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Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2024 2024.

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

Theses/Projects/Dissertations from 2023 2023

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

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

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

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

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

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

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

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

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

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

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

Symmetric Generation , Ana Gonzalez

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

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

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

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

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

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

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

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

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

Permutation and Monomial Progenitors , Crystal Diaz

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

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

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

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

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

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

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

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

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

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

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

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

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

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

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

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

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

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

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

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

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

LIFE EXPECTANCY , Ali R. Hassanzadah

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

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

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

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

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

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

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

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

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

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

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

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

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

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How to Effectively Write a Mathematics Research Paper

Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on logic and a specific type of language, including symbols and regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition .

Structure and Style

Formal Exposition

The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper. The theorem-proof format, definitions, and logic fall under this style.

Informal Exposition

Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper . Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style.

Conventions of Mathematics

Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. Each area incorporates detailed guidelines to assist the authors.

Related: Do you have questions on language, grammar, or manuscript drafting? Get personalized answers on the FREE Q&A Forum!

Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.

Definitions

In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.

Theorems and Equations

Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end.

Math Symbols and Notations

Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers. Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources .

Protocols for mathematics writing get very specific – fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.

LaTeX and Wolfram

Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.

LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.

Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.

The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.

Good Writing and Logical Constructions

Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Authors must apply a strict, logical construct when writing a mathematics research paper.

There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.

• Concept of a math paper
• Title, acknowledgment, and list of authors
• Introduction
• Body of the work
• Conclusion, appendix, and references
• Publication of a math paper
• Preprint archive
• Choice of the journal, submission
• Publication

The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.

Good advice. For me, writing an essay on mathematics was very difficult. I did not have enough time and knowledge to write a quality essay. I worked a lot in the library and read many articles on the Internet. I studied information about essay writing. But I couldn’t finish the essay in full. I had to look for professional writers on the subject of mathematics. He helped me finish a few paragraphs. The work was delivered on time and on an excellent assessment.

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

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

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

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

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1 An international survey in two rounds

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

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

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

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

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

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

2 Methodological approach

2.1 themes for the coming decade (2019).

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

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

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

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

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

Artistic impression of the future themes

2.2 Has the pandemic changed your view? (2020)

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

3 The themes

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

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

3.1 Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

3.1.1 Teaching strategies

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

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

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

3.1.2 Curriculum

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

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

3.2 Goals of mathematics education

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

3.2.1 Societal goals

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

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

3.2.2 Educational goals

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

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

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

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

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

3.3 Relation of mathematics education to other practices

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

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

3.4 Teacher professional development

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

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

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

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

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

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

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

3.5 Technology

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

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

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

3.6 Equity, diversity, and inclusion

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

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

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

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

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

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

3.8 Assessment

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

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

4 Mathematics education research itself

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

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

4.2 Methodology

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

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

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

4.3 Reflection on our discipline

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

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

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

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

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

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

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

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

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

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

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

4.4 Interdisciplinarity and transdisciplinarity

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

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

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

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

5 The themes in their coherence: an artistic impression

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

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

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

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

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

6 Research challenges

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

6.1 Aligning new goals, curricula, and teaching approaches

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

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

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

6.2 Researching mathematics education across contexts

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

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

6.3 Focusing teacher professional development

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

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

6.4 Using low-tech resources

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

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

6.5 Staying in touch online

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

6.6 Studying and improving equity without perpetuating inequality

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

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

6.7 Assessing online

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

6.8 Doing and publishing interdisciplinary research

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

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

7 Concluding remarks

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

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

Akkerman, S. F., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research , 81 (2), 132–169. https://doi.org/10.3102/0034654311404435

Article   Google Scholar  

Arendt, H. (1958/1998). The human condition (2nd ed.). University of Chicago Press.

Backes, B., & Cowan, J. (2019). Is the pen mightier than the keyboard? The effect of online testing on measured student achievement. Economics of Education Review , 68 , 89–103. https://doi.org/10.1016/j.econedurev.2018.12.007

Bakkenes, I., Vermunt, J. D., & Wubbels, T. (2010). Teacher learning in the context of educational innovation: Learning activities and learning outcomes of experienced teachers. Learning and Instruction , 20 (6), 533–548. https://doi.org/10.1016/j.learninstruc.2009.09.001

Bakker, A. (2019). What is worth publishing? A response to Niss. For the Learning of Mathematics , 39 (3), 43–45.

Google Scholar  

Bakker, A., & Gravemeijer, K. P. (2006). An historical phenomenology of mean and median. Educational Studies in Mathematics , 62 (2), 149–168. https://doi.org/10.1007/s10649-006-7099-8

Bakx, A., Bakker, A., Koopman, M., & Beijaard, D. (2016). Boundary crossing by science teacher researchers in a PhD program. Teaching and Teacher Education , 60 , 76–87. https://doi.org/10.1016/j.tate.2016.08.003

Battey, D. (2013). Access to mathematics: “A possessive investment in whiteness”. Curriculum Inquiry , 43 (3), 332–359.

Bawa, P. (2020). Learning in the age of SARS-COV-2: A quantitative study of learners’ performance in the age of emergency remote teaching. Computers and Education Open , 1 , 100016. https://doi.org/10.1016/j.caeo.2020.100016

Beckers, D., & Beckers, A. (2019). ‘Newton was heel exact wetenschappelijk – ook in zijn chemische werk’. Nederlandse wetenschapsgeschiedenis in niet-wetenschapshistorische tijdschriften, 1977–2017. Studium , 12 (4), 185–197. https://doi.org/10.18352/studium.10203

Bessot, A., & Ridgway, J. (Eds.). (2000). Education for mathematics in the workplace . Springer.

Bickerton, R. T., & Sangwin, C. (2020). Practical online assessment of mathematical proof. arXiv preprint:2006.01581 . https://arxiv.org/pdf/2006.01581.pdf .

Bikner-Ahsbahs, A., & Prediger, S. (Eds.). (2014). Networking of theories as a research practice in mathematics education . Springer.

Bini, G., Robutti, O., & Bikner-Ahsbahs, A. (2020). Maths in the time of social media: Conceptualizing the Internet phenomenon of mathematical memes. International Journal of Mathematical Education in Science and Technology , 1–40. https://doi.org/10.1080/0020739x.2020.1807069

Bosch, M., Dreyfus, T., Primi, C., & Shiel, G. (2017, February). Solid findings in mathematics education: What are they and what are they good for? CERME 10 . Ireland: Dublin https://hal.archives-ouvertes.fr/hal-01849607

Bowker, G. C., & Star, S. L. (2000). Sorting things out: Classification and its consequences . MIT Press. https://doi.org/10.7551/mitpress/6352.001.0001

Burkhardt, H. (2019). Improving policy and practice. Educational Designer , 3 (12) http://www.educationaldesigner.org/ed/volume3/issue12/article46/

Cai, J., & Hwang, S. (2019). Constructing and employing theoretical frameworks in (mathematics) education research. For the Learning of Mathematics , 39 (3), 44–47.

Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and U.S. elementary mathematics textbooks. International Journal of Science and Mathematics Education , 15 (8), 1521–1540. https://doi.org/10.1007/s10763-016-9758-2

Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics , 105 , 287–301. https://doi.org/10.1007/s10649-020-10008-x

Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., … Hiebert, J. (2020). Improving the impact of research on practice: Capitalizing on technological advances for research. Journal for Research in Mathematics Education , 51 (5), 518–529 https://pubs.nctm.org/view/journals/jrme/51/5/article-p518.xml

Chronaki, A. (2019). Affective bodying of mathematics, children and difference: Choreographing ‘sad affects’ as affirmative politics in early mathematics teacher education. ZDM-Mathematics Education , 51 (2), 319–330. https://doi.org/10.1007/s11858-019-01045-9

Civil, M., & Bernier, E. (2006). Exploring images of parental participation in mathematics education: Challenges and possibilities. Mathematical Thinking and Learning , 8 (3), 309–330. https://doi.org/10.1207/s15327833mtl0803_6

Cobb, P., Gresalfi, M., & Hodge, L. L. (2009). An interpretive scheme for analyzing the identities that students develop in mathematics classrooms. Journal for Research in Mathematics Education , 40 ( 1 ), 40–68 https://pubs.nctm.org/view/journals/jrme/40/1/article-p40.xml

Darragh, L. (2016). Identity research in mathematics education. Educational Studies in Mathematics , 93 (1), 19–33. https://doi.org/10.1007/s10649-016-9696-5

de Abreu, G., Bishop, A., & Presmeg, N. C. (Eds.). (2006). Transitions between contexts of mathematical practices . Kluwer.

de Freitas, E., Ferrara, F., & Ferrari, G. (2019). The coordinated movements of collaborative mathematical tasks: The role of affect in transindividual sympathy. ZDM-Mathematics Education , 51 (2), 305–318. https://doi.org/10.1007/s11858-018-1007-4

Deng, Z. (2018). Contemporary curriculum theorizing: Crisis and resolution. Journal of Curriculum Studies , 50 (6), 691–710. https://doi.org/10.1080/00220272.2018.1537376

Dobie, T. E., & Sherin, B. (2021). The language of mathematics teaching: A text mining approach to explore the zeitgeist of US mathematics education. Educational Studies in Mathematics .  https://doi.org/10.1007/s10649-020-10019-8

Eames, C., & Eames, R. (1977). Powers of Ten [Film]. YouTube. https://www.youtube.com/watch?v=0fKBhvDjuy0

Engelbrecht, J., Borba, M. C., Llinares, S., & Kaiser, G. (2020). Will 2020 be remembered as the year in which education was changed? ZDM-Mathematics Education , 52 (5), 821–824. https://doi.org/10.1007/s11858-020-01185-3

English, L. (2008). Setting an agenda for international research in mathematics education. In L. D. English (Ed.), Handbook of international research in mathematics education (2nd ed., pp. 3–19). Routledge.

Ernest, P. (2020). Unpicking the meaning of the deceptive mathematics behind the COVID alert levels. Philosophy of Mathematics Education Journal , 36 http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/pome36/index.html

Freudenthal, H. (1986). Didactical phenomenology of mathematical structures . Springer.

Gilmore, C., Göbel, S. M., & Inglis, M. (2018). An introduction to mathematical cognition . Routledge.

Goos, M., & Beswick, K. (Eds.). (2021). The learning and development of mathematics teacher educators: International perspectives and challenges . Springer. https://doi.org/10.1007/978-3-030-62408-8

Gorard, S. (Ed.). (2020). Getting evidence into education. Evaluating the routes to policy and practice . Routledge.

Gravemeijer, K., Stephan, M., Julie, C., Lin, F.-L., & Ohtani, M. (2017). What mathematics education may prepare students for the society of the future? International Journal of Science and Mathematics Education , 15 (1), 105–123. https://doi.org/10.1007/s10763-017-9814-6

Hannula, M. S. (2019). Young learners’ mathematics-related affect: A commentary on concepts, methods, and developmental trends. Educational Studies in Mathematics , 100 (3), 309–316. https://doi.org/10.1007/s10649-018-9865-9

Hilt, L. T. (2015). Included as excluded and excluded as included: Minority language pupils in Norwegian inclusion policy. International Journal of Inclusive Education , 19 (2), 165–182.

Hodgen, J., Taylor, B., Jacques, L., Tereshchenko, A., Kwok, R., & Cockerill, M. (2020). Remote mathematics teaching during COVID-19: Intentions, practices and equity . UCL Institute of Education https://discovery.ucl.ac.uk/id/eprint/10110311/

Horn, I. S. (2017). Motivated: Designing math classrooms where students want to join in . Heinemann.

Hoyles, C., Noss, R., & Pozzi, S. (2001). Proportional reasoning in nursing practice. Journal for Research in Mathematics Education , 32 (1), 4–27. https://doi.org/10.2307/749619

Ito, M., Martin, C., Pfister, R. C., Rafalow, M. H., Salen, K., & Wortman, A. (2018). Affinity online: How connection and shared interest fuel learning . NYU Press.

Jackson, K. (2011). Approaching participation in school-based mathematics as a cross-setting phenomenon. The Journal of the Learning Sciences , 20 (1), 111–150. https://doi.org/10.1080/10508406.2011.528319

Jansen, A., Herbel-Eisenmann, B., & Smith III, J. P. (2012). Detecting students’ experiences of discontinuities between middle school and high school mathematics programs: Learning during boundary crossing. Mathematical Thinking and Learning , 14 (4), 285–309. https://doi.org/10.1080/10986065.2012.717379

Johnson, L. F., Smith, R. S., Smythe, J. T., & Varon, R. K. (2009). Challenge-based learning: An approach for our time (pp. 1–38). The New Media Consortium https://www.learntechlib.org/p/182083

Jullien, F. (2018). Living off landscape: Or the unthought-of in reason . Rowman & Littlefield.

Kazima, M. (2019). What is proven to work in successful countries should be implemented in other countries: The case of Malawi and Zambia. In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 73–78). PME.

Kim, H. (2019). Ask again, “why should we implement what works in successful countries?” In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 79–82). PME.

Kolovou, A., Van Den Heuvel-Panhuizen, M., & Bakker, A. (2009). Non-routine problem solving tasks in primary school mathematics textbooks—a needle in a haystack. Mediterranean Journal for Research in Mathematics Education , 8 (2), 29–66.

Kwon, O. N., Han, C., Lee, C., Lee, K., Kim, K., Jo, G., & Yoon, G. (2021). Graphs in the COVID-19 news: A mathematics audit of newspapers in Korea. Educational Studies in Mathematics . https://doi.org/10.1007/s10649-021-10029-0

Lefebvre, H. (2004). Rhythmanalysis: Space, time and everyday life (Original 1992; Translation by S. Elden & G. Moore) . Bloomsbury Academic. https://doi.org/10.5040/9781472547385 .

Li, Y. (2019). Should what works in successful countries be implemented in other countries? In M. Graven, H. Venkat, A. A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the international group for the Psychology of Mathematics Education (Vol. 1, pp. 67–72). PME.

Martin, D., Gholson, M., & Leonard, J. (2010). Mathematics as gatekeeper: Power and privilege in the production of power. Journal of Urban Mathematics Education , 3 (2), 12–24.

Masschelein, J., & Simons, M. (2019). Bringing more ‘school’ into our educational institutions. Reclaiming school as pedagogic form. In A. Bikner-Ahsbahs & M. Peters (Eds.), Unterrichtsentwicklung macht Schule (pp. 11–26) . Springer. https://doi.org/10.1007/978-3-658-20487-7_2

Meeter, M., Bele, T., den Hartogh, C., Bakker, T., de Vries, R. E., & Plak, S. (2020). College students’ motivation and study results after COVID-19 stay-at-home orders. https://psyarxiv.com .

Nemirovsky, R., Kelton, M. L., & Civil, M. (2017). Toward a vibrant and socially significant informal mathematics education. In J. Cai (Ed.), Compendium for Research in Mathematics Education (pp. 968–979). National Council of Teachers of Mathematics.

Niss, M. (2019). The very multi-faceted nature of mathematics education research. For the Learning of Mathematics , 39 (2), 2–7.

OECD. (2020). Back to the Future of Education: Four OECD Scenarios for Schooling. Educational Research and Innovation . OECD Publishing. https://doi.org/10.1787/20769679

Potari, D., Psycharis, G., Sakonidis, C., & Zachariades, T. (2019). Collaborative design of a reform-oriented mathematics curriculum: Contradictions and boundaries across teaching, research, and policy. Educational Studies in Mathematics , 102 (3), 417–434. https://doi.org/10.1007/s10649-018-9834-3

Proulx, J., & Maheux, J. F. (2019). Effect sizes, epistemological issues, and identity of mathematics education research: A commentary on editorial 102(1). Educational Studies in Mathematics , 102 (2), 299–302. https://doi.org/10.1007/s10649-019-09913-7

Roos, H. (2019). Inclusion in mathematics education: An ideology, A way of teaching, or both? Educational Studies in Mathematics , 100 (1), 25–41. https://doi.org/10.1007/s10649-018-9854-z

Saenz, M., Medina, A., & Urbine Holguin, B. (2020). Colombia: La prender al onda (to turn on the wave). Education continuity stories series . OECD Publishing https://oecdedutoday.com/wp-content/uploads/2020/12/Colombia-a-prender-la-onda.pdf

Schindler, M., & Bakker, A. (2020). Affective field during collaborative problem posing and problem solving: A case study. Educational Studies in Mathematics , 105 , 303–324. https://doi.org/10.1007/s10649-020-09973-0

Schoenfeld, A. H. (1999). Looking toward the 21st century: Challenges of educational theory and practice. Educational Researcher , 28 (7), 4–14. https://doi.org/10.3102/0013189x028007004

Schukajlow, S., Rakoczy, K., & Pekrun, R. (2017). Emotions and motivation in mathematics education: Theoretical considerations and empirical contributions. ZDM-Mathematics Education , 49 (3), 307–322. https://doi.org/10.1007/s11858-017-0864-6

Sfard, A. (2005). What could be more practical than good research? Educational Studies in Mathematics , 58 (3), 393–413. https://doi.org/10.1007/s10649-005-4818-5

Shimizu, Y., & Vithal, R. (Eds.). (2019). ICMI Study 24 Conference Proceedings. School mathematics curriculum reforms: Challenges, changes and opportunities . ICMI: University of Tsukuba & ICMI http://www.human.tsukuba.ac.jp/~icmi24/

Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics , 10 (3), 24–41.

Stephan, M. L., Chval, K. B., Wanko, J. J., Civil, M., Fish, M. C., Herbel-Eisenmann, B., … Wilkerson, T. L. (2015). Grand challenges and opportunities in mathematics education research. Journal for Research in Mathematics Education , 46 (2), 134–146. https://doi.org/10.5951/jresematheduc.46.2.0134

Suazo-Flores, E., Alyami, H., Walker, W. S., Aqazade, M., & Kastberg, S. E. (2021). A call for exploring mathematics education researchers’ interdisciplinary research practices. Mathematics Education Research Journal , 1–10. https://doi.org/10.1007/s13394-021-00371-0

Svensson, P., Meaney, T., & Norén, E. (2014). Immigrant students’ perceptions of their possibilities to learn mathematics: The case of homework. For the Learning of Mathematics , 34 (3), 32–37.

UNESCO. (2015). Teacher policy development guide . UNESCO, International Task Force on Teachers for Education 2030. https://teachertaskforce.org/sites/default/files/2020-09/370966eng_0_1.pdf .

Van den Heuvel-Panhuizen, M. (2005). Can scientific research answer the ‘what’ question of mathematics education? Cambridge Journal of Education , 35 (1), 35–53. https://doi.org/10.1080/0305764042000332489

Wittmann, E. C. (1995). Mathematics education as a ‘design science’. Educational Studies in Mathematics , 29 (4), 355–374.

Yoon, H., Byerley, C. O. N., Joshua, S., Moore, K., Park, M. S., Musgrave, S., Valaas, L., & Drimalla, J. (2021). United States and South Korean citizens’ interpretation and assessment of COVID-19 quantitative data. The Journal of Mathematical Behavior . https://doi.org/10.1016/j.jmathb.2021.100865 .

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Acknowledgments

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

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Appendix 1: Explanation of Fig. 1

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

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

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Seating unfriendly customers, a combinatorics problem.

By Lisa Honeyman February 12, 2002

The Problem

In a certain coffee shop, the customers are grouchy in the early morning and none of them wishes to sit next to another at the counter.

1. Suppose there are ten seats at the counter. How many different ways can three early morning customers sit at the counter so that no one sits next to anyone else?

2. What if there are n seats at the counter?

3. What if we change the number of customers?

4. What if, instead of a counter, there was a round table and people refused to sit next to each other?

Assumptions

I am assuming that the order in which the people sit matters. So, if three people occupy the first, third and fifth seats, there are actually 6 (3!) different ways they can do this. I will explain more thoroughly in the body of my report.

Body of the Report

At first there are 10 seats available for the 3 people to sit in. But once the first person sits down, that limits where the second person can sit. Not only can’t he sit in the now-occupied seat, he can’t sit next to it either. What confused me at first was that if the first person sat at one of the ends, then there were 8 seats left for the second person to chose from. But if the 1 st person sat somewhere else, there were only 7 remaining seats available for the second person. I decided to look for patterns. By starting with a smaller number of seats, I was able to count the possibilities more easily. I was hoping to find a pattern so I could predict how many ways the 10 people could sit without actually trying to count them all. I realized that the smallest number of seats I could have would be 5. Anything less wouldn’t work because people would have to sit next to each other. So, I started with 5 seats. I called the customers A, B, and C.

With 5 seats there is only one configuration that works.

As I said in my assumptions section, I thought that the order in which the people sit is important. Maybe one person prefers to sit near the coffee maker or by the door. These would be different, so I decided to take into account the different possible ways these 3 people could occupy the 3 seats shown above. I know that ABC can be arranged in 3! = 6 ways. (ABC, ACB, BAC, BCA, CAB, CBA). So there are 6 ways to arrange 3 people in 5 seats with spaces between them. But, there is only one configuration of seats that can be used. (The 1 st , 3 rd , and 5 th ).

Next, I tried 6 seats. I used a systematic approach to show that there are 4 possible arrangements of seats. This is how my systematic approach works:

Assign person A to the 1 st seat. Put person B in the 3 rd seat, because he can’t sit next to person A. Now, person C can sit in either the 5 th or 6 th positions. (see the top two rows in the chart, below.) Next suppose that person B sits in the 4 th seat (the next possible one to the right.) That leaves only the 6 th seat free for person C. (see row 3, below.) These are all the possible ways for the people to sit if the 1 st seat is used. Now put person A in the 2 nd seat and person B in the 4 th . There is only one place where person C can sit, and that’s in the 6 th position. (see row 4, below.) There are no other ways to seat the three people if person A sits in the 2 nd seat. So, now we try putting person A in the 3 rd seat. If we do that, there are only 4 seats that can be used, but we know that we need at least 5, so there are no more possibilities.

Possible seats 3 people could occupy if there are 6 seats

Once again, the order the people sit in could be ABC, BAC, etc. so there are 4 * 6 = 24 ways for the 3 customers to sit in 6 seats with spaces between them.

I continued doing this, counting how many different groups of seats could be occupied by the three people using the systematic method I explained. Then I multiplied that number by 6 to account for the possible permutations of people in those seats. I created the following table of what I found.

Next I tried to come up with a formula. I decided to look for a formula using combinations or permutations. Since we are looking at 3 people, I decided to start by seeing what numbers I would get if I used n C 3 and n P 3 .

3 C 3 = 1   4 C 3 = 4   5 C 3 = 10   6 C 3 = 20

3 P 3 = 6   4 P 3 = 24   5 P 3 = 60   6 P 3 = 120

Surprisingly enough, these numbers matched the numbers I found in my table. However, the n in n P r and n C r seemed to be two less than the total # of seats I was investigating. 

Conjecture 1:

Given n seats at a lunch counter, there are n -2 C 3 ways to select the three seats in which the customers will sit such that no customer sits next to another one. There are n -2 P 3 ways to seat the 3 customers in such a way than none sits next to another.

After I found a pattern, I tried to figure out why n -2 C 3 works. (If the formula worked when order didn’t matter it could be easily extended to when the order did, but the numbers are smaller and easier to work with when looking at combinations rather than permutations.)

In order to prove Conjecture 1 convincingly, I need to show two things:

(1) Each n – 2 seat choice leads to a legal n seat configuration.

(2) Each n seat choice resulted from a unique n – 2 seat configuration.

To prove these two things I will show

And then conclude that these two procedures are both functions and therefore 1—1.

Claim (1): Each ( n – 2) -seat choice leads to a legal n seat configuration.

Suppose there were only n – 2 seats to begin with. First we pick three of them in which to put people, without regard to whether or not they sit next to each other. But, in order to guarantee that they don’t end up next to another person, we introduce an empty chair to the right of each of the first two people. It would look like this:

We don’t need a third “new” seat because once the person who is farthest to the right sits down, there are no more customers to seat. So, we started with n – 2 chairs but added two for a total of n chairs. Anyone entering the restaurant after this procedure had been completed wouldn’t know that there had been fewer chairs before these people arrived and would just see three customers sitting at a counter with n chairs. This procedure guarantees that two people will not end up next to each other. Thus, each ( n – 2)-seat choice leads to a unique, legal n seat configuration.

Therefore, positions s 1 ' s 2 ', and s 3 ' are all separated by at least one vacant seat.

This is a function that maps each combination of 3 seats selected from n – 2 seats onto a unique arrangement of n seats with 3 separated customers. Therefore, it is invertible.

Claim (2): Each 10-seat choice has a unique 8-seat configuration.

Given a legal 10-seat configuration, each of the two left-most diners must have an open seat to his/her right. Remove it and you get a unique 8-seat arrangement. If, in the 10-seat setting, we have q 1 > q 2 , q 3 ; q 3 – 1 > q 2 , and q 2 – 1 > q 1 , then the 8 seat positions are q 1 ' = q 2 , q 2 ' = q 2 – 1, and q 3 ' = q 3 – 2. Combining these equations with the conditions we have

q 2 ' = q 2 – 1 which implies q 2 ' > q 1 = q 1 '

q 3 ' = q 3 – 2 which implies q 3 ' > q 2 – 1 = q 2 '

Since q 3 ' > q 2 ' > q 1 ', these seats are distinct. If the diners are seated in locations q 1 , q 2 , and q 3 (where q 3 – 1 > q 2 and q 2 – 1 > q 1 ) and we remove the two seats to the right of q 1 and q 2 , then we can see that the diners came from q 1 , q 2 – 1, and q 3 – 2. This is a function that maps a legal 10-seat configuration to a unique 8-seat configuration.

The size of a set can be abbreviated s ( ). I will use the abbreviation S to stand for n separated seats and N to stand for the n – 2 non-separated seats.

therefore s ( N ) = s ( S ).

Because the sets are the same size, these functions are 1—1.

Using the technique of taking away and adding empty chairs, I can extend the problem to include any number of customers. For example, if there were 4 customers and 10 seats there would be 7 C 4 = 35 different combinations of chairs to use and 7 P 4 = 840 ways for the customers to sit (including the fact that order matters). You can imagine that three of the ten seats would be introduced by three of the customers. So, there would only be 7 to start with.

In general, given n seats and c customers, we remove c- 1 chairs and select the seats for the c customers. This leads to the formula n -( c -1) C c = n - c +1 C c for the number of arrangements.

Once the number of combinations of seats is found, it is necessary to multiply by c ! to find the number of permutations. Looking at the situation of 3 customers and using a little algebraic manipulation, we get the n P 3 formula shown below.

This same algebraic manipulation works if you have c people rather than 3, resulting in n - c +1 P c

Answers to Questions

• With 10 seats there are 8 P 3 = 336 ways to seat the 3 people.
• My formula for n seats and 3 customers is: n -2 P 3 .
• My general formula for n seats and c customers, is: n -( c -1) P c = n - c +1 P c

_________________________________________________________________ _

After I finished looking at this question as it applied to people sitting in a row of chairs at a counter, I considered the last question, which asked would happen if there were a round table with people sitting, as before, always with at least one chair between them.

I went back to my original idea about each person dragging in an extra chair that she places to her right, barring anyone else from sitting there. There is no end seat, so even the last person needs to bring an extra chair because he might sit to the left of someone who has already been seated. So, if there were 3 people there would be 7 seats for them to choose from and 3 extra chairs that no one would be allowed to sit in. By this reasoning, there would be 7 C 3 = 35 possible configurations of chairs to choose and 7 P 3 = 840 ways for 3 unfriendly people to sit at a round table.

Conjecture 2: Given 3 customers and n seats there are n -3 C 3 possible groups of 3 chairs which can be used to seat these customers around a circular table in such a way that no one sits next to anyone else.

My first attempt at a proof: To test this conjecture I started by listing the first few numbers generated by my formula:

When n = 6    6-3 C 3 = 3 C 3 = 1

When n = 7    7-3 C 3 = 4 C 3 = 4

When n = 8    8-3 C 3 = 5 C 3 = 10

When n = 9    9-3 C 3 = 6 C 3 = 20

Then I started to systematically count the first few numbers of groups of possible seats. I got the numbers shown in the following table. The numbers do not agree, so something is wrong — probably my conjecture!

I looked at a circular table with 8 people and tried to figure out the reason this formula doesn’t work. If we remove 3 seats (leaving 5) there are 10 ways to select 3 of the 5 remaining chairs. ( 5 C 3 ).

The circular table at the left in the figure below shows the n – 3 (in this case 5) possible chairs from which 3 will be randomly chosen. The arrows point to where the person who selects that chair could end up. For example, if chair A is selected, that person will definitely end up in seat #1 at the table with 8 seats. If chair B is selected but chair A is not, then seat 2 will end up occupied. However, if chair A and B are selected, then the person who chose chair B will end up in seat 3 . The arrows show all the possible seats in which a person who chose a particular chair could end. Notice that it is impossible for seat #8 to be occupied. This is why the formula 5 C 3 doesn’t work. It does not allow all seats at the table of 8 to be chosen.

The difference is that in the row-of-chairs-at-a-counter problem there is a definite “starting point” and “ending point.” The first chair can be identified as the one farthest to the left, and the last one as the one farthest to the right. These seats are unique because the “starting point” has no seat to the left of it and the “ending point” has no seat to its right. In a circle, it is not so easy.

Using finite differences I was able to find a formula that generates the correct numbers:

Proof: We need to establish a “starting point.” This could be any of the n seats. So, we select one and seat person A in that seat. Person B cannot sit on this person’s left (as he faces the table), so we must eliminate that as a possibility. Also, remove any 2 other chairs, leaving ( n – 4) possible seats where the second person can sit. Select another seat and put person B in it. Now, select any other seat from the ( n – 5) remaining seats and put person C in that. Finally, take the two seats that were previously removed and put one to the left of B and one to the left of C.

The following diagram should help make this procedure clear.

In a manner similar to the method I used in the row-of-chairs-at-a-counter problem, this could be proven more rigorously.

An Idea for Further Research:

Consider a grid of chairs in a classroom and a group of 3 very smelly people. No one wants to sit adjacent to anyone else. (There would be 9 empty seats around each person.) Suppose there are 16 chairs in a room with 4 rows and 4 columns. How many different ways could 3 people sit? What if there was a room with n rows and n columns? What if it had n rows and m columns?

References:

Abrams, Joshua. Education Development Center, Newton, MA. December 2001 - February 2002. Conversations with my mathematics mentor.

Brown, Richard G. 1994. Advanced Mathematics . Evanston, Illinois. McDougal Littell Inc. pp. 578-591

The Oral Presentation

Giving an oral presentation about your mathematics research can be very exciting! You have the opportunity to share what you have learned, answer questions about your project, and engage others in the topic you have been studying. After you finish doing your mathematics research, you may have the opportunity to present your work to a group of people such as your classmates, judges at a science fair or other type of contest, or educators at a conference. With some advance preparation, you can give a thoughtful, engaging talk that will leave your audience informed and excited about what you have done.

Planning for Your Oral Presentation

In most situations, you will have a time limit of between 10 and 30 minutes in which to give your presentation. Based upon that limit, you must decide what to include in your talk. Come up with some good examples that will keep your audience engaged. Think about what vocabulary, explanations, and proofs are really necessary in order for people to understand your work. It is important to keep the information as simple as possible while accurately representing what you’ve done. It can be difficult for people to understand a lot of technical language or to follow a long proof during a talk. As you begin to plan, you may find it helpful to create an outline of the points you want to include. Then you can decide how best to make those points clear to your audience.

You must also consider who your audience is and where the presentation will take place. If you are going to give your presentation to a single judge while standing next to your project display, your presentation will be considerably different than if you are going to speak from the stage in an auditorium full of people! Consider the background of your audience as well. Is this a group of people that knows something about your topic area? Or, do you need to start with some very basic information in order for people to understand your work? If you can tailor your presentation to your audience, it will be much more satisfying for them and for you.

No matter where you are presenting your speech and for whom, the structure of your presentation is very important. There is an old bit of advice about public speaking that goes something like this: “Tell em what you’re gonna tell ’em. Tell ’em. Then tell ’em what you told ’em.” If you use this advice, your audience will find it very easy to follow your presentation. Get the attention of the audience and tell them what you are going to talk about, explain your research, and then following it up with a re-cap in the conclusion.

Writing Your Introduction

Your introduction sets the stage for your entire presentation. The first 30 seconds of your speech will either capture the attention of your audience or let them know that a short nap is in order. You want to capture their attention. There are many different ways to start your speech. Some people like to tell a joke, some quote famous people, and others tell stories.

Here are a few examples of different types of openers.

You can use a quote from a famous person that is engaging and relevant to your topic. For example:

• Benjamin Disraeli once said, “There are three kinds of lies: lies, damn lies, and statistics.” Even though I am going to show you some statistics this morning, I promise I am not going to lie to you! Instead, . . .

• The famous mathematician, Paul Erdös, said, “A Mathematician is a machine for turning coffee into theorems.” Today I’m here to show you a great theorem that I discovered and proved during my mathematics research experience. And yes, I did drink a lot of coffee during the project!

• According to Stephen Hawking, “Equations are just the boring part of mathematics.” With all due respect to Dr. Hawking, I am here to convince you that he is wrong. Today I’m going to show you one equation that is not boring at all!

Some people like to tell a short story that leads into their discussion.

“Last summer I worked at a diner during the breakfast shift. There were 3 regular customers who came in between 6:00 and 6:15 every morning. If I tell you that you didn’t want to talk to these folks before they’ve had their first cup of coffee, you’ll get the idea of what they were like. In fact, these people never sat next to each other. That’s how grouchy they were! Well, their anti-social behavior led me to wonder, how many different ways could these three grouchy customers sit at the breakfast counter without sitting next to each other? Amazingly enough, my summer job serving coffee and eggs to grouchy folks in Boston led me to an interesting combinatorics problem that I am going to talk to you about today.”

A short joke related to your topic can be an engaging way to start your speech.

It has been said that there are three kinds of mathematicians: those who can count and those who can’t.

All joking aside, my mathematics research project involves counting. I have spent the past 8 weeks working on a combinatorics problem.. . .

To find quotes to use in introductions and conclusions try: http://www.quotationspage.com/

To find some mathematical quotes, consult the Mathematical Quotation Server: http://math.furman.edu/~mwoodard/mquot.html

To find some mathematical jokes, you can look at the “Profession Jokes” web site: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm

There is a collection of math jokes compiled by the Canadian Mathematical Society at http://camel.math.ca/Recreation/

After you have the attention of your audience, you must introduce your research more formally. You might start with a statement of the problem that you investigated and what lead you to choose that topic. Then you might say something like this,

“Today I will demonstrate how I came to the conclusion that there are n ( n  – 4)( n  – 5) ways to seat 3 people at a circular table with n seats in such a way that no two people sit next to each other. In order to do this I will first explain how I came up with this formula and then I will show you how I proved it works. Finally, I will extend this result to tables with more than 3 people sitting at them.”

By providing a brief outline of your talk at the beginning and reminding people where you are in the speech while you are talking, you will be more effective in keeping the attention of your audience. It will also make it much easier for you to remember where you are in your speech as you are giving it.

The Middle of Your Presentation

Because you only have a limited amount of time to present your work, you need to plan carefully. Decide what is most important about your project and what you want people to know when you are finished. Outline the steps that people need to follow in order to understand your research and then think carefully about how you will lead them through those steps. It may help to write your entire speech out in advance. Even if you choose not to memorize it and present it word for word, the act of writing will help you clarify your ideas. Some speakers like to display an outline of their talk throughout their entire presentation. That way, the audience always knows where they are in the presentation and the speaker can glance at it to remind him or herself what comes next.

An oral presentation must be structured differently than a written one because people can’t go back and “re-read” a complicated section when they are at a talk. You have to be extremely clear so that they can understand what you are saying the first time you say it. There is an acronym that some presenters like to remember as they prepare a talk: “KISS.” It means, “Keep It Simple, Student.” It may sound silly, but it is good advice. Keep your sentences short and try not to use too many complicated words. If you need to use technical language, be sure to define it carefully. If you feel that it is important to present a proof, remember that you need to keep things easy to understand. Rather than going through every step, discuss the main points and the conclusion. If you like, you can write out the entire proof and include it in a handout so that folks who are interested in the details can look at them later. Give lots of examples! Not only will examples make your talk more interesting, but they will also make it much easier for people to follow what you are saying.

It is useful to remember that when people have something to look at, it helps to hold their attention and makes it easier for them to understand what you are saying. Therefore, use lots of graphs and other visual materials to support your work. You can do this using posters, overhead transparencies, models, or anything else that helps make your explanations clear.

Using Materials

As you plan for your presentation, consider what equipment or other materials you might want use. Find out what is available in advance so you don’t spend valuable time creating materials that you will not be able to use. Common equipment used in talks include an over-head projector, VCR, computer, or graphing calculator. Be sure you know how to operate any equipment that you plan to use. On the day of your talk, make sure everything is ready to go (software loaded, tape at the right starting point etc.) so that you don’t have “technical difficulties.”

Visual aides can be very useful in a presentation. (See Displaying Your Results for details about poster design.) If you are going to introduce new vocabulary, consider making a poster with the words and their meanings to display throughout your talk. If people forget what a term means while you are speaking, they can refer to the poster you have provided. (You could also write the words and meanings on a black/white board in advance.) If there are important equations that you would like to show, you can present them on an overhead transparency that you prepare prior to the talk. Minimize the amount you write on the board or on an overhead transparency during your presentation. It is not very engaging for the audience to sit watching while you write things down. Prepare all equations and materials in advance. If you don’t want to reveal all of what you have written on your transparency at once, you can cover up sections of your overhead with a piece of paper and slide it down the page as you move along in your talk. If you decide to use overhead transparencies, be sure to make the lettering large enough for your audience to read. It also helps to limit how much you put on your transparencies so they are not cluttered. Lastly, note that you can only project approximately half of a standard 8.5" by 11" page at any one time, so limit your information to displays of that size.

Presenters often create handouts to give to members of the audience. Handouts may include more information about the topic than the presenter has time to discuss, allowing listeners to learn more if they are interested. Handouts may also include exercises that you would like audience members to try, copies of complicated diagrams that you will display, and a list of resources where folks might find more information about your topic. Give your audience the handout before you begin to speak so you don’t have to stop in the middle of the talk to distribute it. In a handout you might include:

• A proof you would like to share, but you don’t have time to present entirely.

• Copies of important overhead transparencies that you use in your talk.

• Diagrams that you will display, but which may be too complicated for someone to copy down accurately.

• Resources that you think your audience members might find useful if they are interested in learning more about your topic.

The Conclusion

Ending your speech is also very important. Your conclusion should leave the audience feeling satisfied that the presentation was complete. One effective way to conclude a speech is to review what you presented and then to tie back to your introduction. If you used the Disraeli quote in your introduction, you might end by saying something like,

I hope that my presentation today has convinced you that . . . Statistical analysis backs up the claims that I have made, but more importantly, . . . . And that’s no lie!

Getting Ready

After you have written your speech and prepared your visuals, there is still work to be done.

• Prepare your notes on cards rather than full-size sheets of paper. Note cards will be less likely to block your face when you read from them. (They don’t flop around either.) Use a large font that is easy for you to read. Write notes to yourself on your notes. Remind yourself to smile or to look up. Mark when to show a particular slide, etc.
• Practice! Be sure you know your speech well enough that you can look up from your notes and make eye contact with your audience. Practice for other people and listen to their feedback.
• Time your speech in advance so that you are sure it is the right length. If necessary, cut or add some material and time yourself again until your speech meets the time requirements. Do not go over time!
• Anticipate questions and be sure you are prepared to answer them.
• Make a list of all materials that you will need so that you are sure you won’t forget anything.
• If you are planning to provide a handout, make a few extras.
• If you are going to write on a whiteboard or a blackboard, do it before starting your talk.

The Delivery

How you deliver your speech is almost as important as what you say. If you are enthusiastic about your presentation, it is far more likely that your audience will be engaged. Never apologize for yourself. If you start out by saying that your presentation isn’t very good, why would anyone want to listen to it? Everything about how you present yourself will contribute to how well your presentation is received. Dress professionally. And don’t forget to smile!

Here are a few tips about delivery that you might find helpful.

• Make direct eye contact with members of your audience. Pick a person and speak an entire phrase before shifting your gaze to another person. Don’t just “scan” the audience. Try not to look over their heads or at the floor. Be sure to look at all parts of the room at some point during the speech so everyone feels included.
• Speak loudly enough for people to hear and slowly enough for them to follow what you are saying.
• Do not read your speech directly from your note cards or your paper. Be sure you know your speech well enough to make eye contact with your audience. Similarly, don’t read your talk directly off of transparencies.
• Avoid using distracting or repetitive hand gestures. Be careful not to wave your manuscript around as you speak.
• Move around the front of the room if possible. On the other hand, don’t pace around so much that it becomes distracting. (If you are speaking at a podium, you may not be able to move.)
• Keep technical language to a minimum. Explain any new vocabulary carefully and provide a visual aide for people to use as a reference if necessary.
• Be careful to avoid repetitive space-fillers and slang such as “umm”, “er”, “you know”, etc. If you need to pause to collect your thoughts, it is okay just to be silent for a moment. (You should ask your practice audiences to monitor this habit and let you know how you did).
• Leave time at the end of your speech so that the audience can ask questions.

Displaying Your Results

When you create a visual display of your work, it is important to capture and retain the attention of your audience. Entice people to come over and look at your work. Once they are there, make them want to stay to learn about what you have to tell them. There are a number of different formats you may use in creating your visual display, but the underlying principle is always the same: your work should be neat, well-organized, informative, and easy to read.

It is unlikely that you will be able to present your entire project on a single poster or display board. So, you will need to decide which are the most important parts to include. Don’t try to cram too much onto the poster. If you do, it may look crowded and be hard to read! The display should summarize your most important points and conclusions and allow the reader to come away with a good understanding of what you have done.

A good display board will have a catchy title that is easy to read from a distance. Each section of your display should be easily identifiable. You can create posters such as this by using headings and also by separating parts visually. Titles and headings can be carefully hand-lettered or created using a computer. It is very important to include lots of examples on your display. It speeds up people’s understanding and makes your presentation much more effective. The use of diagrams, charts, and graphs also makes your presentation much more interesting to view. Every diagram or chart should be clearly labeled. If you include photographs or drawings, be sure to write captions that explain what the reader is looking at.

In order to make your presentation look more appealing, you will probably want to use some color. However, you must be careful that the color does not become distracting. Avoid florescent colors, and avoid using so many different colors that your display looks like a patch-work quilt. You want your presentation to be eye-catching, but you also want it to look professional.

People should be able to read your work easily, so use a reasonably large font for your text. (14 point is a recommended minimum.) Avoid writing in all-capitals because that is much harder to read than regular text. It is also a good idea to limit the number of different fonts you use on your display. Too many different fonts can make your poster look disorganized.

Notice how each section on the sample poster is defined by the use of a heading and how the various parts of the presentation are displayed on white rectangles. (Some of the rectangles are blank, but they would also have text or graphics on them in a real presentation.) Section titles were made with pale green paper mounted on red paper to create a boarder. Color was used in the diagrams to make them more eye-catching. This poster would be suitable for hanging on a bulletin board.

If you are planning to use a poster, such as this, as a visual aid during an oral presentation, you might consider backing your poster with foam-core board or corrugated cardboard. A strong board will not flop around while you are trying to show it to your audience. You can also stand a stiff board on an easel or the tray of a classroom blackboard or whiteboard so that your hands will be free during your talk. If you use a poster as a display during an oral presentation, you will need to make the text visible for your audience. You can create a hand-out or you can make overhead transparencies of the important parts. If you use overhead transparencies, be sure to use lettering that is large enough to be read at a distance when the text is projected.

If you are preparing your display for a science fair, you will probably want to use a presentation board that can be set up on a table. You can buy a pre-made presentation board at an office supply or art store or you can create one yourself using foam-core board. With a presentation board, you can often use the space created by the sides of the board by placing a copy of your report or other objects that you would like people to be able to look at there. In the illustration, a black trapezoid was cut out of foam-core board and placed on the table to make the entire display look more unified. Although the text is not shown in the various rectangles in this example, you will present your information in spaces such as these.

Don’t forget to put your name on your poster or display board. And, don’t forget to carefully proof-read your work. There should be no spelling, grammatical or typing mistakes on your project. If your display is not put together well, it may make people wonder about the quality of the work you did on the rest of your project.

For more information about creating posters for science fair competitions, see

http://school.discovery.com/sciencefaircentral/scifairstudio/handbook/display.html ,

http://www.siemens-foundation.org/science/poster_guidelines.htm ,

Robert Gerver’s book, Writing Math Research Papers , (published by Key Curriculum Press) has an excellent section about doing oral presentations and making posters, complete with many examples.

References Used

American Psychological Association . Electronic reference formats recommended by the American Psychological Association . (2000, August 22). Washington, DC: American Psychological Association. Retrieved October 6, 2000, from the World Wide Web: http://www.apastyle.org/elecsource.html

Bridgewater State College. (1998, August 5 ). APA Style: Sample Bibliographic Entries (4th ed) . Bridgewater, MA: Clement C. Maxwell Library. Retrieved December 20, 2001, from the World Wide Web: http://www.bridgew.edu/dept/maxwell/apa.htm

Crannell, Annalisa. (1994). A Guide to Writing in Mathematics Classes . Franklin & Marshall College. Retrieved January 2, 2002, from the World Wide Web: http://www.fandm.edu/Departments/Mathematics/writing_in_math/guide.html

Gerver, Robert. 1997. Writing Math Research Papers . Berkeley, CA: Key Curriculum Press.

Moncur, Michael. (1994-2002 ). The Quotations Page . Retrieved April 9, 2002, from the World Wide Web: http://www.quotationspage.com/

Public Speaking -- Be the Best You Can Be . (2002). Landover, Hills, MD: Advanced Public Speaking Institute. Retrieved April 9, 2002, from the World Wide Web: http://www.public-speaking.org/

Recreational Mathematics. (1988) Ottawa, Ontario, Canada: Canadian Mathematical Society. Retrieved April 9, 2002, from the World Wide Web: http://camel.math.ca/Recreation/

Shay, David. (1996). Profession Jokes — Mathematicians. Retrieved April 5, 2001, from the World Wide Web: http://www.geocities.com/CapeCanaveral/4661/projoke22.htm

Sieman’s Foundation. (2001). Judging Guidelines — Poster . Retrieved April 9, 2002, from the World Wide Web: http://www.siemens-foundation.org/science/poster_guidelines.htm ,

VanCleave, Janice. (1997). Science Fair Handbook. Discovery.com. Retrieved April 9, 2002, from the World Wide Web: http://school.discovery.com/sciencefaircentral/scifairstudio/handbook/display.html ,

Woodward, Mark. (2000) . The Mathematical Quotations Server . Furman University. Greenville, SC. Retrieved April 9, 2002, from the World Wide Web: http://math.furman.edu/~mwoodard/mquot.html

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Computing the theta function

A quick estimate for the volume of a polyhedron (with m. rudelson), when a system of real quadratic equations has a solution (with m. rudelson), smoothed counting of 0-1 points in polyhedra, testing systems of real quadratic equations for approximate solutions, more on zeros and approximation of the ising partition function (with n. barvinok), a remark on approximating permanents of positive definite matrices, integrating products of quadratic forms, testing for dense subsets in a graph via the partition function (with a. della pella), approximating real-rooted and stable polynomials, with combinatorial applications, stability and complexity of mixed discriminants, computing permanents of complex diagonally dominant matrices and tensors, weighted counting of solutions to sparse systems of equations (with g. regts), approximating permanents and hafnians, concentration of the mixed discriminant of well-conditioned matrices, computing the partition function of a polynomial on the boolean cube, computing the partition function for graph homomorphisms with multiplicities (with p. soberon), computing the partition function for graph homomorphisms (with p. soberon), computing the partition function for cliques in a graph, computing the permanent of (some) complex matrices, on testing hamiltonicity of graphs, convexity of the image of a quadratic map via the relative entropy distance, thrifty approximations of convex bodies by polytopes, approximations of convex bodies by polytopes and by projections of spectrahedra, explicit constructions of centrally symmetric k -neighborly polytopes and large strictly antipodal sets (with s.j. lee and i. novik), a bound for the number of vertices of a polytope with applications, centrally symmetric polytopes with many faces (with s.j. lee and i. novik), neighborliness of the symmetric moment curve (with s.j. lee and i. novik), matrices with prescribed row and column sums, computing the partition function for perfect matchings in a hypergraph (with a. samorodnitsky), the number of graphs and a random graph with a given degree sequence (with j.a. hartigan), an asymptotic formula for the number of non-negative integer matrices with prescribed row and column sums (with j.a. hartigan), maximum entropy gaussian approximation for the number of integer points and volumes of polytopes (with j.a. hartigan), what does a random contingency table look like, on the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries, an approximation algorithm for counting contingency tables (with z. luria, a. samorodnitsky and a. yong), asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes, a centrally symmetric version of the cyclic polytope (with i. novik), the computational complexity of convex bodies (with e. veomett), brunn-minkowski inequalities for contingency tables and integer flows, the complexity of generating functions for integer points in polyhedra and beyond, enumerating contingency tables via random permanents, approximating orthogonal matrices by permutation matrices, computing the ehrhart quasi-polynomial of a rational simplex, integration and optimization of multivariate polynomials by restriction onto a random subspace, lattice points, polyhedra, and complexity, convex geometry of orbits (with g. blekherman), c++ codes for estimating permanents, hafnians and the number of forests in a graph, random weighting, asymptotic counting, and inverse isoperimetry (with a. samorodnitsky), short rational generating functions for lattice point problems (with k. woods), estimating l- infinity norms by l 2k norms for functions on orbits, approximating a norm by a polynomial, the distribution of values in the quadratic assignment problem (with t. stephen), the maximum traveling salesman problem (with e.kh. gimadi and a.i. serdyukov), new permanent estimators via non-commutative determinants, a c++ code to compute bounds for the permanent of a 0-1 matrix by the ``average distance'' approach, the distance approach to approximate combinatorial counting (with a. samorodnitsky), a remark on the rank of positive semidefinite matrices subject to affine constraints, polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor, finding maximum length tours under polyhedral norms (with d. johnson, g. woeginger, and r. woodroofe), an algorithmic theory of lattice points in polyhedra (with j. pommersheim).

• Open access
• Published: 11 March 2019

Enhancing achievement and interest in mathematics learning through Math-Island

• Charles Y. C. Yeh   ORCID: orcid.org/0000-0003-4581-6575 1 ,
• Hercy N. H. Cheng 2 ,
• Zhi-Hong Chen 3 ,
• Calvin C. Y. Liao 4 &
• Tak-Wai Chan 5  

Research and Practice in Technology Enhanced Learning volume  14 , Article number:  5 ( 2019 ) Cite this article

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Conventional teacher-led instruction remains dominant in most elementary mathematics classrooms in Taiwan. Under such instruction, the teacher can rarely take care of all students. Many students may then continue to fall behind the standard of mathematics achievement and lose their interest in mathematics; they eventually give up on learning mathematics. In fact, students in Taiwan generally have lower interest in learning mathematics compared to many other regions/countries. Thus, how to enhance students’ mathematics achievement and interest are two major problems, especially for those low-achieving students. This paper describes how we designed a game-based learning environment, called Math-Island , by incorporating the mechanisms of a construction management game into the knowledge map of the elementary mathematics curriculum. We also report an experiment conducted with 215 elementary students for 2 years, from grade 2 to grade 3. In this experiment, in addition to teacher-led instruction in the classroom, students were directed to learn with Math-Island by using their own tablets at school and at home. As a result of this experiment, we found that there is an increase in students’ mathematics achievement, especially in the calculation and word problems. Moreover, the achievements of low-achieving students in the experimental school outperformed the low-achieving students in the control school (a control group in another school) in word problems. Moreover, both the low-achieving students and the high-achieving students in the experimental school maintained a rather high level of interest in mathematics and in the system.

Introduction

Mathematics has been regarded as a fundamental subject because arithmetic and logical reasoning are the basis of science and technology. For this reason, educational authorities emphasize students’ proficiency in computational skills and problem-solving. Recently, the results of the Program for International Student Assessment (PISA) and the Trends in Mathematics and Science Study (TIMSS) in 2015 (OECD 2016 ; Mullis et al. 2016 ) revealed a challenge for Taiwan. Although Taiwanese students had higher average performance in mathematics literacy compared to students in other countries, there was still a significant percentage of low-achieving students in Taiwan. Additionally, most Taiwanese students show low levels of interest and confidence in learning mathematics (Lee 2012 ).

The existence of a significant percentage of low-achieving students is probably due to teacher-led instruction, which still dominates mathematics classrooms in most Asian countries. It should be noted that students in every classroom possess different abilities and hence demonstrate different achievements. Unfortunately, in teacher-led instruction, all the students are required to learn from the teacher in the same way at the same pace (Hwang et al. 2012 ). Low-achieving students, without sufficient time, are forced to receive knowledge passively. Barr and Tagg ( 1995 ) pointed out that it is urgent for low-achieving students to have more opportunities to learn mathematics at their own pace. Researchers suggested one-to-one technology (Chan et al. 2006 ) through which every student is equipped with a device to learn in school or at home seamlessly. Furthermore, they can receive immediate feedback from Math-Island, which supports their individualized learning actively and productively. Thus, this may provide more opportunities for helping low-achieving students improve their achievement.

The low-interest problem for almost all students in Taiwan is usually accompanied by low motivation (Krapp 1999 ). Furthermore, students with continuously low performance in mathematics may eventually lose their interest and refuse to learn further (Schraw et al. 2001 ). This is a severe problem. To motivate students to learn, researchers design educational games to provide enjoyable and engaging learning experiences (Kiili and Ketamo 2007 ). Some of these researchers found that game-based learning may facilitate students’ learning in terms of motivation and learning effects (Liu and Chu 2010 ), spatial abilities and attention (Barlett et al. 2009 ), situated learning, and problem-solving (Li and Tsai 2013 ). Given these positive results, we hope that our educational game can enhance and sustain the student’s interest in learning mathematics.

In fact, many researchers who endeavored to develop educational games for learning mathematics have shown that their games could facilitate mathematics performance, enjoyment, and self-efficacy (Ku et al. 2014 ; McLaren et al. 2017 ). Although some of the studies were conducted for as many as 4 months (e.g., Hanus and Fox 2015 ), one may still criticize them for the possibility that the students’ interest could be a novelty effect—meaning their interest will decrease as the feeling of novelty diminishes over time (Koivisto and Hamari 2014 ). Due to the limitations of either experimental time or sample sizes, most studies could not effectively exclude the novelty effect of games, unless they were conducted in a natural setting for a long time.

In this study, we collaborated with an experimental elementary school for more than 2 years. The mathematics teachers in the school adopted our online educational game, Math-Island . The students used their own tablet PCs to learn mathematics from the game in class or at home at their own pace. In particular, low-achieving students might have a chance to catch up with the other students and start to feel interested in learning mathematics. Most importantly, because the online educational game was a part of the mathematics curriculum, the students could treat the game as their ordinary learning materials like textbooks. In this paper, we reported a 2-year study, in which 215 second graders in the school adopted the Math-Island game in their daily routine. More specifically, the purpose of this paper was to investigate the effect of the game on students’ mathematics achievement. Additionally, we were also concerned about how well the low-achieving students learned, whether they were interested in mathematics and the game, and how their interest in mathematics compared with that of high-achieving students. In such a long-term study with a large sample size, it was expected that the novelty effect would be considerably reduced, allowing us to evaluate the effect of the educational game on students’ achievement and interest.

The paper is organized as follows. In the “ Related works ” section, we review related studies on computer-supported mathematics learning and educational games. In the “ Design ” section, the game mechanism and the system design are presented. In the “ Method ” section, we describe the research method and the procedures of this study. In the “ Results ” section, the research results about students’ achievement and interest are presented. In the “ Discussion on some features of this study ” section, we discuss the long-term study, knowledge map design, and the two game mechanisms. Finally, the summary of the current situation and potential future work is described in the “ Conclusion and future work ” section.

Related works

Computer-supported mathematics learning.

The mathematics curriculum in elementary schools basically includes conceptual understanding, procedural fluency, and strategic competence in terms of mathematical proficiency (see Kilpatrick et al. 2001 ). First, conceptual understanding refers to students’ comprehension of mathematical concepts and the relationships between concepts. Researchers have designed various computer-based scaffolds and feedback to build students’ concepts and clarify potential misconceptions. For example, for guiding students’ discovery of the patterns of concepts, Yang et al. ( 2012 ) adopted an inductive discovery learning approach to design online learning materials in which students were provided with similar examples with a critical attribute of the concept varied. McLaren et al. ( 2017 ) provided students with prompts to correct their common misconceptions about decimals. They conducted a study with the game adopted as a replacement for seven lessons of regular mathematics classes. Their results showed that the educational game could facilitate better learning performance and enjoyment than a conventional instructional approach.

Second, procedural fluency refers to the skill in carrying out calculations correctly and efficiently. For improving procedural fluency, students need to have knowledge of calculation rules (e.g., place values) and practice the procedure without mistakes. Researchers developed various digital games to overcome the boredom of practice. For example, Chen et al. ( 2012a , 2012b ) designed a Cross Number Puzzle game for practicing arithmetic expressions. In the game, students could individually or collaboratively solve a puzzle, which involved extensive calculation. Their study showed that the low-ability students in the collaborative condition made the most improvement in calculation skills. Ku et al. ( 2014 ) developed mini-games to train students’ mental calculation ability. They showed that the mini-games could not only improve students’ calculation performance but also increase their confidence in mathematics.

Third, strategic competence refers to mathematical problem-solving ability, in particular, word problem-solving in elementary education. Some researchers developed multilevel computer-based scaffolds to help students translate word problems to equations step by step (e.g., González-Calero et al. 2014 ), while other researchers noticed the problem of over-scaffolding. Specifically, students could be too scaffolded and have little space to develop their abilities. To avoid this situation, many researchers proposed allowing students to seek help during word problem-solving (Chase and Abrahamson 2015 ; Roll et al. 2014 ). For example, Cheng et al. ( 2015 ) designed a Scaffolding Seeking system to encourage elementary students to solve word problems by themselves by expressing their thinking first, instead of receiving and potentially abusing scaffolds.

Digital educational games for mathematics learning

Because mathematics is an abstract subject, elementary students easily lose interest in it, especially low-achieving students. Some researchers tailored educational games for learning a specific set of mathematical knowledge (e.g., the Decimal Points game; McLaren et al. 2017 ), so that students could be motivated to learn mathematics. However, if our purpose was to support a complete mathematics curriculum for elementary schools, it seemed impractical to design various educational games for all kinds of knowledge. A feasible approach is to adopt a gamified content structure to reorganize all learning materials. For example, inspired by the design of most role-playing games, Chen et al. ( 2012a , 2012b ) proposed a three-tiered framework of game-based learning—a game world, quests, and learning materials—for supporting elementary students’ enjoyment and goal setting in mathematics learning. Furthermore, while a game world may facilitate students’ exploration and participation, quests are the containers of learning materials with specific goals and rewards. In the game world, students receive quests from nonplayer virtual characters, who may enhance social commitments. To complete the quests, students have to make efforts to undertake learning materials. Today, quests have been widely adopted in the design of educational games (e.g., Azevedo et al. 2012 ; Hwang et al. 2015 ).

However, in educational games with quests, students still play the role of receivers rather than active learners. To facilitate elementary students’ initiative, Lao et al. ( 2017 ) designed digital learning contracts, which required students to set weekly learning goals at the beginning of a week and checked whether they achieved the goals at the end of the week. More specifically, when setting weekly goals, students had to decide on the quantity of learning materials that they wanted to undertake in the coming week. Furthermore, they also had to decide the average correctness of the tests that followed the learning materials. To help them set reasonable and feasible goals, the system provided statistics from the past 4 weeks. As a result, the students may reflect on how well they learned and then make appropriate decisions. After setting goals, students are provided with a series of learning materials for attempting to accomplish those goals. At the end of the week, they may reflect on whether they achieved their learning goals in the contracts. In a sense, learning contracts may not only strengthen the sense of commitment but also empower students to take more control of their learning.

In textbooks or classrooms, learning is usually predefined as a specific sequence, which students must follow to learn. Nevertheless, the structure of knowledge is not linear, but a network. If we could reorganize these learning materials according to the structure of knowledge, students could explore knowledge and discover the relationships among different pieces of knowledge when learning (Davenport and Prusak 2000 ). Knowledge mapping has the advantage of providing students concrete content through explicit knowledge graphics (Ebener et al. 2006 ). Previous studies have shown that the incorporation of knowledge structures into educational games could effectively enhance students’ achievement without affecting their motivation and self-efficacy (Chu et al. 2015 ). For this reason, this study attempted to visualize the structure of knowledge in an educational game. In other words, a knowledge map was visualized and gamified so that students could make decisions to construct their own knowledge map in games.

To enhance students’ mathematics achievement and interests, we designed the Math-Island online game by incorporating a gamified knowledge map of the elementary mathematics curriculum. More specifically, we adopt the mechanisms of a construction management game , in which every student owns a virtual island (a city) and plays the role of the mayor. The goal of the game is to build their cities on the islands by learning mathematics.

System architecture

The Math-Island game is a Web application, supporting cross-device interactions among students, teachers, and the mathematics content structure. The system architecture of the Math-Island is shown in Fig.  1 . The pedagogical knowledge and learning materials are stored in the module of digital learning content, organized by a mathematical knowledge map. The students’ portfolios about interactions and works are stored in the portfolio database and the status database. When a student chooses a goal concept in the knowledge map, the corresponding digital learning content is arranged and delivered to his/her browser. Besides, when the student is learning in the Math-Island, the feedback module provides immediate feedback (e.g., hints or scaffolded solutions) for guidance and grants rewards for encouragement. The learning results can also be shared with other classmates by the interaction module. In addition to students, their teachers can also access the databases for the students’ learning information. Furthermore, the information consists of the students’ status (e.g., learning performance or virtual achievement in the game) and processes (e.g., their personal learning logs). In the Math-Island, it is expected that students can manage their learning and monitor the learning results by the construction management mechanism. In the meantime, teachers can also trace students’ learning logs, diagnose their weaknesses from portfolio analysis, and assign students with specific tasks to improve their mathematics learning.

The system architecture of Math-Island

• Knowledge map

To increase students’ mathematics achievement, the Math-Island game targets the complete mathematics curriculum of elementary schools in Taiwan, which mainly contains the four domains: numerical operation , quantity and measure , geometry , and statistics and probability (Ministry of Education of R.O.C. 2003 ). Furthermore, every domain consists of several subdomains with corresponding concepts. For instance, the domain of numerical operation contains four subdomains: numbers, addition, and subtraction for the first and second graders. In the subdomain of subtraction, there are a series of concepts, including the meaning of subtraction, one-digit subtraction, and two-digit subtraction. These concepts should be learned consecutively. In the Math-Island system, the curriculum is restructured as a knowledge map, so that they may preview the whole structure of knowledge, recall what they have learned, and realize what they will learn.

More specifically, the Math-Island system uses the representational metaphor of an “island,” where a virtual city is located and represents the knowledge map. Furthermore, the island comprises areas, roads, and buildings, which are the embodiments of domains, subdomains, and concepts in the curriculum, respectively. As shown in Fig.  2 , for example, in an area of numeral operation in Math-Island, there are many roads, such as an addition road and a subtraction road. On the addition road, the first building should be the meaning of addition, followed by the buildings of one-digit addition and then two-digit addition. Students can choose these buildings to learn mathematical concepts. In each building, the system provides a series of learning tasks for learning the specific concept. Currently, Math-Island provides elementary students with more than 1300 learning tasks from the first grade to the sixth grade, with more than 25,000 questions in the tasks.

The knowledge map

In Math-Island, a learning task is an interactive page turner, including video clips and interactive exercises for conceptual understanding, calculation, and word problem-solving. In each task, the learning procedure mainly consists of three steps: watching demonstrations, practicing examples, and getting rewards. First, students learn a mathematical concept by watching videos, in which a human tutor demonstrates examples, explains the rationale, and provides instructions. Second, students follow the instructions to answer a series of questions related to the examples in the videos. When answering questions, students are provided with immediate feedback. Furthermore, if students input wrong answers, the system provides multilevel hints so that they could figure out solutions by themselves. Finally, after completing learning tasks, students receive virtual money according to their accuracy rates in the tasks. The virtual money is used to purchase unique buildings to develop their islands in the game.

Game mechanisms

In the Math-Island game, there are two game mechanisms: construction and sightseeing (as shown in Fig.  3 ). The former is designed to help students manage their learning process, whereas the latter is designed to facilitate social interaction, which may further motivate students to better develop their cities. By doing so, the Math-Island can be regarded as one’s learning portfolio, which is a complete record that purposely collects information about one’s learning processes and outcomes (Arter and Spandel 2005 ). Furthermore, learning portfolios are a valuable research tool for gaining an understanding about personal accomplishments (Birgin and Baki 2007 ), because learning portfolios can display one’s learning process, attitude, and growth after learning (Lin and Tsai 2001 ). The appearance of the island reflects what students have learned and have not learned from the knowledge map. When students observe their learning status in an interesting way, they may be concerned about their learning status with the enhanced awareness of their learning portfolios. By keeping all activity processes, students can reflect on their efforts, growth, and achievements. In a sense, with the game mechanisms, the knowledge map can be regarded as a manipulatable open learner model, which not only represents students’ learning status but also invites students to improve it (Vélez et al. 2009 ).

Two game mechanisms for Math-Island

First, the construction mechanism allows students to plan and manage their cities by constructing and upgrading buildings. To do so, they have to decide which buildings they want to construct or upgrade. Then, they are required to complete corresponding learning tasks in the building to determine which levels of buildings they can construct. As shown in Fig.  4 , the levels of buildings depend on the completeness of a certain concept, compared with the thresholds. For example, when students complete one third of the learning tasks, the first level of a building is constructed. Later, when they complete two thirds of the tasks, the building is upgraded to the second level. After completing all the tasks in a building, they also complete the final level and are allowed to construct the next building on the road. Conversely, if students failed the lowest level of the threshold, they might need to watch the video and/or do the learning tasks again. By doing so, students can make their plans to construct the buildings at their own pace. When students manage their cities, they actually attempt to improve their learning status. In other words, the construction mechanism offers an alternative way to guide students to regulate their learning efforts.

Screenshots of construction and sightseeing mechanisms in Math-Island

Second, the sightseeing mechanism provides students with a social stage to show other students how well their Math-Islands have been built. This mechanism is implemented as a public space, where other students play the role of tourists who visit Math-Island. In other words, this sightseeing mechanism harnesses social interaction to improve individual learning. As shown in Fig.  4 , because students can construct different areas or roads, their islands may have different appearances. When students visit a well-developed Math-Island, they might have a positive impression, which may facilitate their self-reflection. Accordingly, they may be willing to expend more effort to improve their island. On the other hand, the student who owns the island may also be encouraged to develop their island better. Furthermore, when students see that they have a completely constructed building on a road, they may perceive that they are good at these concepts. Conversely, if their buildings are small, the students may realize their weaknesses or difficulties in these concepts. Accordingly, they may be willing to make more effort for improvement. On the other hand, the student who owns the island may also be encouraged to develop their island better. In a word, the visualization may play the role of stimulators, so that students may be motivated to improve their learning status.

This paper reported a 2-year study in which the Math-Island system was adopted in an elementary school. The study addressed the following two research questions: (1) Did the Math-Island system facilitate students’ mathematics achievement in terms of conceptual understanding, calculating, and word problem-solving? In particular, how was the mathematics achievement of the low-achieving students? (2) What was students’ levels of interest in mathematics and the system, particularly that of low-achieving students?

Participants

The study, conducted from June 2013 to June 2015, included 215 second graders (98 females and 117 males), whose average age was 8 years old, in an elementary school located in a suburban region of a northern city in Taiwan. The school had collaborated with our research team for more than 2 years and was thus chosen as an experimental school for this study. In this school, approximately one third of the students came from families with a low or middle level of socioeconomic status. It was expected that the lessons learned from this study could be applicable to other schools with similar student populations in the future. The parents were supportive of this program and willing to provide personal tablets for their children (Liao et al. 2017 ). By doing so, the students in the experimental school were able to use their tablets to access the Math-Island system as a learning tool at both school and home. To compare the students’ mathematics achievement with a baseline, this study also included 125 second graders (63 females and 62 males) from another school with similar socioeconomic backgrounds in the same region of the city as a control school. The students in the control school received only conventional mathematics instruction without using the Math-Island system during the 2-year period.

Before the first semester, a 3-week training workshop was conducted to familiarize the students with the basic operation of tablets and the Math-Island system. By doing so, it was ensured that all participants had similar prerequisite skills. The procedure of this study was illustrated in Table  1 . At the beginning of the first semester, a mathematics achievement assessment was conducted as a pretest in both the experimental and the control school to examine the students’ initial mathematics ability as second graders. From June 2013 to June 2015, while the students in the control school learned mathematics in a conventional way, the students in the experimental school learned mathematics not only in mathematics classes but also through the Math-Island system. Although the teachers in the experimental school mainly adopted lectures in mathematics classes, they used the Math-Island system as learning materials at school and for homework. At the same time, they allowed the students to explore the knowledge map at their own pace. During the 2 years, every student completed 286.78 learning tasks on average, and each task took them 8.86 min. Given that there were 344 tasks for the second and third graders, the students could finish 83.37% of tasks according to the standard progress. The data also showed that the average correctness rate of the students was 85.75%. At the end of the second year, another mathematics achievement assessment was administered as a posttest in both schools to evaluate students’ mathematics ability as third graders. Additionally, an interest questionnaire was employed in the experimental school to collect the students’ perceptions of mathematics and the Math-Island system. To understand the teachers’ opinions of how they feel about the students using the system, interviews with the teachers in the experimental school were also conducted.

Data collection

Mathematics achievement assessment.

To evaluate the students’ mathematics ability, this study adopted a standardized achievement assessment of mathematics ability (Lin et al. 2009 ), which was developed from a random sample of elementary students from different counties in Taiwan to serve as a norm with appropriate reliability (the internal consistency was 0.85, and the test-retest reliability was 0.86) and validity (the correlation by domain experts in content validity was 0.92, and the concurrent validity was 0.75). As a pretest, the assessment of the second graders consisted of 50 items, including conceptual understanding (23 items), calculating (18 items), and word problem-solving (9 items). As a posttest, the assessment of the third graders consisted of 60 items, including conceptual understanding (18 items), calculating (27 items), and word problem-solving (15 items). The scores of the test ranged from 0 to 50 points. Because some students were absent during the test, this study obtained 209 valid tests from the experimental school and 125 tests from the control school.

Interest questionnaire

The interest questionnaire comprised two parts: students’ interest in mathematics and the Math-Island system. Regarding the first part, this study adopted items from a mathematics questionnaire of PISA and TIMSS 2012 (OECD 2013 ; Mullis et al. 2012 ), the reliability of which was sound. This part included three dimensions: attitude (14 items, Cronbach’s alpha = .83), initiative (17 items, Cronbach’s alpha = .82), and confidence (14 items Cronbach’s alpha = .72). Furthermore, the dimension of attitude was used to assess the tendency of students’ view on mathematics. For example, a sample item of attitudes was “I am interested in learning mathematics.” The dimension of initiatives was used to assess how students were willing to learn mathematics actively. A sample item of initiatives was “I keep studying until I understand mathematics materials.” The dimension of confidences was used to assess students’ perceived mathematics abilities. A sample item was “I am confident about calculating whole numbers such as 3 + 5 × 4.” These items were translated to Chinese for this study. Regarding the second part, this study adopted self-made items to assess students’ motivations for using the Math-Island system. This part included two dimensions: attraction (8 items) and satisfaction (5 items). The dimension of attraction was used to assess how well the system could attract students’ attention. A sample item was “I feel Math-island is very appealing to me.” The dimension of satisfaction was used to assess how the students felt after using the system. A sample item was “I felt that upgrading the buildings in my Math-Island brought me much happiness.” These items were assessed according to a 4-point Likert scale, ranging from “strongly disagreed (1),” “disagreed (2),” “agreed (3),” and “strongly agreed (4)” in this questionnaire. Due to the absences of several students on the day the questionnaire was administered, there were only 207 valid questionnaires in this study.

Teacher interview

This study also included teachers’ perspectives on how the students used the Math-Island system to learn mathematics in the experimental school. This part of the study adopted semistructured interviews of eight teachers, which comprised the following three main questions: (a) Do you have any notable stories about students using the Math-Island system? (b) Regarding Math-Island, what are your teaching experiences that can be shared with other teachers? (c) Do you have any suggestions for the Math-Island system? The interview was recorded and transcribed verbatim. The transcripts were coded and categorized according to the five dimensions of the questionnaire (i.e., the attitude, initiative, and confidence about mathematics, as well as the attraction and satisfaction with the system) as additional evidence of the students’ interest in the experimental school.

Data analysis

For the first research question, this study conducted a multivariate analysis of variance (MANOVA) with the schools as a between-subject variable and the students’ scores (conceptual understanding, calculating, and word problem-solving) in the pre/posttests as dependent variables. Moreover, this study also conducted a MANOVA to compare the low-achieving students from both schools. In addition, the tests were also carried out to compare achievements with the norm (Lin et al. 2009 ). For the second research question, several z tests were used to examine how the interests of the low-achieving students were distributed compared with the whole sample. Teachers’ interviews were also adopted to support the results of the questionnaire.

Mathematics achievement

To examine the homogeneity of the students in both schools in the first year, the MANOVA of the pretest was conducted. The results, as shown in Table  2 , indicated that there were no significant differences in their initial mathematics achievements in terms of conceptual understanding, calculating, and word problem-solving (Wilks’ λ  = 0.982, F (3330) = 2.034, p  > 0.05). In other words, the students of both schools had similar mathematics abilities at the time of the first mathematics achievement assessment and could be fairly compared.

At the end of the fourth grade, the students of both schools received the posttest, the results of which were examined by a MANOVA. As shown in Table  3 , the effect of the posttest on students’ mathematics achievement was significant (Wilks’ λ  = 0.946, p  < 0.05). The results suggested that the students who used Math-Island for 2 years had better mathematics abilities than those who did not. The analysis further revealed that the univariate effects on calculating and word problem-solving were significant, but the effect on conceptual understanding was insignificant. The results indicated that the students in the experimental school outperformed their counterparts in terms of the procedure and application of arithmetic. The reason may be that the system provided students with more opportunities to do calculation exercises and word problems, and the students were more willing to do these exercises in a game-based environment. Furthermore, they were engaged in solving various exercises with the support of immediate feedback until they passed the requirements of every building in their Math-Island. However, the students learned mathematical concepts mainly by watching videos in the system, which provided only demonstrations like lectures in conventional classrooms. For this reason, the effect of the system on conceptual understanding was similar to that of teachers’ conventional instruction.

Furthermore, to examine the differences between the low-achieving students in both schools, another MANOVA was also conducted on the pretest and the posttest. The pretest results indicated that there were no significant differences in their initial mathematics achievement in terms of conceptual understanding, calculating, and word problem-solving (Wilks’ λ  = 0.943, F (3110) = 2.210, p  > 0.05).

The MANOVA analysis of the posttest is shown in Table  4 . The results showed that the effect of the system on the mathematics achievement of low-achieving students was significant (Wilks’ λ  = 0.934, p  < 0.05). The analysis further revealed that only the univariate effect on word problem-solving was significant. The results suggested that the low-achieving students who used Math-Island for 2 years had better word problem-solving ability than those students in the control school, but the effect on conceptual understanding and procedural fluency was insignificant. The results indicated that the Math-Island system could effectively enhance low-achieving students’ ability to solve word problems.

Because the mathematics achievement assessment was a standardized achievement assessment (Lin et al. 2009 ), the research team did a further analysis of the assessments by comparing the results with the norm. In the pretest, the average score of the control school was the percentile rank of a score (PR) 55, but their average score surprisingly decreased to PR 34 in the posttest. The results confirmed the fact that conventional mathematics teaching in Taiwan might result in an M-shape distribution, suggesting that low-achieving students required additional learning resources. Conversely, the average score of the experimental school was PR 48 in the pretest, and their score slightly decreased to PR 44 in the posttest. Overall, both PR values were decreasing, because the mathematics curriculum became more and more difficult from the second grade to the fourth grade. However, it should be noted that the experimental school has been less affected, resulting in a significant difference compared with the control school (see Table  5 ). Notably, the average score of word problem-solving in the posttest of the experimental school was PR 64, which was significantly higher than the nationwide norm ( z  = 20.8, p  < .05). The results were consistent with the univariate effect of the MANOVA on word problem-solving, suggesting that the Math-Island system could help students learn to complete word problems better. This may be because the learning tasks in Math-Island provided students with adequate explanations for various types of word problems and provided feedback for exercises.

To examine whether the low-achieving students had low levels of interest in mathematics and the Math-Island system, the study adopted z tests on the data of the interest questionnaire. Table  5 shows the descriptive statistics and the results of the z tests. Regarding the interest in mathematics, the analysis showed that the interest of the low-achieving students was similar to that of the whole sample in terms of attitude, initiative, and confidence. The results were different from previous studies asserting that low-achieving students tended to have lower levels of interest in mathematics (Al-Zoubi and Younes 2015 ). The reason was perhaps that the low-achieving students were comparably motivated to learn mathematics in the Math-Island system. As a result, a teacher ( #T-301 ) said, “some students would like to go to Math-Island after school, and a handful of students could even complete up to forty tasks (in a day),” implying that the students had a positive attitude and initiative related to learning mathematics.

Another teacher ( T-312 ) also indicated “some students who were frustrated with math could regain confidence when receiving the feedback for correct answers in the basic tasks. Thanks to this, they would not feel high-pressure when moving on to current lessons.” In a sense, the immediate feedback provided the low-achieving students with sufficient support and may encourage them to persistently learn mathematics. Furthermore, by learning individually after class, they could effectively prepare themselves for future learning. The results suggested that the system could serve as a scaffolding on conventional instruction for low-achieving students. The students could benefit from such a blended learning environment and, thus, build confidence in mathematics by learning at their own paces.

The low-achieving students as a whole were also attracted to the system and felt satisfaction from it. Teacher ( #T-307 ) said that, “There was a hyperactive and mischievous student in my class. However, when he was alone, he would go on to Math-Island, concentrating on the tasks quietly. He gradually came to enjoy learning mathematics. It seemed that Math-Island was more attractive to them than a lecture by a teacher. I believed that students could be encouraged, thus improve their ability and learn happily.” Another teacher ( #T-304 ) further pointed out that, “For students, they did not only feel like they were learning mathematics because of the game-based user interface. Conversely, they enjoyed the contentment when completing a task, as if they were going aboard to join a competition.” In teachers’ opinions, such a game-based learning environment did not disturb their instruction. Instead, the system could help the teachers attract students’ attention and motivate them to learn mathematics actively because of its appealing game and joyful learning tasks. Furthermore, continuously overcoming the tasks might bring students a sense of achievement and satisfaction.

Discussion on some features of this study

In addition to the enhancement of achievement and interest, we noticed that there are some features in this study and our design worth some discussion.

The advantages of building a long-term study

Owing to the limitations of deployment time and sample sizes, it is hard for most researchers to conduct a longitudinal study. Fortunately, we had a chance to maintain a long-term collaboration with an experimental school for more than 2 years. From this experiment, we notice that there are two advantages to conducting a long-term study.

Obtaining substantial evidence from the game-based learning environment

The research environment was a natural setting, which could not be entirely controlled and manipulated like most experiments in laboratories. However, this study could provide long-term evidence to investigate how students learned in a game-based learning environment with their tablets. It should be noted that we did not aim to replace teachers in classrooms with the Math-Island game. Instead, we attempted to establish an ordinary learning scenario, in which the teachers and students regarded the game as one of the learning resources. For example, teachers may help low-achieving students to improve their understanding of a specific concept in the Math-Island system. When students are learning mathematics in the Math-Island game, teachers may take the game as a formative assessment and locate students’ difficulties in mathematics.

Supporting teachers’ instructions and facilitating students’ learning

The long-term study not only proved the effectiveness of Math-Island but also offered researchers an opportunity to determine teachers’ roles in such a computer-supported learning environment. For example, teachers may encounter difficulties in dealing with the progress of both high- and low-achieving students. How do they take care of all students with different abilities at the same time? Future teachers may require more teaching strategies in such a self-directed learning environment. Digital technology has an advantage in helping teachers manage students’ learning portfolios. For example, the system can keep track of all the learning activities. Furthermore, the system should provide teachers with monitoring functions so that they know the average status of their class’s and individuals’ learning progress. Even so, it is still a challenge for researchers to develop a well-designed visualization tool to support teachers’ understanding of students’ learning conditions and their choice of appropriate teaching strategies.

Incorporating a gamified knowledge map of the elementary mathematics curriculum

Providing choices of learning paths.

Math-Island uses a representational metaphor of an “island,” where a virtual city is located and represents the knowledge map. Furthermore, the island comprises areas, roads, and buildings, which are the embodiments of domains, subdomains, and concepts in the curriculum, respectively. Because the gamified knowledge map provides students with multiple virtual roads to learn in the system, every student may take different routes. For instance, some students may be more interested in geometry, while others may be confident in exploring the rules of arithmetic. In this study, we noticed that the low-achieving students needed more time to work on basic tasks, while high-achieving students easily passed those tasks and moved on to the next ones. As a result, some of the high-achieving students had already started to learn the materials for the next grade level. This was possibly because high-achieving students were able to respond well to challenging assignments (Singh 2011 ). Therefore, we should provide high-achieving students with more complex tasks to maintain their interest. For example, Math-Island should provide some authentic mathematical problems as advanced exercises.

Visualizing the learning portfolio

In this study, we demonstrated a long-term example of incorporating a gamified knowledge map in an elementary mathematical curriculum. In the Math-Island game, the curriculum is visualized as a knowledge map instead of a linear sequence, as in textbooks. By doing so, students are enabled to explore relationships in the mathematics curriculum represented by the knowledge map; that is, the structure of the different roads on Math-Island. Furthermore, before learning, students may preview what will be learned, and after learning, students may also reflect on how well they learned. Unlike traditional lectures or textbooks, in which students could only follow a predefined order to learn knowledge without thinking why they have to learn it, the knowledge map allows students to understand the structure of knowledge and plan how to achieve advanced knowledge. Although the order of knowledge still remains the same, students take primary control of their learning. In a sense, the knowledge map may liberate elementary students from passive learning.

Adopting the mechanisms of a construction management game

This 2-year study showed that the adaptation of two game mechanisms, construction and sightseeing, into the elementary mathematical curriculum could effectively improve students’ learning achievement. The reason may be that students likely developed interests in using Math-Island to learn mathematics actively, regardless of whether they are high- and low-achieving students.

Gaining a sense of achievement and ownership through the construction mechanism

Regardless of the construction mechanism, Math-Island allows students to plan and manage their cities by constructing and upgrading buildings. Math-Island took the advantages of construction management games to facilitate elementary students’ active participation in their mathematical learning. Furthermore, students may manage their knowledge by planning and constructing of buildings on their virtual islands. Like most construction management games, students set goals and make decisions so that they may accumulate their assets. These assets are not only external rewards but also visible achievements, which may bring a sense of ownership and confidence. In other words, the system gamified the process of self-directed learning.

Demonstrating learning result to peers through the sightseeing mechanism

As for the sightseeing mechanism, in conventional instruction, elementary students usually lack the self-control to learn knowledge actively (Duckworth et al. 2014 ) or require a social stage to show other students, resulting in low achievement and motivation. On the other hand, although previous researchers have already proposed various self-regulated learning strategies (such as Taub et al. 2014 ), it is still hard for children to keep adopting specific learning strategies for a long time. For these reasons, this study uses the sightseeing mechanism to engage elementary students in a social stage to show other students how well their Math-Islands have been built. For example, in Math-Island, although the students think that they construct buildings in their islands, they plan the development of their knowledge maps. After learning, they may also reflect on their progress by observing the appearance of the buildings.

In brief, owing to the construction mechanism, the students are allowed to choose a place and build their unique islands by learning concepts. During the process, students have to do the learning task, get feedback, and get rewards, which are the three major functions of the construction functions. In the sightseeing mechanism, students’ unique islands (learning result) can be shared and visited by other classmates. The student’s Math-Island thus serves as a stage for showing off their learning results. The two mechanisms offer an incentive model connected to the game mechanism’s forming a positive cycle: the more the students learn, the more unique islands they can build, with more visitors.

Conclusion and future work

This study reported the results of a 2-year experiment with the Math-Island system, in which a knowledge map with extensive mathematics content was provided to support the complete elementary mathematics curriculum. Each road in Math-Island represents a mathematical topic, such as addition. There are many buildings on each road, with each building representing a unit of the mathematics curriculum. Students may learn about the concept and practice it in each building while being provided with feedback by the system. In addition, the construction management online game mechanism is designed to enhance and sustain students’ interest in learning mathematics. The aim of this study was not only to examine whether the Math-Island system could improve students’ achievements but also to investigate how much the low-achieving students would be interested in learning mathematics after using the system for 2 years.

As for enhancing achievement, the result indicated that the Math-Island system could effectively improve the students’ ability to calculate expressions and solve word problems. In particular, the low-achieving students outperformed those of the norm in terms of word problem-solving. For enhancing interest, we found that both the low-achieving and the high-achieving students in the experimental school, when using the Math-Island system, maintained a rather high level of interest in learning mathematics and using the system. The results of this study indicated some possibility that elementary students could be able to learn mathematics in a self-directed learning fashion (Nilson 2014 ; Chen et al. 2012a , b ) under the Math-Island environment. This possibility is worthy of future exploration. For example, by analyzing student data, we can investigate how to support students in conducting self-directed learning. Additionally, because we have already collected a considerable amount of student data, we are currently employing machine learning techniques to improve feedback to the students. Finally, to provide students appropriate challenges, the diversity, quantity, and difficulty of content may need to be increased in the Math-Island system.

Abbreviations

Program for International Student Assessment

The percentile rank of a score

Trends in Mathematics and Science Study

Al-Zoubi, S. M., & Younes, M. A. B. (2015). Low academic achievement: causes and results. Theory and Practice in Language Studies, 5 (11), 2262.

Google Scholar  

Arter, J. A., & Spandel, V. (2005). Using portfolios of student work in instruction and assessment. Educational Measurement Issues and Practice, 11 (1), 36–44.

Azevedo, R., Feyzi-Behnagh, R., Duffy, M., Harley, J., & Trevors, G. (2012). Metacognition and self-regulated learning in student-centered leaning environments. In D. Jonassen & S. Land (Eds.), Theoretical foundations of student-centered learning environments (pp. 171–197). New York: Routledge.

Barlett, C. P., Anderson, C. A., & Swing, E. L. (2009). Video game effects confirmed, suspected and speculative: a review of the evidence. Simulation & Gaming, 40 (3), 377–403.

Barr, R. B., & Tagg, J. (1995). From teaching to learning—a new paradigm for undergraduate education. Change The Magazine of Higher Learning, 27 (6), 12–26.

Birgin, O., & Baki, A. (2007). The use of portfolio to assess student’s performance. Journal of Turkish Science Education, 4 (2), 75–90.

Chan, T. W., Roschelle, J., Hsi, S., Kinshuk, Sharples, M., Brown, T., et al. (2006). One-to-one technology-enhanced learning: an opportunity for global research collaboration. Research and Practice in Technology Enhanced Learning, 1 (01), 3–29.

Chase, K., & Abrahamson, D. (2015). Reverse-scaffolding algebra: empirical evaluation of design architecture. ZDM Mathematics Education, 47 (7), 1195–1209.

Chen, Y. H., Looi, C. K., Lin, C. P., Shao, Y. J., & Chan, T. W. (2012a). Utilizing a collaborative cross number puzzle game to develop the computing ability of addition and subtraction. Educational Technology & Society, 15 (1), 354–366.

Chen, Z. H., Liao, C. C., Cheng, H. N., Yeh, C. Y., & Chan, T. W. (2012b). Influence of game quests on pupils’ enjoyment and goal-pursuing in math learning. Journal of Educational Technology & Society, 15 (2), 317–327.

Cheng, H. N. H., Yang, E. F. Y., Liao, C. C. Y., Chang, B., Huang, Y. C. Y., & Chan, T. W. (2015). Scaffold seeking: a reverse design of scaffolding in computer-supported word problem solving. Journal of Educational Computing Research, 53 (3), 409–435.

Chu, H. C., Yang, K. H., & Chen, J. H. (2015). A time sequence-oriented concept map approach to developing educational computer games for history courses. Interactive Learning Environments, 23 (2), 212–229.

Davenport, T. H. & Prusak, L. (2000). Working knowledge: How organizations manage what they know . Boston: Harvard Business School Press.

Duckworth, A. L., Gendler, T. S., & Gross, J. J. (2014). Self-control in school-age children. Educational Psychologist, 49 (3), 199–217.

Ebener, S., Khan, A., Shademani, R., Compernolle, L., Beltran, M., Lansang, M. A., & Lippman, M. (2006). Knowledge mapping as a technique to support knowledge translation. Bulletin of the World Health Organization, 84 , 636–642.

González-Calero, J. A., Arnau, D., Puig, L., & Arevalillo-Herráez, M. (2014). Intensive scaffolding in an intelligent tutoring system for the learning of algebraic word problem solving. British Journal of Educational Technology, 46 (6), 1189–1200.

Hanus, M. D., & Fox, J. (2015). Assessing the effects of gamification in the classroom: a longitudinal study on intrinsic motivation, social comparison, satisfaction, effort, and academic performance. Computers & Education, 80 , 152–161.

Hwang, G. J., Chiu, L. Y., & Chen, C. H. (2015). A contextual game-based learning approach to improving students’ inquiry-based learning performance in social studies courses. Computers & Education, 81 , 13–25.

Hwang, G. J., Su, J. M., & Chen, N. S. (2012). E-learning introduction and practice . Taiwan: Drmaste.

Kiili, K., & Ketamo, H. (2007). Exploring the learning mechanism in educational games. Journal of Computing and Information Technology, 15 (4), 319–324.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: helping children learn mathematics . Washington, DC: National Academies Press.

Koivisto, J., & Hamari, J. (2014). Demographic differences in perceived benefits from gamification. Computers in Human Behavior, 35 , 179–188.

Krapp, A. (1999). Interest, motivation and learning: an educational-psychological perspective. European Journal of Psychology of Education, 14 (1), 23–40.

Ku, O., Chen, S. Y., Wu, D. H., Lao, A. C., & Chan, T. W. (2014). The effects of game-based learning on mathematical confidence and performance: high ability vs. low ability. Journal of Educational Technology & Society, 17 (3), 65–78.

Lao, A. C. C., Cheng, H. N., Huang, M. C., Ku, O., & Chan, T. W. (2017). Examining motivational orientation and learning strategies in computer-supported self-directed learning (CS-SDL) for mathematics: the perspective of intrinsic and extrinsic goals. Journal of Educational Computing Research, 54 (8), 1168–1188.

Lee, Y. M. (2012). Discriminating math low-achievement motivation patterns: comparing disadvantaged and other students in elementary and junior high school. Journal of Research in Education Sciences, 57 (4), 39–71. https://doi.org/10.3966/2073753X2012125704002 .

Li, M.-C., & Tsai, C.-C. (2013). Game-based learning in science education: a review of relevant research. Journal of Science Education and Technology, 22 (6), 877–898. https://doi.org/10.1007/s10956-013-9436-x .

Liao, C. C., Cheng, H. N., Chang, W. C., & Chan, T. W. (2017). Supporting parental engagement in a BYOD (bring your own device) school. Journal of Computers in Education, 4 (2), 107–125.

Lin, B. G., Li, R. P., & Huang, Y. Z. (2009). Instructional manual of mathematical ability test for the school-aged . Taipei: Ministry of Education.

Lin, P. J., & Tsai, W. H. (2001). Using research-based cases to enhance prospective teachers’ understanding of teaching mathematics and their reflections. In F. L. Lin (Ed.), Common sense in mathematics education. Proceedings of 2001 the Netherlands and Taiwan Conference on Common Sense in Mathematics Education (pp. 231–272). Taipei: Taiwan.

Liu, T. Y., & Chu, Y. L. (2010). Using ubiquitous games in an English listening and speaking course: impact on learning outcomes and motivation. Computers & Education, 55 (2), 630–643. https://doi.org/10.1016/j.compedu.2010.02.023 .

McLaren, B. M., Adams, D. M., Mayer, R. E., & Forlizzi, J. (2017). A computer-based game that promotes mathematics learning more than a conventional approach. International Journal of Game-Based Learning, 7 (1), 36–56.

Ministry of Education. (2003). Guidelines of grades 1-9 curriculum of elementary and junior high school education . Retrieved from https://www.k12ea.gov.tw/92_sid17/%E6%96%B0%E7%B8%BD%E7%B6%B1%E8%8B%B1%E6%96%87%E7%89%88.pdf .

Mullis, I. V. S., Martin, M. O., Foy, P., & Drucker, K. T. (2012). PIRLS 2011 international results in reading . Chestnut Hill: TIMSS & PIRLS International Study Center, Boston College.

Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). TIMSS 2015 International Results in Mathematics. Retrieved from http://timssandpirls.bc.edu/timss2015/international-results/

Nilson, L. B. (2014). The secret of self-regulated learning. In Invited article for Faculty Focus: Higher Ed Teaching Strategies from Magna Publications .

OECD. (2013). PISA 2012 results in focus: what 15-year-olds know and what they can do with what they know: key results from PISA 2012.

OECD. (2016). PISA 2015 results in focus. Retrieved from: https://www.oecd.org/pisa/pisa-2015-results-in-focus.pdf .

Roll, I., Baker, R. S. J. D., Aleven, V., & Koedinger, K. R. (2014). On the benefits of seeking (and avoiding) help in online problem-solving environments. Journal of the Learning Sciences, 23 (4), 537–560.

Schraw, G., Flowerday, T., & Lehman, S. (2001). Increasing situational interest in the classroom. Educational Psychology Review, 13 (3), 211–224.

Singh, K. (2011). Study of achievement motivation in relation to academic achievement of students. International Journal of Educational Planning and Administration, 1 (2), 161–171.

Taub, M., Azevedo, R., Bouchet, F., & Khosravifar, B. (2014). Can the use of cognitive and metacognitive self-regulated learning strategies be predicted by learners’ levels of prior knowledge in hypermedia-learning environments? Computers in Human Behavior, 39 , 356–367.

Vélez, J., Fabregat, R., Bull, S., & Hueva, D. (2009). The potential for open learner models in adaptive virtual learning environments. In S. D. Craig & D. Dicheva (Eds.), AIED 2009: 14th International Conference on Artificial Intelligence in Education Workshops Proceedings Volume 8 (pp. 11–20). Brighton: International AIED Society.

Yang, E. F. Y., Cheng, H. N. H., Ching, E., & Chan, T. W. (2012). Variation based discovery learning design in 1 to 1 mathematics classroom. In G. Biswas, L.-H. Wong, T. Hirashima, & W. Chen (Eds.), Proceedings of the 20th International Conference on Computers in Education (pp. 811–815). Singapore: Asia-Pacific Society for Computers in Education.

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Acknowledgements

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to thank the Ministry of Science and Technology of the Republic of China, Taiwan, for financial support (MOST 106-2511-S-008-003-MY3), and Research Center for Science and Technology forLearning, National Central University, Taiwan.

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CYCY contributed to the study design, data acquisition and analysis, mainly drafted the manuscript and execution project. HNHC was involved in data acquisition, revision of the manuscript and data analysis.ZHC was contributed to the study idea and drafted the manuscript. CCYL of this research was involved in data acquisition and revision of the manuscript. TWC was project manager and revision of the manuscript. All authors read and approved the final manuscript.

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Charles Y.C. Yeh is currently an PhD student in Graduate Institute of Network Learning Technology at National Central University. The research interests include one-to-one learning environments and game-based learning.

Hercy N. H. Cheng is currently an associate professor and researcher in National Engineering Research Center for E-Learning at Central China Normal University, China. His research interests include one-to-one learning environments and game-based learning.

Zhi-Hong Chen is an associate professor in Graduate Institute of Information and Computer Education at National Taiwan Normal University. His research interests focus on learning technology and interactive stories, technology intensive language learning and game-based learning.

Calvin C. Y. Liao is currently an Assistant Professor and Dean’s Special Assistant in College of Nursing at National Taipei University of Nursing and Health Sciences in Taiwan. His research focuses on computer-based language learning for primary schools. His current research interests include a game-based learning environment and smart technology for caregiving & wellbeing.

Tak-Wai Chan is Chair Professor of the Graduate Institute of Network Learning Technology at National Central University in Taiwan. He has worked on various areas of digital technology supported learning, including artificial intelligence in education, computer supported collaborative learning, digital classrooms, online learning communities, mobile and ubiquitous learning, digital game based learning, and, most recently, technology supported mathematics and language arts learning.

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Yeh, C.Y.C., Cheng, H.N.H., Chen, ZH. et al. Enhancing achievement and interest in mathematics learning through Math-Island. RPTEL 14 , 5 (2019). https://doi.org/10.1186/s41039-019-0100-9

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

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

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

Successful Math Majors Generate Their Own Examples

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

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

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

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

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

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

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

Being Asked For Examples Can Be Disconcerting

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

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

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

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

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

Generating Counterexamples That Are Explanatory

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

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

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

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

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

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

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

and tried to check commutativity using

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

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

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

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Title: integrability and combinatorics.

Abstract: We discuss the use of methods coming from integrable systems to study problems of enumerative and algebraic combinatorics, and develop two examples: the enumeration of Alternating Sign Matrices and related combinatorial objects, and the theory of symmetric polynomials.

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Mathematics Research Paper

View sample mathematics research paper. Browse other  research paper examples and check the list of history research paper topics for more inspiration. If you need a history research paper written according to all the academic standards, you can always turn to our experienced writers for help. This is how your paper can get an A! Feel free to contact our custom writing service for professional assistance. We offer high-quality assignments for reasonable rates.

The twenty-first-century approach to the history of mathematics looks beyond the once-traditional Eurocentric views that credited the Greeks for the most advanced developments; it now embraces a global view of mathematical accomplishment through the ages and crosscultural exchanges in the field.

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The traditional history of mathematics took shape at the turn of the twentieth century, reflecting the European influence of that era. According to this traditional history the first mathematics worthy of the name arose in ancient Greece and led directly, but for the Dark Ages, toward the modern mathematics of Western universities today. Since the mid-twentieth century historians of mathematics have broadened their subject’s horizons. New historical evidence and changing attitudes toward other cultures have made this history of mathematics more complex and more global. Equally important have been changes in what is considered mathematics. The key to the study of a truly global history of mathematics has been flexibility in the definition of mathematics itself. The term ethnomathematics refers to the mathematical practices of a particular society and includes mathematical processes beyond the narrow academic sense, in recreation, art, numeration, and other fields. A more conceptually inclusive definition of mathematics has led directly to a more geographically inclusive history of mathematics. We can see the greatest diversity in mathematical practices by comparing counting systems around the world, such as the Incan quipu, a knot-tying system for recording numbers.

Toward a Hellenistic Synthesis

Hundreds of well-preserved clay tablets (most dating from c. 1700 BCE) disclose the sexigesimal (base sixty) numeration system of ancient Mesopotamia, a system sufficient for the development of an ancient algebra. The sources for early Egyptian mathematics are far fewer. The earliest, the Ahmes Papyrus, is a copy of an earlier version from about 1900 BCE. In addition to developing an arithmetic distinctive for its reliance on unit fractions (1?4, 1?3, 1?2, etc.), Egyptians elaborated a practical computational geometry with methods for calculating the volume of a truncated pyramid and approximating the area of a circle.

Only recently have historians begun favorably to reevaluate the achievements of Mesopotamia and Egypt because none of their earliest mathematics adopted the deductive and theoretical approach of the Greeks—an approach that has become enshrined in modern definitions of mathematics. The Greeks themselves, however, explicitly acknowledged their debt to the Egyptians. Moreover, after the conquests of Alexander of Macedon (Alexander the Great, 356–323 BCE) spread Greek thought throughout southwestern Eurasia, a shared Hellenistic culture united the mathematicians of the new world order, and the great Greek geometer Euclid (early third century BCE) taught in the Egyptian capital Alexandria. The Hellenistic world witnessed an extraordinary synthesis of Greek deductive geometry with the algebraic and empirical mathematics of neighboring peoples.

Indian Mathematics

The earliest Indian mathematics was expressly religious in purpose. The geometry developed in the Sulbasutras (technical appendices to Hindu sacred writings, the Vedas, c. 3000 BCE) adopted sophisticated techniques, including the Pythagorean theorem, for the proper orientation and construction of effective sacrificial altars. A hymn from the Atharavaveda (the name translates to Sacred Knowledge of Magical Forms) describes a complex design of overlapping triangles for meditation.

Possibly during the Vedic period, and certainly by the seventh century CE (perhaps inspired by the counting boards brought by Chinese travelers), a decimal place-value system arose that in principle is like our own: distinct numerals represented the numbers 0 through 9, and each instance of a numeral would be associated with a decimal scale based on its position in the number. (The first dated inscription of such a system, however, comes from Cambodia in 683.) Most early mathematical knowledge was preserved orally in verse.

The decline in Vedic sacrifices after about 500 BCE shifted the focus of Indian mathematics. The developing Buddhist and Jaina traditions involved large numbers. (The religion of Jainism, originating in sixth-century-BCE India, taught liberation of the soul.) The collection of Buddhist texts Samyutta Nikaya (The Grouped Discourses), for example, defines a kalpa as a unit of time longer than that needed to erode a rock of one cubic mile by wiping it with silk once a century. The Jainas worked with series and permutations, and they theorized that not all infinite numbers are equal, an insight that would elude Western mathematics until the nineteenth century. Only with the practical, commercial algebraic examples of the Bakhshali manuscript, composed in the first centuries CE, did Indian mathematics lose its primarily religious applications.

Later Indian mathematics would achieve breakthroughs in indeterminate equations, as well as in trigonometry and its application to astronomy. Algebraic work used Sanskrit letters abbreviating the names of colors to stand for unknown qualities, like the modern x and y, and until the modern period the use of letters of the alphabet to represent numerals would reinforce the poetry of Indian mathematics. The research on power series (series—sum of a sequence of terms—in which the terms follow a certain pattern and are infinite) for pi and trigonometric functions by fifteenth- and sixteenth-century mathematicians in Kerala in India anticipated many of the results achieved by the English physicist and mathematician Isaac Newton and the German philosopher and mathematician G. W. Leibniz—independently, or perhaps through the Portuguese—in the seventeenth century. Proofs and derivations of formulas were quite rare throughout the Indian mathematical tradition.

Chinese Mathematics

The first existing numerals appeared on Chinese Shang dynasty (1766–1045 BCE) tortoise shells used for divination, with symbols to indicate 1 through 10, 20s, 100s, and 1000s. The oldest treatise is the Zhoubi Suanjing (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (500– 200 BCE), an archaic astronomy text that contains a demonstration of the Pythagorean theorem. In China counting rods were used for elementary arithmetical operations at least by the Qin dynasty (221–206 BCE). China also boasts the earliest known magic squares, which are matrixes of numbers in which every row, column, and major diagonal produces the same sum. Interest in magic squares as protective talismans encouraged their spread to Mongolia and Tibet, although they were limited in size to three by three until Yang Hui’s work in the thirteenth century.

The principal classic source was the Jiuzhang Suanshu (Nine Chapters of the Mathematical Art), composed during the Han dynasty (206 BCE–220 CE). Eminently practical, its nine chapters included 246 problems treating land surveying, proportions of millet and rice, volumes of three-dimensional solids, and tax assessment. Some results were precocious, and the approaches to simultaneous linear equations closely approximated the methods later developed by the German mathematician and astronomer C. F. Gauss (1777–1855). The preeminence of this work sparked during later centuries numerous commentaries elucidating the often obscure explanations. If for some people the commentaries inhibited creative work, other people used the commentaries to introduce original research.

Although geometry largely stagnated at the level of the Jiuzhang Suanshu, Chinese scholars made important advances in algebra and arithmetic. During the fifth century Zu Chongzhi calculated pi to seven decimal places, an achievement first surpassed by the Indian mathematician Madhava around 1400. In 1247 the “0” sign for zero first appeared, and later in the century Guo Shoujing became the first Chinese mathematician to tackle spherical trigonometry. Both developments may have occurred under Arab or Indian influence. Zhu Shijie (1280–1303) presented the French scientist and philosopher Blaise Pascal’s triangle and algebraic methods for solving simultaneous equations and equations of high degree. Only in the eighteenth century would European algebra catch up with Chinese.

Islamic Mathematics

The Islamic ‘Abbasid caliphate (the office or dominion of a Muslim spiritual leader) (750–1258 CE) sponsored missions to gather astronomical texts (and scholars) from a variety of cultures. Islamic mathematics had two major sets of sources. The first consisted of Persian and Indian sources that featured prominently astronomical tables and an algebraic approach. The second set of sources was Hellenistic, sometimes transmitted through Syrian intermediaries and translations. In their deductive and abstract approach these works in the second, Hellenistic set of sources betray their Greek origins.

The most famous developer of the first Islamic mathematical tradition was al-Khwarizmi (c. 780–c. 850). His treatise Hisab al-jabr w’al-muqabala (Calculation by Completion and Balancing) gives us the word algebra, and his Algorithimi introduced the word algorithm and the Indian positional number system—which had probably first moved out of India with Nestorian (relating to a church separating from Byzantine Christianity after 431) Christians, who found it useful in calculating the date of Easter. In the second mathematical tradition Thabit ibn Qurra (c. 836–901) proved the Pythagorean theorem, attempted a proof of Euclid’s parallel postulate, and made discoveries in mensuration (geometry applied to the computation of lengths, areas, or volumes from given dimensions or angles) and spherical trigonometry.

In addition to preserving these earlier texts in Arabic translation and making further progress within these two traditions, several mathematicians of the Islamic world worked toward a synthesis of Greek deductive geometry and Indian algebra. The fusion of the two streams is best represented by the application of geometry to algebra evident in the geometric solution to quadratic and cubic equations pursued by al- Khwarizmi and the Persian poet and mathematician Omar Khayyam (c. 1040–1123). In addition to pursuing further work on the parallel postulate, Omar Khayyam pursued calendar reform and a theory of proportions useful for the inheritance problems of Islamic law. Jemshid Al-Kashi (1380–1429) relied on a near-circular polygon of 805,306,368 sides to approximate pi to sixteen decimal places. Arab mathematicians enhanced the Indian number system by introducing in the tenth century a small vertical line over the units place to serve the same purpose as the modern decimal point. Nasir al-Din al-Tusi (1201– 1274) anticipated the mathematical improvements of the model of the cosmos developed by the Polish astronomer Nicolaus Copernicus (1473–1543).

Toward a European Synthesis

Indian numbers reached Europe by the tenth century, were popularized by the Liber abaci (Book of Calculation) of the Italian mathematician Leonardo Fibonacci (1170–1250), and slowly overcame prohibitions by various governments (concerned that the system facilitated fraud) that lasted until King Charles XII of Sweden (1682–1718). The Hindu-Arabic system of our modern numbers attracted merchants with its usefulness in calculations—and attracted swindlers with its easily altered numbers. In Europe resistance to mathematics was not limited to the new number system. The German religious reformer Martin Luther considered mathematics hostile to theology, and the establishment of the Savilian chair in geometry at Oxford University in England prompted concerned parents to withdraw their sons, lest they be exposed to diabolical mathematics.

During the early modern period, as Europeans increasingly exerted influence across the globe, they also began to dominate a mathematical tradition that itself was absorbing compatible elements from other cultures. The dominance of a single stream of mathematics, perhaps fed by many cultural tributaries, is characteristic of modern mathematics.

Europe’s mathematical exchange with China illustrates how global connections created this single stream. In 1607 the Jesuit Matteo Ricci and the Christian convert Xu Guangxi (1562–1633) translated Euclid’s Elements into Chinese. Although Euclid’s deductions contrasted sharply with the inductive approach of the Jiuzhang Suanshu, the translation proved highly influential in subsequent Chinese mathematics. Many of the technical geometric terms coined by Xu remain in use today. Other cross-cultural mathematical connections are less certain. Mathematics from Kerala may have traveled to China, and the Elements may have already been introduced by Arabs. In the other direction, the Jesuits possibly reported some advanced Chinese algebra back to Italy.

Under Chinese imperial sponsorship the text Shu Li Jing Yun (Collected Basic Principles of Mathematics, 1723) marked the end of the integration of Western techniques into Chinese mathematics. During the next emperor’s reign a new closed-door policy restricted access to Western research. Some Chinese worked on critical editions of ancient mathematical texts, while others conducted research in isolation from the West. In 1859 Dai Xu (1805–1860) proved the binomial theorem independently of Newton, whose own 1676 proof had not yet reached China.

Outside the mainstream of Western mathematics stood the Indian prodigy Srinivasa Ramanujan (1887–1920) and his discoveries in the theory of numbers. Although his genius had no contemporary rival, his knowledge of mathematics was limited. Some of his theorems (for example, on the theory of prime numbers) have thus been described as brilliant but wrong. His cavalier attitude toward proof was once attributed to this background, although it in fact fits the Indian pedagogical tradition of leaving demonstration and commentary to students.

The twentieth century has brought a remarkable international tradition of mathematical scholarship often transcending geopolitical divisions. In 1897, 208 mathematicians met in Zurich, Switzerland, as the first International Congress of Mathematics (ICM). Representing sixteen countries, they included twelve from Russia, seven from the United States, and four women. The Cambridge ICM of 1912 counted 82 non-Europeans among its 574 participants. Founded just after World War I, the International Mathematical Union (IMU) excluded the defeated Germany until the IMU adopted a policy of principled neutrality in the 1920s—a policy that would endure despite the challenges of World War II and the Cold War. The mainstream did divide somewhat as Communist and Western mathematicians developed results in parallel, at a lag in either direction, often with different methods. The Commission on Development and Exchange (known before 1978 as the “Commission for Exchange of Mathematicians”) was established in 1959 to facilitate the development of mathematics beyond industrialized countries. The African Mathematical Union formed in 1976, although the exodus of mathematicians from underdeveloped nations to Western research institutes continues to concern some observers.

As historians discover new sources and question old attitudes, the global history of mathematics will continue to expand. The nature of mathematics perhaps makes it more likely to move across cultural boundaries intact, and serious inquiry into the global dissemination of mathematical ideas has only begun. Unfortunately, because so much mathematics has evolved in traditions that do not value the citation of sources, exact lines of transmission will be hard to re-create.

Bibliography:

• Ascher, M. (1991). Ethnomathematics: A multicultural view of mathematical ideas. Pacific Grove, CA: Brooks/Cole Publishing.
• Ascher, M. (2004). Mathematics elsewhere: An exploration of ideas across cultures. Princeton, NJ: Princeton University Press.
• Cohen, P. C. (1982). A calculating people: The spread of numeracy in early America. Chicago: University of Chicago Press.
• D’Ambrosio, U. (2006). Ethnomathematics: Link between traditions and modernity (A. Keppel, Trans.). Rotterdam, The Netherlands: Sense Publishers. (Original work published 2001)
• Ifrah, G. (2000). The universal history of numbers: From prehistory to the invention of the computer (D. Bellos, Trans.). New York: J. Wiley.
• Joseph, G. G. (2000). The crest of the peacock: Non-European roots of mathematics. Princeton, NJ: Princeton University Press.
• Kanigal, R. (1991). The man who knew infinity: A life of the genius Ramanujan. New York: Charles Scribner’s Sons.
• Katz, V. J. (1998). A history of mathematics: An introduction (2nd ed.). Reading, MA: Addison-Wesley Longman.
• Lehto, O. (1998). Mathematics without borders: A history of the international mathematical union. New York: Springer.
• Li, Y. (1987). Chinese mathematics: A concise history (J. N. Crossley & W-C. L. Anthony, Trans.). Oxford, U.K.: Clarendon Press.
• Powell, A. B., & Frankenstein, M. (Eds.). (1997). Ethnomathematics: Challenging Eurocentrism in mathematics education. Albany: State University of New York Press.
• Selin, H. (Ed.). (2000). Mathematics across cultures: The history of non-Western mathematics. Boston: Kluwer Academic Publishers.
• Zaslavsky, C. (1973). Africa counts: Number and pattern in African culture. Boston: Prindle, Weber & Schmidt.

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• 5 Steps to Getting Started with Llama 2
• The Llama Ecosystem: Past, Present, and Future
• Introducing Code Llama, a state-of-the-art large language model for coding
• Meta and Microsoft Introduce the Next Generation of Llama
• Today, we’re introducing Meta Llama 3, the next generation of our state-of-the-art open source large language model.
• Llama 3 models will soon be available on AWS, Databricks, Google Cloud, Hugging Face, Kaggle, IBM WatsonX, Microsoft Azure, NVIDIA NIM, and Snowflake, and with support from hardware platforms offered by AMD, AWS, Dell, Intel, NVIDIA, and Qualcomm.
• We’re dedicated to developing Llama 3 in a responsible way, and we’re offering various resources to help others use it responsibly as well. This includes introducing new trust and safety tools with Llama Guard 2, Code Shield, and CyberSec Eval 2.
• In the coming months, we expect to introduce new capabilities, longer context windows, additional model sizes, and enhanced performance, and we’ll share the Llama 3 research paper.
• Meta AI, built with Llama 3 technology, is now one of the world’s leading AI assistants that can boost your intelligence and lighten your load—helping you learn, get things done, create content, and connect to make the most out of every moment. You can try Meta AI here .

Today, we’re excited to share the first two models of the next generation of Llama, Meta Llama 3, available for broad use. This release features pretrained and instruction-fine-tuned language models with 8B and 70B parameters that can support a broad range of use cases. This next generation of Llama demonstrates state-of-the-art performance on a wide range of industry benchmarks and offers new capabilities, including improved reasoning. We believe these are the best open source models of their class, period. In support of our longstanding open approach, we’re putting Llama 3 in the hands of the community. We want to kickstart the next wave of innovation in AI across the stack—from applications to developer tools to evals to inference optimizations and more. We can’t wait to see what you build and look forward to your feedback.

Our goals for Llama 3

With Llama 3, we set out to build the best open models that are on par with the best proprietary models available today. We wanted to address developer feedback to increase the overall helpfulness of Llama 3 and are doing so while continuing to play a leading role on responsible use and deployment of LLMs. We are embracing the open source ethos of releasing early and often to enable the community to get access to these models while they are still in development. The text-based models we are releasing today are the first in the Llama 3 collection of models. Our goal in the near future is to make Llama 3 multilingual and multimodal, have longer context, and continue to improve overall performance across core LLM capabilities such as reasoning and coding.

State-of-the-art performance

Our new 8B and 70B parameter Llama 3 models are a major leap over Llama 2 and establish a new state-of-the-art for LLM models at those scales. Thanks to improvements in pretraining and post-training, our pretrained and instruction-fine-tuned models are the best models existing today at the 8B and 70B parameter scale. Improvements in our post-training procedures substantially reduced false refusal rates, improved alignment, and increased diversity in model responses. We also saw greatly improved capabilities like reasoning, code generation, and instruction following making Llama 3 more steerable.

*Please see evaluation details for setting and parameters with which these evaluations are calculated.

In the development of Llama 3, we looked at model performance on standard benchmarks and also sought to optimize for performance for real-world scenarios. To this end, we developed a new high-quality human evaluation set. This evaluation set contains 1,800 prompts that cover 12 key use cases: asking for advice, brainstorming, classification, closed question answering, coding, creative writing, extraction, inhabiting a character/persona, open question answering, reasoning, rewriting, and summarization. To prevent accidental overfitting of our models on this evaluation set, even our own modeling teams do not have access to it. The chart below shows aggregated results of our human evaluations across of these categories and prompts against Claude Sonnet, Mistral Medium, and GPT-3.5.

Preference rankings by human annotators based on this evaluation set highlight the strong performance of our 70B instruction-following model compared to competing models of comparable size in real-world scenarios.

Our pretrained model also establishes a new state-of-the-art for LLM models at those scales.

To develop a great language model, we believe it’s important to innovate, scale, and optimize for simplicity. We adopted this design philosophy throughout the Llama 3 project with a focus on four key ingredients: the model architecture, the pretraining data, scaling up pretraining, and instruction fine-tuning.

Model architecture

In line with our design philosophy, we opted for a relatively standard decoder-only transformer architecture in Llama 3. Compared to Llama 2, we made several key improvements. Llama 3 uses a tokenizer with a vocabulary of 128K tokens that encodes language much more efficiently, which leads to substantially improved model performance. To improve the inference efficiency of Llama 3 models, we’ve adopted grouped query attention (GQA) across both the 8B and 70B sizes. We trained the models on sequences of 8,192 tokens, using a mask to ensure self-attention does not cross document boundaries.

Training data

To train the best language model, the curation of a large, high-quality training dataset is paramount. In line with our design principles, we invested heavily in pretraining data. Llama 3 is pretrained on over 15T tokens that were all collected from publicly available sources. Our training dataset is seven times larger than that used for Llama 2, and it includes four times more code. To prepare for upcoming multilingual use cases, over 5% of the Llama 3 pretraining dataset consists of high-quality non-English data that covers over 30 languages. However, we do not expect the same level of performance in these languages as in English.

To ensure Llama 3 is trained on data of the highest quality, we developed a series of data-filtering pipelines. These pipelines include using heuristic filters, NSFW filters, semantic deduplication approaches, and text classifiers to predict data quality. We found that previous generations of Llama are surprisingly good at identifying high-quality data, hence we used Llama 2 to generate the training data for the text-quality classifiers that are powering Llama 3.

We also performed extensive experiments to evaluate the best ways of mixing data from different sources in our final pretraining dataset. These experiments enabled us to select a data mix that ensures that Llama 3 performs well across use cases including trivia questions, STEM, coding, historical knowledge, etc.

Scaling up pretraining

To effectively leverage our pretraining data in Llama 3 models, we put substantial effort into scaling up pretraining. Specifically, we have developed a series of detailed scaling laws for downstream benchmark evaluations. These scaling laws enable us to select an optimal data mix and to make informed decisions on how to best use our training compute. Importantly, scaling laws allow us to predict the performance of our largest models on key tasks (for example, code generation as evaluated on the HumanEval benchmark—see above) before we actually train the models. This helps us ensure strong performance of our final models across a variety of use cases and capabilities.

We made several new observations on scaling behavior during the development of Llama 3. For example, while the Chinchilla-optimal amount of training compute for an 8B parameter model corresponds to ~200B tokens, we found that model performance continues to improve even after the model is trained on two orders of magnitude more data. Both our 8B and 70B parameter models continued to improve log-linearly after we trained them on up to 15T tokens. Larger models can match the performance of these smaller models with less training compute, but smaller models are generally preferred because they are much more efficient during inference.

To train our largest Llama 3 models, we combined three types of parallelization: data parallelization, model parallelization, and pipeline parallelization. Our most efficient implementation achieves a compute utilization of over 400 TFLOPS per GPU when trained on 16K GPUs simultaneously. We performed training runs on two custom-built 24K GPU clusters . To maximize GPU uptime, we developed an advanced new training stack that automates error detection, handling, and maintenance. We also greatly improved our hardware reliability and detection mechanisms for silent data corruption, and we developed new scalable storage systems that reduce overheads of checkpointing and rollback. Those improvements resulted in an overall effective training time of more than 95%. Combined, these improvements increased the efficiency of Llama 3 training by ~three times compared to Llama 2.

Instruction fine-tuning

To fully unlock the potential of our pretrained models in chat use cases, we innovated on our approach to instruction-tuning as well. Our approach to post-training is a combination of supervised fine-tuning (SFT), rejection sampling, proximal policy optimization (PPO), and direct preference optimization (DPO). The quality of the prompts that are used in SFT and the preference rankings that are used in PPO and DPO has an outsized influence on the performance of aligned models. Some of our biggest improvements in model quality came from carefully curating this data and performing multiple rounds of quality assurance on annotations provided by human annotators.

Learning from preference rankings via PPO and DPO also greatly improved the performance of Llama 3 on reasoning and coding tasks. We found that if you ask a model a reasoning question that it struggles to answer, the model will sometimes produce the right reasoning trace: The model knows how to produce the right answer, but it does not know how to select it. Training on preference rankings enables the model to learn how to select it.

Building with Llama 3

Our vision is to enable developers to customize Llama 3 to support relevant use cases and to make it easier to adopt best practices and improve the open ecosystem. With this release, we’re providing new trust and safety tools including updated components with both Llama Guard 2 and Cybersec Eval 2, and the introduction of Code Shield—an inference time guardrail for filtering insecure code produced by LLMs.

We’ve also co-developed Llama 3 with torchtune , the new PyTorch-native library for easily authoring, fine-tuning, and experimenting with LLMs. torchtune provides memory efficient and hackable training recipes written entirely in PyTorch. The library is integrated with popular platforms such as Hugging Face, Weights & Biases, and EleutherAI and even supports Executorch for enabling efficient inference to be run on a wide variety of mobile and edge devices. For everything from prompt engineering to using Llama 3 with LangChain we have a comprehensive getting started guide and takes you from downloading Llama 3 all the way to deployment at scale within your generative AI application.

A system-level approach to responsibility

We have designed Llama 3 models to be maximally helpful while ensuring an industry leading approach to responsibly deploying them. To achieve this, we have adopted a new, system-level approach to the responsible development and deployment of Llama. We envision Llama models as part of a broader system that puts the developer in the driver’s seat. Llama models will serve as a foundational piece of a system that developers design with their unique end goals in mind.

Instruction fine-tuning also plays a major role in ensuring the safety of our models. Our instruction-fine-tuned models have been red-teamed (tested) for safety through internal and external efforts. ​​Our red teaming approach leverages human experts and automation methods to generate adversarial prompts that try to elicit problematic responses. For instance, we apply comprehensive testing to assess risks of misuse related to Chemical, Biological, Cyber Security, and other risk areas. All of these efforts are iterative and used to inform safety fine-tuning of the models being released. You can read more about our efforts in the model card .

Llama Guard models are meant to be a foundation for prompt and response safety and can easily be fine-tuned to create a new taxonomy depending on application needs. As a starting point, the new Llama Guard 2 uses the recently announced MLCommons taxonomy, in an effort to support the emergence of industry standards in this important area. Additionally, CyberSecEval 2 expands on its predecessor by adding measures of an LLM’s propensity to allow for abuse of its code interpreter, offensive cybersecurity capabilities, and susceptibility to prompt injection attacks (learn more in our technical paper ). Finally, we’re introducing Code Shield which adds support for inference-time filtering of insecure code produced by LLMs. This offers mitigation of risks around insecure code suggestions, code interpreter abuse prevention, and secure command execution.

With the speed at which the generative AI space is moving, we believe an open approach is an important way to bring the ecosystem together and mitigate these potential harms. As part of that, we’re updating our Responsible Use Guide (RUG) that provides a comprehensive guide to responsible development with LLMs. As we outlined in the RUG, we recommend that all inputs and outputs be checked and filtered in accordance with content guidelines appropriate to the application. Additionally, many cloud service providers offer content moderation APIs and other tools for responsible deployment, and we encourage developers to also consider using these options.

Deploying Llama 3 at scale

Llama 3 will soon be available on all major platforms including cloud providers, model API providers, and much more. Llama 3 will be everywhere .

Our benchmarks show the tokenizer offers improved token efficiency, yielding up to 15% fewer tokens compared to Llama 2. Also, Group Query Attention (GQA) now has been added to Llama 3 8B as well. As a result, we observed that despite the model having 1B more parameters compared to Llama 2 7B, the improved tokenizer efficiency and GQA contribute to maintaining the inference efficiency on par with Llama 2 7B.

For examples of how to leverage all of these capabilities, check out Llama Recipes which contains all of our open source code that can be leveraged for everything from fine-tuning to deployment to model evaluation.

What’s next for Llama 3?

The Llama 3 8B and 70B models mark the beginning of what we plan to release for Llama 3. And there’s a lot more to come.

Our largest models are over 400B parameters and, while these models are still training, our team is excited about how they’re trending. Over the coming months, we’ll release multiple models with new capabilities including multimodality, the ability to converse in multiple languages, a much longer context window, and stronger overall capabilities. We will also publish a detailed research paper once we are done training Llama 3.

To give you a sneak preview for where these models are today as they continue training, we thought we could share some snapshots of how our largest LLM model is trending. Please note that this data is based on an early checkpoint of Llama 3 that is still training and these capabilities are not supported as part of the models released today.

We’re committed to the continued growth and development of an open AI ecosystem for releasing our models responsibly. We have long believed that openness leads to better, safer products, faster innovation, and a healthier overall market. This is good for Meta, and it is good for society. We’re taking a community-first approach with Llama 3, and starting today, these models are available on the leading cloud, hosting, and hardware platforms with many more to come.

Try Meta Llama 3 today

We’ve integrated our latest models into Meta AI, which we believe is the world’s leading AI assistant. It’s now built with Llama 3 technology and it’s available in more countries across our apps.

You can use Meta AI on Facebook, Instagram, WhatsApp, Messenger, and the web to get things done, learn, create, and connect with the things that matter to you. You can read more about the Meta AI experience here .

Visit the Llama 3 website to download the models and reference the Getting Started Guide for the latest list of all available platforms.

You’ll also soon be able to test multimodal Meta AI on our Ray-Ban Meta smart glasses.

As always, we look forward to seeing all the amazing products and experiences you will build with Meta Llama 3.

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