects of attentiveness to children’s experiences and knowledge and the related choices of tasks (Clements et al. 2020 ; Grando and Lopes 2020 ). Nevertheless, essential to studying intervention success or failure is how learning outcomes are measured and interpreted, which is also an important aspect of early childhood mathematics education research (Li et al. 2020 ).

How teaching is framed to present mathematical content to young children, in order for it to be meaningful to them, and in order to be attentive to children’s experiences and knowledge, is investigated and discussed by Grando and Lopes ( 2020 ). Through narratives provided by early childhood teachers, they find insights into how teachers chose to frame the subjects of statistics and probability in ways that engaged children and were responsive to the children’s own experiences, rather than using materials provided by textbooks. Unconventional teaching methods whereby teachers turned their mathematics classroom into a space of creative insubordination are discussed in this paper in relation to the opportunities they offer children to become equipped with critical thinking. The authors argue that the specific content—statistics and probability—demands problematizing activities and experimentation with uncertain outcomes of problems in order to develop probabilistic thinking. This study highlights an essential issue in didactical research: that the content to be taught is not indifferent to how the teaching is designed. The study particularly raises concerns that the design of teaching cannot be random but rather has to be linked to the educational environment and the students attending that particular environment. Consequently, the generalizability of intervention programmes and teaching methods has to be taken into serious consideration if they are to be implemented in different educational settings.

Clements et al. ( 2020 ) set out to investigate the efficacy of implementing an intervention programme in which instructions and progression are grounded in a research-based learning trajectory. Even though the programme itself had previously been found to have positive outcomes for preschool children’s mathematics learning, the goal of the current study was to investigate how to teach in the most successful way. For this purpose, the authors used the same programme but adapted the choices of the tasks’ difficulty level to the children’s current knowledge levels. How to teach was then related to what to teach individual children. Results indicate that skipping difficulty levels to shorten the steps to the learning goals was not successful. This thorough investigation of teaching by adapting the complexity of the content to the child’s ability to learn best what is intended draws attention to the delicate work of teaching in early childhood education. The study supports child-centred approaches that are sensitive to the individual needs and potential of the child, while simultaneously aiming for the learning goals set by the curriculum.

While Clements et al. investigated the effects of an intervention programme covering broader numerical knowledge, Paliwal and Baroody ( 2020 ) aimed to investigate what conditions for learning the cardinality principle are most effective and how subitizing abilities impact on cardinality knowledge achievement. Their efforts were directed towards a fine-grained analysis of how to teach this aspect of the number concept, and what learning processes different approaches elicit in children. What stands out in their study is that they used a highly advanced research design, which allowed them to examine the effects of different ways of directing children’s attention to seeing numbers’ cardinality. In their paper, they point out the importance of directing children’s attention to various ways of seeing numbers’ cardinality, as follows: as a constructing act by adding units to get a number; as an act starting from naming the whole set with a counting word and then differentiating the added units by counting; and a third condition, attending only to single units in a counting act. Thus, their intervention was designed with explicit rigour as to what was made possible for the children to experience, and their investigation concerned the learning outcomes of the different conditions. While this attention in Paliwal and Baroody’s study to the different conditions can at first glance be considered subtle and far from the instruction children encounter in their mathematics education, the study offers insight into the importance of teachers’ awareness of their way of directing children’s attention to certain meanings of the content.

In another paper focusing on the effects of an intervention programme, Mulligan et al. ( 2020 ) analysed children’s written answers to pattern tasks in order to identify differences and changes in their structural awareness. They found a positive effect on the children’s development of awareness of mathematical pattern and structure (AMPS), and showed how the levels changed as an effect of a 37-week intervention programme. Mulligan et al. add to the field of early childhood mathematics knowledge of a particular ability (structural awareness), how it can be identified among young children, and also how the ability changes over a prolonged period of time (during an intervention), which may provide insight into what children actually learn while taking part in an intervention programme.

Children’s learning is of course at the centre of attention in intervention studies, and Li et al. ( 2020 ) pay explicit attention to how to interpret results from a pre- and post-diagnostic test. In their study, Li et al. investigated the development of mathematics problem-solving skills among kindergarteners by analysing their responses to a cognitive diagnostic test. As in most large-scale analyses, it can be shown in quantitative terms how children develop in producing correct answers that indicate growth in knowledge within certain domains that are tested for. However, Li et al. take a step further in their inquiry and illustrate how two children who scored similarly on the cognitive diagnostic test before an intervention had made different progress during the intervention period. Li et al. suggest that the reason for this difference may lie in how children understand and approach tasks, indicating different understanding even though similar answers are produced. Quantitative measures alone do not reveal such differences. The study thus shows the significance of paying attention to how children reason in order to solve a task. Based on their study, Li et al. recommend that children’s learning outcomes from participating in interventions be seen in the light of how the effects of interventions are measured, as it is observed that some developed skills do not endure over time and similar outcomes among children may conceal different learning paths.

3.2 What facilitates children’s learning and development?

Today, it is undisputed that the development of mathematical skills and the teaching of emerging skills in the early years are essential for mathematics education and developmental progress in the long term (Aunio and Niemivirta 2010 ; Duncan et al. 2007 ; Krajewski and Schneider 2009 ). However, in contrast to this perspective, a recent overview of the long-term effects of preschool mathematics education and interventions (Watts et al. 2018 ) challenges this almost taken-for-granted assumption, as most early interventions have a substantial fadeout effect. Thus, there is a need to revisit our current knowledge of teaching and learning, and scrutinize what seems to make a difference. Some of the papers in the special issue particularly consider this issue in their efforts to ascertain what facilitates children’s mathematical learning and development, and focus on influential aspects found in play settings (Reikerås 2020 ; Tirosh et al. 2020 ), verbal communication in teaching practices (Hundeland et al. 2020 ), and the home numeracy environment (Rathé et al. 2020 ).

Hundeland et al. ( 2020 ) raise the question of how children learn to use and understand the canonical language of mathematics, and study this aspect in terms of mathematical discourses taking place in kindergarten teaching sessions. They take a sociocultural stance (see Vygotsky 1987 ), seeing communication as the link between internal communication (thinking) and external communication (interaction). Therefore, children’s opportunities to contribute ideas and arguments are vital for their (mathematical) learning processes. Earlier research has also shown that care-takers’ talk influences not only children’s vocabulary but also, for instance, their spatial problem-solving (Pruden et al. 2011 ). The deeper knowledge that the study by Hundeland et al. ( 2020 ) provides regarding the quantity and quality of mathematical talk in which children are involved, offers us better opportunities also to organize supportive and stimulating conditions for knowledge growth.

What differs in the study by Hundeland et al. compared to most others with similar research questions is their focus on the kind of interaction that the mathematical discourse induces, which, based on the chosen sociocultural theoretical framework, should be crucial for positive learning outcomes. However, what they study and compare is the impact on the mathematical discourse that a certain in-service training has. This places mathematics in the spotlight of mathematics education research. While psychological and cognitive research provides us with important knowledge of mental processes and developmental advancement, studies like the one by Hundeland et al. have a clear direction towards understanding, and not least improving, the conditions for children’s learning and development, either by implementing teachers’ professional development or through curriculum improvements.

It is commonly agreed that young children’s learning is often situated in play. In a large-scale observation study, Reikerås ( 2020 ) conducted a thorough examination of the kind of play in which toddlers engage, for the purpose of learning how play skills may be related to early mathematical skills. It was found that competencies that allow the child to be active in solitary and parallel play, as well as children’s ability to initiate and remain in a play activity, correlated positively with the toddlers’ mathematical skills. The kind of play skills that showed the highest correlation with mathematical skills was their competence to interact in play. General social play skills thus seem to have an impact on mathematical learning, but Reikerås’ study cannot reveal how these are connected or any causal effects. An effort to better understand the interaction going on in toddlers’ play is made by Tirosh et al. ( 2020 ), investigating the challenges toddlers may face as they practise one-to-one correspondence in a playful context, and how different individuals participate in the playful mathematical context. Here, interaction and social skills become one issue with an impact on the learning opportunities arising in the play.

In many cases, the messy context of children’s play is a methodological challenge. It is not possible to control influencing variables to the same extent as in an experimental design. On the other hand, findings from the messy settings are more likely to bring to the fore aspects that were not anticipated, which raises new questions for research and theory development. Design research supports this kind of knowledge contribution, as several cycles are conducted, each developed based on insights from the previous cycle. These cycles adhere to children’s initiatives such as practising one-to-one correspondence in a setting the table task by putting one spoon inside each cup instead of placing one spoon beside each cup (see Tirosh et al. 2020 ); thus, the child is expressing an understanding of the concept, but is expressing it differently than how the task suggests. This highlights the importance of directing attention to instructions used in research studies, and particularly to the language of mathematics and the spatial aspects of props used in a task, related to the possibilities involved as young children interpret and execute a task.

Children take part in cultural life, where today numerical aspects are an inevitable part of the everyday environment. Nevertheless, there are differences in the extent to which children attend to these aspects, and consequently in how they learn the meaning of numbers, graphical representations of numbers, and how to use numbers. A common assumption is that home numeracy environment is a strong factor (LeFevre et al. 2009 ; Skwarchuk et al. 2014 ), which is reflected not least in the abundance of studies regarding socio-cultural background and demographic factors as a pre-cursor for learning progress. Rathé et al. ( 2020 ) put the common assumption to the test—that home environment has an influence on children’s progress in mathematical development—by comparing young children’s tendency to focus spontaneously on numeracy and numerical symbols in their home numeracy environment. Concerning this specific directionality to numbers, which is assumed to have an impact on children’s arithmetic skills in later years (see McMullen et al. 2015 ), based on their study they propose that the home numeracy environment does not seem to have any significant impact.

3.3 What mathematical key concepts can be observed in children?

A great deal of research in the field of early childhood mathematics education studies what mathematics children understand and how this understanding evolves. This knowledge is crucial in designing teaching that contributes to more advanced thinking and problem-solving strategies that support conceptual growth. Therefore, children’s utterances and how they act are the centre of interest for many researchers. Also, in this special issue, much attention is paid to the mathematical key concepts that can be attributed to children’s thinking, resulting in papers addressing children’s understanding of similarity in mathematical objects (Palmér and Van Bommel 2020 ), their understanding and use of structures (Sprenger and Bentz 2020 ; Kullberg and Björklund 2020 ), their understanding of the concept of cardinality and ordinality (Askew and Venkat 2020 ), and the underlying structure of their quantitative competencies (Van den Heuvel-Panhuizen and Elia 2020 ).

Children’s expressions, and how they are allowed to express themselves, are critical for our understanding of the learning of mathematics. Children’s problem posing is one aspect that can tell us about their understanding of mathematics (Cai et al. 2015 ). In the special issue, this is particularly addressed in the paper by Palmér and Van Bommel ( 2020 ), who investigated children’s understanding of similarity in mathematical objects. They analysed how children themselves created tasks in three-dimensional geometry that were similar to a previous problem-solving task they had worked on. It is suggested that this finding sheds light on the children’s interpretation of the specific mathematical features of the original task.

How children perceive structure has been shown to play an important role in how they, for example, determine a number of objects or solve an arithmetic problem (Ellemor-Collins and Wright 2009 ; Resnick 1983 ). In line with these earlier studies, Sprenger and Bentz ( 2020 ) investigated how 5-year-olds perceive structures in visually presented sets. By having the children determine the number of eggs in a 10-egg box while using an eye-tracking device (and recording the children’s utterances and gestures), they were able to analyse the children’s gaze when determining the cardinality of the set, and thereby gain insight into the process of perception. The eye-tracking data showed, for example, that many of the children were able to see structures (e.g. 4 + 1 or 3 + 2) and use them to determine a quantity without having to count all the objects. The authors argue that children’s ability to perceive structures in sets and use them to determine cardinality is central for their further arithmetic learning, as how children perceive sets (e.g., as individual objects, as a composite whole, or in structured part-whole relations) affects the strategies they use for solving arithmetic tasks.

Similar ideas are found in the study by Kullberg and Björklund ( 2020 ), who studied 5-year-olds’ use of finger patterns to structure number relations while solving an arithmetic problem. They identified two major ways of structuring the task: only structuring, and counting and structuring. In the group that both structured using their fingers and counted on some fingers, some ways were found to be more powerful. Children who solved the arithmetic task (3 + _ = 8) by creating a finger pattern of eight raised fingers and simultaneously identifying (‘seeing’) the missing part (5) on two hands (3 + (2 + 3) = 8) were more successful in solving arithmetic tasks, even in a later follow-up assessment. It is suggested that a possible reason for this later success is that these children were able to see numbers as parts included in other numbers, which has been found in earlier research (Resnick 1983 ) to be important for developing arithmetic skills.

Baccaglini-Frank et al. ( 2020 ) also argue that the appropriate use of fingers can contribute to developing children’s number sense. They studied how 4-year-olds interacted (verbally and using finger patterns) when using the application TouchCounts. The app combines multi-touch with audile, visual, and symbolic representation, and several solution strategies are possible, affording the simultaneous experience of, for example, finger patterns on the screen, with the number both seen and spoken. In their paper the authors emphasize how multimodal affordances may encourage children to use different strategies in response to different tasks, and thus experience a broad range of abilities related to number sense, including both cardinality and ordinality.

Askew and Venkat ( 2020 ) examined children’s understanding of the concept of cardinality and ordinality in connection with their awareness of additive and multiplicative number relations. To investigate this topic, first graders (6- and 7-year-olds) in South Africa were asked to position the numerals 1–9 on a bounded 0–10 number line. The children were able to do this in the correct order, with the fewest errors at the upper and lower ends of the number range. Furthermore, evidence was found that awareness of ordinality and that of cardinality develop alongside each other. However, the logarithmic scale, predicted in earlier research, which is considered to indicate a multiplicative structuring of number relationships, was not confirmed in the South African data. Instead, when the numerals grew larger the intervals became more stretched out rather than compressed. In fact, the children’s responses were closer to the linear model, which is considered to indicate an additive structuring of number relationships. Also, the use of unit sizes that did not take into account the length of the number line, together with the underestimation of the position of 5 on the 0–10 line, offered limited evidence of the children’s awareness of the multiplicative structure of the cardinality of numbers. More research is needed to disclose the deep interconnections between children’s understanding of cardinality and ordinality, and their understanding of multiplicative and additive number relations.

Another effort to unravel the complex nature of children’s early number understanding was carried out by Van den Heuvel-Panhuizen and Elia ( 2020 ), investigating the structure of the quantitative competence repertoire of kindergartners. Based on a literature review, they arrived at a model consisting of two constituent parts: quantification (the ability to connect a number to a given collection of objects) and quantitative reasoning (the ability to think and operate with quantities). Quantification was split up into counting and subitizing, and quantitative reasoning into additive and multiplicative reasoning. Although this model is partly in line with models found in earlier research, it also extends previously developed models by including multiplicative reasoning. Data were collected in the Netherlands and Cyprus. A series of confirmatory factor analyses showed that the hypothesized four-factor model fitted the empirical data of the Netherlands, but not those of Cyprus, which clearly challenges the model’s generalizability. A comparison of the component performances in the Dutch sample revealed that, in accordance with other studies, the lowest scores were found for multiplicative reasoning and that the competence of subitizing seems to develop before counting. This was partly confirmed by a statistical implicative analysis at item level. Although this analysis resulted in different implicative chains in the two countries, in both samples the multiplicative reasoning and conceptual subitizing items were found at the top of the chain and the counting and perceptual subitizing items at the end. Also, more research is necessary here, particularly concerning the generalizability of the model to other countries.

4 Future directions for research on early mathematics teaching and learning

After the Research Forum at PME42 we concluded that to move early childhood mathematics education research forward, more efforts are needed to bring together the state of the art within this field. Thus, we proposed a special issue on the theme Research on early childhood mathematics teaching and learning for the purpose of opening up further discussion and inquiry. In this article, the 15 papers included in the special issue are synthesized and discussed in terms of their contribution to the current field of research in early mathematics teaching and learning along with recent research presented at international mathematics education research conferences. Naturally, these do not cover the worldwide field of research, but they at least give a general idea of the current research interests and challenges.

All the papers in this special issue address aspects of early mathematics education and its underlying theories and research methodologies. They share common interests and challenges concerning how to gain knowledge of the youngest children’s mathematical development, and they identify prosperous teaching approaches. Our appeal to researchers participating in the special issue was to cover the broad span of mathematical ideas that are relevant in early childhood education. Nevertheless, we see a strong direction towards research on the learning and teaching of number concepts and basic arithmetic. This is in line with Alpaslan and Erden ( 2015 ) review of early mathematics research published in 2000–2013 in high-ranked scientific journals in the field of mathematics education, in which the most frequently reported research topics were number systems and arithmetic. The same trend is also found in the research addressed in the latest meetings of ICME, ERME, and POEM. We believe further research should widen this scope, and consider and investigate mathematical topics that are currently less highlighted. There is a need for deeper insight into what mathematics means to young children, and also how the foundations can be laid for the domains of spatial and geometric thinking and measurement, as well as for the domains of structures and patterns, data handling, problem-solving and mathematical reasoning.

Moving an educational field forward, however, is not solely based in covering a broad field of content. To strengthen the field, we need to scrutinize the research designs and methods that are used and the knowledge that is generated. Here, new technologies may open up opportunities for designing tools for investigating children’s competencies. However, this initiative goes beyond choosing digital tools or concrete building blocks; it concerns children’s opportunities to express themselves within different environments and make use of tools and manipulatives that may reveal new insights into their competencies and open up for innovative research questions to be posed. What is made available to experience surely has an impact on children’s expressions of knowledge. And expressions in both words and gestures are important keys here to interpreting the youngest children’s knowledge and skills. We can see this in the recent ICME, ERME, and POEM meetings’ presentation of a large variety of research designs and in the papers of this special issue. Many innovative research designs have been developed that allow thorough investigation of children’s mathematical competence and understanding. What we see, for example, is that subtle differences in expression (e.g. gaze, finger use, or ways of posing questions) reveal new and important insights for developing knowledge of children’s mathematical learning. These innovations in methodology allow for the thorough investigation of key features of learning mathematics that go beyond the broad content areas and highlight how mathematical aspects such as cardinality, ordinality, and number structure are experienced by children. Several of the papers in the special issue particularly attend to these aspects, and do so by creating and using new methodologies and technologies.

The consensus in the field of early mathematics education, reflected in the papers and conference presentations, is strong concerning the impact of early interventions on children’s opportunities to thrive as mathematics learners. From longitudinal studies, we know that early knowledge and skills seem to follow through the child’s development; that is, weak mathematical skills in early childhood years are likely to predict weak mathematics performance in later school years (Reikerås and Salomonsen 2019 ; Hannula-Sormunen et al. 2015 ). This means that early intervention and knowledge of how to offer all children a good start for their mathematical learning are essential to the field of early childhood mathematics education. However, it cannot be assumed that simply participating in education, whether it is framed as free or guided play or problem-solving, or stimulating interactive environments, will result in successful learning outcomes, even though most interventions do have a positive impact and most children develop their knowledge to some extent (Wang et al. 2016 ). Common research objectives, therefore, concern intervention implementation, and analyses of children’s learning outcomes from participating in differently designed activities. These studies are of high importance, as they connect the teaching to the learning and provide insights into what seem to be key aspects in the teaching practice. Nevertheless, researching interventions is delicate work, and it is essential to maintain scientific rigor in the design and analysis. Because early childhood education most often takes place in dynamic settings, the conditions under which children learn vary greatly. This diversity is observed in many studies in which children’s engagement in play, both self-initiated and guided, is used as data for analysing their mathematics competencies and learning of mathematics. This phenomenon means that the conditions offered to explore mathematical concepts and principles should be critically examined, along with how learning from interventions is measured and valued. There is a need to determine what works, what seems critical, and what aspects serve as particular challenges. In research, also special attention has to be given to the nature of the teaching practices. What we learn from intervention studies, both those included in the special issue and those in other contemporary research, is the importance of situating research in the current field of knowledge and the context in which the research is conducted. Each study broadens the picture of the teaching–learning relationship, which is by no means one-directional. There are many aspects to consider that potentially influence this relationship, and all of them cannot be included in one study alone.

Early childhood mathematics education research often attends to the opportunities and conditions that are offered for learning. There is no doubt that children’s activities and interaction with others, already from an early age, offer many opportunities to learn mathematical concepts and basic principles, but our ability to discern what children actually learn from the mathematical learning environments offered to them places high demands on the interpretation process. How to understand the processes going on in play and interaction, and what impacts the children’s learning outcomes—what is made possible to learn—often remains an unsolved issue, as the interaction between teacher and children is dynamic, and particularly as play is multidirectional in nature. Studies of interaction in both formal and informal contexts are nevertheless important, as they are conducted in the complex of social and cultural settings that do influence, through norms and individuals’ experiences, what is possible for children to learn.

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Björklund, C., van den Heuvel-Panhuizen, M. & Kullberg, A. Research on early childhood mathematics teaching and learning. ZDM Mathematics Education 52 , 607–619 (2020). https://doi.org/10.1007/s11858-020-01177-3

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Accepted : 18 June 2020

Published : 23 June 2020

Issue Date : August 2020

DOI : https://doi.org/10.1007/s11858-020-01177-3

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Dr. Bailey receives National Dissertation Award

Posted in: Awards & Recognition

Dr. Nina Bailey with her AMTE award along with her dissertation chair

Dr. Nina Bailey, Assistant Professor, and Dr. Siddhi Desai of Fairleigh Dickinson University are the 2024 co-recipients of the Association of Mathematics Teacher Educators Dissertation Award , and they were recognized at the annual conference earlier this month. This national award highlights mathematics education research that helps the field better understand the connections between teaching and learning and social, historical, and institutional contexts.

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