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Trends in mathematics education and insights from a meta-review and bibliometric analysis of review studies

• Original Paper
• Open access
• Published: 15 May 2024

Cite this article

You have full access to this open access article

• Mustafa Cevikbas   ORCID: orcid.org/0000-0002-7844-4707 1 ,
• Gabriele Kaiser 2 , 3 &
• Stanislaw Schukajlow 4  

1 Altmetric

Review studies are vital for advancing knowledge in many scientific fields, including mathematics education, amid burgeoning publications. Based on an extensive consideration of existing review typologies, we conducted a meta-review and bibliometric analysis to provide a comprehensive overview of and deeper insights into review studies within mathematics education. After searching Web of Science, we identified 259 review studies, revealing a significant increase in such studies over the last five years. Systematic reviews were the most prevalent type, followed by meta-analyses, generic literature reviews, and scoping reviews. On average, the review studies had a sample size of 99, with the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) guidelines commonly employed. Despite certain studies offering nuanced distinctions among review types, ambiguity persisted. Only about a quarter of the studies explicitly reported employing specific theoretical frameworks (particularly, technology, knowledge, and competence models). Co-authored publications were most common within American institutions and the leading countries are the United States, Germany, China, Australia, and England in publishing most review studies. Educational review journals, educational psychology journals, special education journals, educational technology journals, and mathematics education journals provided platforms for review studies, and prominent research topics included digital technologies, teacher education, mathematics achievement, and learning disabilities. In this study, we synthesised a range of reviews to facilitate readers’ comprehension of conceptual congruities and disparities across various review types, as well as to track current research trends. The results suggest that there is a need for discipline-specific standards and guidelines for different types of mathematics education reviews, which may lead to more high-quality review studies to enhance progress in mathematics education.

Avoid common mistakes on your manuscript.

1 Introduction

Comprehensive literature reviews serve as foundational pillars for advancing scholarly discourse, offering critical insights into existing research and shaping future inquiries across disciplines. In the realm of academic writing, spanning from journal articles to dissertations, literature reviews are highly regarded for their capacity to synthesize knowledge, identify gaps, and provide a cohesive framework for understanding complex topics (Boote & Beile, 2005 ). Moreover, reviews play a significant role in academia by setting new research agendas and informing decision-making processes in practice, policy, and society (Kunisch et al., 2023 ).

As empirical and theoretical research burgeons in diverse fields, the need for literature review studies has become even more pronounced, facilitating a deeper understanding of specific research areas or themes (Hart, 2018 ; Nane et al., 2023 ). Additional factors contributing to the popularity of review studies in recent years include the rise of specialized review journals (Kunisch et al., 2023 ), challenges associated with conducting various types of empirical studies during the prolonged COVID-19 crisis (Cevikbas & Kaiser, 2023 ), and a competitive research climate wherein factors such as impact factors and citations hold significant weight (Ketcham & Crawford, 2007 ). Review studies are particularly attractive as they often garner a substantial number of citations, thereby enhancing researchers’ visibility and scholarly impact (Grant & Booth, 2009 ; Taherdoost, 2023 ).

The importance of review studies has been duly acknowledged in mathematics education, as evidenced by the inclusion of review papers in thematically oriented special issues of journals such as ZDM– Mathematics Education (Kaiser & Schukajlow, 2024 ), which has been originally founded as review journal. Several upcoming or already published special issues of ZDM– Mathematics Education , which emphasise ‘reviews on important themes in mathematics education’, highlight the importance of review studies as valuable contributions to the field.

The proliferation of literature reviews has increased interest in developing typologies to categorise them and understand different literature review approaches (Grant & Booth, 2009 ; Paré et al., 2015 ; Schryen & Sperling, 2023 ). Despite its significance, there remains a notable lack of research aimed at comprehensively understanding review studies within the field of mathematics education from a meta-perspective. In response to this gap, we conducted a systematic meta-review with the aim of providing an overview of different types of review studies in mathematics education over the past few decades and consolidating insights from multiple high-level review studies (Becker & Oxman, 2008 ; Schryen & Sperling, 2023 ). Meta-reviews offer concise yet comprehensive synopses and curated lists of pertinent reviews, adeptly addressing the perennial challenge of balancing thorough coverage with focused specificity (Grant & Booth, 2009 ).

In addition, we applied bibliometric analysis as a valuable tool for identifying research trends, progress, reliable sources, and future directions within the field. The bibliometric analysis aids in identifying hot research topics and trends (Song et al., 2019 ), assessing progress, identifying reliable sources, recognising major contributors, and predicting future research success (Geng et al., 2017 ). Furthermore, it helps researchers to pinpoint potential topics, suitable institutions for cooperation, and potential scholars for scientific collaboration (Martínez et al., 2015 ). By combining a meta-review and bibliometric analysis, we aim to offer a comprehensive overview of and deeper insights into state-of-the-art review studies within mathematics education.

Specifically, we seek to understand how the distribution and development of literature review studies in mathematics education have evolved over the years, examining factors such as publication years, publishers, review types, sample sizes, and the use of theoretical or conceptual frameworks. Additionally, we aim to assess adherence to review study guidelines and protocols, providing insights into the rigor and quality of research methodologies employed, particularly in light of the lack of clear guidance on producing rigorous and impactful literature reviews (Kunisch et al., 2023 ).

Furthermore, we endeavour to identify authors who have made contribution to the field of mathematics education through review studies, as well as those whose work is most frequently cited. We also identify co-authorship network analysis as understanding research networks allows researchers to identify potential collaborators and build partnerships with other scholars in various countries. Collaborative research endeavours can lead to enhanced research outcomes, broader dissemination of findings, and increased opportunities for funding and professional development. It can also highlight interdisciplinary connections and collaborations within and across fields, leading to innovative approaches and solutions to complex research questions (RQs) that transcend disciplinary boundaries.

Moreover, we analysed the distribution of common keywords across review studies, identifying focal subjects and thematic areas prevalent in mathematics education research. This analysis can provide valuable insights into key topics and trends shaping the field, guiding future research directions and priorities.

Lastly, we identified the most cited review papers in mathematics education and the journals in which they have been published, recognizing seminal works and influential publications that have contributed to the advancement of the field.

Overall, in light of the preceding discourse, we addressed the following RQs to uncover the characteristics of review studies, identify research trends, and delineate future research directions in mathematics education:

How can the distribution and development of review studies in mathematics education over time be characterised according to the number of manuscripts, publishers, review types, sample sizes, the use of theoretical or conceptual frameworks, and adherence to review study guidelines and protocols?

Which authors have contributed the largest number of review studies in mathematics education, and which authors’ review papers are most frequently cited in the literature?

From which countries are the authors of the review studies in mathematics education?

Which author keywords can be identified in the review studies in mathematics education, how are these keywords distributed across the analysed review studies, and which focal topics do these keywords indicate?

What are the most cited review papers in mathematics education, and in which journals have they been published?

2 Literature review studies and review typologies– background information

In this chapter, we provide a thorough analysis of different typologies for review studies, as we seek to elucidate the primary characteristics of various review studies conducted within mathematics education (Sect.  2.1 ). This effort led to the identification of 28 review types presented in Table  1 , which were used in the current study’s literature search processes to access existing review studies and the analysis of identified studies in the field of mathematics education. Furthermore, we discuss the advancement of guidelines and protocols, highlighting their role in shaping the conduct of review studies (Sect. 2.2). Finally, we conclude the chapter by underscoring the importance and potential impact of meta-reviews and bibliometric analyses in the context of mathematics education (Sect. 2.3).

2.1 Literature review typologies

Researchers have defined and emphasized different review types with distinct features, objectives, and methodologies. To address the challenge of ambiguous review categorisations, we conducted an extensive search and analysis of the literature on Web of Science (WoS) using the search strings ‘typology of reviews’ and ‘taxonomy of reviews’ to search the titles of studies. We focused particularly on influential theoretical, conceptual, and review papers discussing the taxonomy and typology of review studies and recent advances driven by scholars across diverse fields.

2.1.1 Seminal work by Grant and Booth ( 2009 ) on the discourse of literature review typologies

The categorisation of literature reviews has been profoundly influenced by the seminal work of Grant and Booth ( 2009 ), on which typologies of literature reviews are often based. Their paper garnered significant attention, with over 10,304 citations as of 20 April 2024 according to Google Scholar. Originally in the field of health information theory and practice, these authors founded their work on earlier approaches, notably Cochrane’s ( 1979 ) approach. Grant and Booth ( 2009 ) claimed that the developed typology could standardise the diverse terminology used. They distinguished 14 review types, which we summarise below, highlighting the main scope and search methodologies (Grant & Booth, 2009 , pp. 94–95):

A critical review ‘goes beyond mere description of identified articles and includes a degree of analysis and conceptual innovation’; no formalised or systematic approach is required because the aim of such a review is ‘to identify conceptual contributions to embody existing or derive new theory’.

A generic literature review incorporates ‘published materials that provide examination of recent of current literature’; comprehensive searching may or may not be necessary.

A mapping review/systematic mapping is used to ‘categorize existing literature’ and identify gaps in the research literature. The completeness of a search is important, but no formal quality assessment is needed.

A meta-analysis is a ‘technique that statistically combines the results of quantitative studies to provide a more precise effect of the results’; a comprehensive search is conducted based on the inclusion and exclusion criteria.

A mixed-studies review/mixed-methods review incorporates ‘a combination of review approaches, for example combining quantitative with qualitative research… and requires a very sensitive search’.

An overview is a generic term describing a ‘summary of the… literature that attempts to survey the literature and describe its characteristics’; it may or may not include comprehensive searching and quality assessment.

A qualitative systematic review/qualitative evidence synthesis is a ‘method for integrating or comparing the findings from qualitative studies’, and it may involve selective sampling.

A rapid review comprises an ‘assessment of what is already known about a policy or practice issue, by using systematic review methods to search and critically appraise existing research’; a characteristic of such a review is that the ‘completeness of searching is determined by time constraints’.

A scoping review is a ‘preliminary assessment of the potential size and scope of available research literature’, with the ‘completeness of searching determined by time/scope constraints’.

A state-of-the-art review ‘tend[s] to address more current matters in contrast to other combined retrospective and current approaches’ and ‘aims for comprehensive searching of current literature’.

A systematic review ‘seeks to systematically search for, appraise and synthesise research evidence’ and should be comprehensive and based on inclusion/exclusion criteria.

A systematic search and review ‘combines [the] strengths of critical review with a comprehensive search process’, typically addressing broad questions to produce ‘best evidence synthesis’ based on ‘exhaustive, comprehensive searching’.

A systematised review ‘include[s] elements of systematic review process while stopping short of systematic review’, ‘typically conducted as postgraduate student assignment’; it ‘may or may not include comprehensive searching’.

An umbrella review ‘specifically refers to review compiling evidence from multiple reviews into one accessible and usable document’ via ‘identification of component reviews, but no search for primary studies’. ‘Primary studies’ refer to original research studies or individual studies conducted by researchers to gather data first-hand.

Booth with colleagues later expanded the typology by introducing the concept of a review family construct and amalgamating various types of reviews for further refinement, such as traditional reviews, systematic reviews, review of reviews, rapid reviews, mixed-methods reviews, and purpose-specific reviews (for details, see Sutton et al., 2019 ).

2.1.2 Further development of the review typologies

Many classifications for review studies have been developed, and in the following section, we present more recent approaches. Paré et al. ( 2015 ), in another highly cited study (2,059 Google Scholar citations as of 20 April 2024) considered seven recurrent dimensions: the goal of the review, the scope of the review questions, the search strategy, the nature of the primary sources, the explicitness of the study selection, quality appraisal, and the methods used to analyse/synthesise the findings. Based on these dimensions, they formulated nine different literature review types: narrative reviews, descriptive reviews, scoping/mapping reviews, meta-analyses, qualitative systematic reviews, umbrella reviews, critical reviews, theoretical reviews, and realist reviews.

In Paré et al.’s ( 2015 ) classification, the review categories that differ from Grant and Booth’s ( 2009 ) classification are theoretical reviews, realist reviews, narrative reviews, and descriptive reviews, which we therefore describe them briefly. A theoretical review draws on conceptual and empirical studies to develop a conceptual framework or model using structured approaches, such as taxonomies, to discover patterns or commonalities. The aim of a realist review (also called a meta-narrative review) is to formulate explanations; such reviews ‘are theory-driven interpretative reviews which were developed to inform, enhance, extend, or alternatively supplement conventional systematic reviews by making sense of heterogeneous evidence about complex interventions applied in diverse contexts in a way that informs policy decision making’ (Paré et al., 2015 , p. 188). The purpose of a narrative review is to survey the existing literature on a particular subject or topic without necessarily seeking generalisations or cumulative insights from the material reviewed (Davies, 2000 ). Typically, such reviews do not detail the underpinning review processes or involve systematic and exhaustive searches of all pertinent literature. This category resembles Grant and Booth’s ( 2009 ) description of ‘literature reviews’ and overlaps with Samnani et al.’s ( 2017 ) narrative reviews, literature reviews, and overviews, resulting in a somewhat ambiguous typology. The aim of a descriptive review is to identify patterns and trends across a set of empirical studies within a specific research field, encompassing pre-existing propositions, theories, methodological approaches, or findings. To accomplish this objective, descriptive reviews collect, structure, and analyse numerical data that reflect the frequency distribution of research elements.

MacEntee ( 2019 ), Samnani et al. ( 2017 ), Schryen et al. ( 2020 ), and Taherdoost ( 2023 ) corroborated Grant and Booth’s ( 2009 ) and Paré et al.’s ( 2015 ) classifications, identifying various common review categories (see Table  1 ). In Samnani et al.’s ( 2017 ) classification, a distinct review type based on the previously mentioned categories is meta-synthesis , the aim of which is to provide explanations for phenomena, in contrast to meta-analysis, which focuses on quantitative outcomes.

Later, Schryen and Sperling ( 2023 ) introduced a slightly revised typology of literature review studies, which they applied to a meta-review of operations research. Their study distinguished nine types of literature reviews, newly introduced categories included tutorial reviews, selective reviews, algorithmic reviews, computational reviews, and meta-reviews. The objective of a tutorial review is to offer a research-oriented summary of principles, mathematical fundamentals, and concepts, aiming to inspire and direct future research endeavours. The authors’ emphasis on foundational aspects has often provided a launching pad for research advances. A selective review typically has a limited scope because it is not based on a thorough search of all relevant literature. This type of review concentrates on specific segments of the literature, such as journals, time periods, methodologies, or issues, to delve deeper into specific questions and phenomena. An algorithmic review focuses on advances in algorithms and frameworks in the literature that address a spectrum of problems. It employs either selective or comprehensive search strategies, predominantly examining algorithm-related sources. A computational review investigates algorithms and/or parameterisations proposed in the literature, largely considering implementations and computational studies, measurement efficiency, effectiveness, and different forms of robustness. Finally, Schryen and Sperling ( 2023 ) defined a meta-review as an overview of systematic reviews or a systematic review of reviews and pointed out that a meta-review can also be called an umbrella review (which is the case by Grant and Booth), again confirming the fuzzy nature of the currently available typologies. According to Schryen and Sperling ( 2023 ), meta-reviews primarily aim to furnish descriptive overviews of literature reviews, serving as tertiary studies that integrate evidence from multiple (qualitative or quantitative) reviews into unified and user-friendly documents (Becker & Oxman, 2008 ; Paré et al., 2015 ). In contrast to the previously mentioned perspectives, Schryen and Sperling ( 2023 ) argued that meta-reviews are not limited to addressing specific research questions but can also address a wide range of enquiries.

Chigbu et al. ( 2023 , pp. 5–6) emphasised that there ‘is a continuum of literature types’ (p. 4) and distinguished twelve different types of literature reviews, six of which were not covered by the classifications provided by previously mentioned studies: integrated reviews, interpretative reviews, iterative reviews, semi-systematic reviews, and bibliometric reviews. According to their approach, an integrative review builds ‘new knowledge based on the existing body of literature following a rationalist perspective’, an interpretative review ‘interprets what other scholars have written to put into specific perspectives’, and an iterative review is an ‘algorithm-based approach performed to collate all studies in a specific field of research’. Moreover, a meta-synthesis review examines and analyses qualitative study findings and is often employed to clarify specific concepts. Additionally, a semi-systematic review analyses the data and findings of other studies to address specific research inquiries, using a partial systematic review methodology. Lastly, a bibliometric review systematically examines the literature on a specific subject or research discipline by quantitatively measuring indicators such as authors, citations, journals, countries, and years of publications.

As previously noted in this paper, this detailed description of review types is instrumental in facilitating our investigation of various review studies in the realm of mathematics education.

2.2 Advancements in guidelines and protocols for review studies

Various researchers have developed guidelines, protocols, and statements to assist authors in conducting, evaluating, and reporting their review studies. This academic endeavour has predominantly focused on enhancing the rigour and transparency of systematic reviews, meta-analyses, and, more recently, scoping reviews. For instance, the population, intervention, comparison, and outcomes (PICO) model, originally conceived to support evidence-based healthcare, serves as a cornerstone for establishing review criteria, crafting research questions and search strategies, and delineating the characteristics of included studies or meta-analyses (Richardson et al., 1995 ). In response to the observed deficiencies in reporting standards within meta-analyses, an international consortium introduced the Quality of Reporting of Meta-Analyses (QUOROM) statement in 1996, primarily to enhance the reporting quality of meta-analyses involving randomised controlled trials (Moher et al., 1999 ). Subsequently, Moher et al. ( 2009 ) updated these guidelines, which are now known as the PRISMA guidelines, and incorporated various conceptual and methodological advances in systematic reviews and meta-analyses. Additionally, Shea et al. ( 2007 ) introduced the Assessment of Multiple Systematic Reviews (AMSTAR) checklist to evaluate methodological quality and guide the conduct of systematic reviews, while Grant and Booth ( 2009 ) developed the search, appraisal, synthesis, and analysis (SALSA) framework to analyse and characterise review types. Most recently, Page et al. ( 2021 ) updated the PRISMA guidelines, providing updated reporting standards that reflect advances in methods for identifying, selecting, appraising, and synthesising studies, with the aim of promoting more transparent, complete, and accurate reporting of systematic reviews and meta-analyses. An extension of PRISMA guidelines for scoping reviews, known as PRISMA-ScR, aids readers in understanding relevant terminology, core concepts, and key items for reporting scoping reviews (Tricco et al., 2018 ). Despite the value of these efforts, further studies are warranted, particularly comprehensive guidelines for each type of review studies.

2.3 Literature reviews in mathematics education

The preceding section delineates various types of review studies, underscoring their key methodological attributes. Within the realm of mathematics education, akin to other disciplines, literature review studies, particularly systematic reviews, and meta-analyses, received considerable attention (Cevikbas et al., 2022 ; Cevikbas & Kaiser, 2023 ; Kaiser & Schukajlow, 2024 ). However, the understanding of the prevailing characteristics of review studies in mathematics education, including prevalent review types, trends, gaps, and avenues for future improvement, remains limited.

Meta-reviews can offer a promising avenue for pinpointing research gaps, evaluating evidence quality, and informing policy and intervention strategies and guiding evidence-based decision-making processes by synthesizing findings from multiple review studies (Schryen & Sperling, 2023 ). In addition to meta-reviews, the bibliometric analyses serve to ascertain the scope of prior research, discern contemporary review trends, identify literature gaps, and propose future research agendas (Chigbu et al., 2023 ). While meta-reviews provide a comprehensive assessment of the literature, bibliometric analyses aid in systematically screening literature on a specific subject, topic, or research discipline by quantitatively measuring various indicators such as authors, citations, journals, countries, and years of publication. These methodological approaches hold promise for instituting a systematic, transparent, and reproducible review process, thereby augmenting the overall quality of reviews in mathematics education. Bibliometric techniques serve as valuable tools in literature reviews, guiding researchers by pinpointing influential works and impartially mapping the research landscape prior to in-depth exploration (Zupic & Cater, 2015 ).

Despite their significance, meta-reviews and bibliometric analyses remain seldom within the domain of mathematics education, signifying a substantial gap in the literature. Our comprehensive literature review underscores an urgent need for meta-review studies encompassing literature review studies in the realm of mathematics education. Additionally, while no bibliometric analysis study specifically focusing on review studies in mathematics education was identified, several bibliometric studies in mathematics education on various topics were noted, such as mathematics anxiety (Radevic & Milovanovic, 2023 ), problem-solving (Suseelan et al., 2022 ), and teacher noticing (Wei et al., 2023 ).

Overall, there exists a compelling need for meta-reviews enriched by bibliometric analyses to explore the current state of literature review research in mathematics education, and the current study aims to address this gap in a timely manner.

3 Methodology

3.1 literature search and manuscript selection process.

In this study, following the latest PRISMA guidelines (Page et al., 2021 ), we aimed to conduct a systematic review of previous review studies in mathematics education. Specifically, we employed the meta-review (umbrella review) method supplemented by bibliometric analyses. We processed the manuscript selection under three stages: identification, screening, and included.

3.1.1 Identification

On 10 January 2024 (last access), we conducted an extensive literature search using the WoS electronic database, which includes publications in high-ranking peer-reviewed journals and is widely acknowledged as a primary source of review and bibliometric data that meet high quality standards (Korom, 2019 ). WoS facilitates effective literature searches, supports various information purposes, and aids research topic mapping, trend monitoring as well as scholarly activity analysis (Birkle et al., 2020 ).

To comprehensively identify potentially relevant review studies in mathematics education, we developed an inclusive search query targeting specific terms in the titles, abstracts, and keywords of papers. The query comprised terms that we extracted from the typologies of literature reviews described in Chap. 2, particularly the more general, commonly used types of reviews:

( TOPIC ) ((literature review*OR literature survey* OR systematic review* OR rapid review* OR scoping review* OR critical review* OR meta-analysis OR narrative review* OR umbrella review* OR meta review* OR meta-review OR bibliometric review OR bibliometric analysis OR mapping review OR mixed-methods review OR integrative review OR interpretative review OR iterative review OR meta-synthesis OR descriptive review OR theoretical review OR realist review OR selective review OR algorithmic review OR computational review)) AND ( TOPIC ) ((math* OR geometry OR algebra OR calculus OR probability OR statistics OR arithmetic).

Based on these search strings, we conducted an online search that initially yielded 63,462 records.

3.1.2 Screening

In this stage, we applied data cleaning filters based on the manuscript inclusion and exclusion criteria (see Table  2 ). First, we electronically filtered the identified records based on language, resulting in the retention of 61,787 papers published in English. Subsequently, we narrowed down the selection to 10,098 papers using the following five categories of research areas within the WoS: ‘education/educational research, psychology, social sciences other topics, mathematics, or science technology other topics’. Following this categorisation, we further refined the dataset by excluding non-review papers and accessing 3,344 records within the ‘review article’ and ‘early access’ categories of the WoS database. We categorised records lacking a final publication date that had undergone peer review and acceptance as ‘early access’. Notably, to comprehensively capture publication trends, we imposed no restrictions on the publication years of the studies. In the subsequent phase, a meticulous manual screening of the titles, abstracts, and keywords of 3,344 papers led to the identification of 357 studies in mathematics education.

3.1.3 Included

Ultimately, after an extensive review of the full-text versions of initially identified 357 papers, 259 eligible review articles remained for analysis as these papers fulfilled our criteria comprehensively (see the Appendix for the list of included studies; see Fig.  1 for the flow diagram of the entire manuscript selection process). Subsequently, as detailed below, the data analysis process commenced with the inclusion of these eligible review papers in mathematics education.

Flow diagram of the manuscript selection process

3.2 Data analysis

After incorporating 259 studies into this meta-review and bibliometric analysis, we compiled the identified records into a marked list on WoS. Subsequently, we exported the records into Excel, EndNote, and plain text file formats for analysis. The analysis consisted of content analysis and bibliometric analysis (see Fig.  2 , adapted from Wei et al., 2023 ).

For the content analysis, we meticulously organised the records using EndNote reference management software and Excel worksheets. We scrutinised the full-text versions of all included articles, coding them based on (1) publication year, (2) publisher, (3) review type, (4) number of included studies (sample size), (5) guidelines and protocols for the article selection process, and (6) the theoretical and conceptual framework of the studies.

Our coding manual, informed by prior studies (Cevikbas et al., 2022 , 2024 ), guided this process (see appendix for a sample of the coding manual). After completing the content analysis coding procedure, 20% of the papers ( n  = 52) were double-coded based on the initial coding protocol. The intercoder reliability, gauged at 0.92, signifies the presence of a coding system that exhibits satisfactory reliability (Creswell, 2013 ). Any discrepancies were addressed through discussions among the coders until consensus was reached.

For the bibliometric analysis, we employed VOSviewer software (version 1.6.20), which is widely recognised and extensively used in various fields, including the educational sciences (van Eck & Waltman, 2010 ). Chigbu et al. ( 2023 ) pointed out that the WoS database plays a pivotal role in facilitating bibliometric analyses across various disciplines. These analyses help establish trends in the development and application of knowledge within specific subjects and disciplines.

In our study, the bibliometric network presented in the results chapter consists of nodes and edges, with nodes representing entities such as publications, journals, researchers, or keywords. Edges denote relationships between pairs of nodes, indicating not only the presence or absence of connections but also conveying the intensity or strength of relationships (van Eck & Waltman, 2010 ). For distance-based approaches, the positioning of nodes in a bibliometric network reflects their approximate relatedness based on proximity.

Utilising VOSviewer software, we conducted (1) co-authorship analysis (authors and countries) to elucidate collaboration patterns and contributions, (2) co-occurrence analysis (focusing Author Keywords) to scrutinise knowledge structures and the distribution and development of key research topics in mathematics education, and (3) citation analysis to delve deeper into research influences and citation networks, drawing insights from the documents and sources.

This multifaceted approach allowed us to gain a comprehensive understanding of the bibliometric landscape and unravel collaborative structures, thematic foci, and the influence of key works on mathematics education.

Analytical process for this study

In this chapter, we present the key results of the meta-review and bibliometric analyses divided into two main categories: an overview of the review studies in mathematics education based on the content analysis, addressing RQ1, and the results of the bibliometric analysis, addressing RQ2 – RQ5.

4.1 Overview of review studies in mathematics education (RQ1)

To discern the research trends and essential attributes of review studies in mathematics education, we conducted a content analysis within our meta-review to examine the 259 included review studies. Our analysis encompassed publication years, publishers, review types, guidelines, protocols used, sample sizes, and the theoretical and conceptual frameworks employed in these review studies. A general overview of the included studies is presented in Table  3 .

Our literature search with no restriction on the publication years yielded review studies published between 1996 and 2023, with a notable increase within the last five years (2019–2023, see Fig.  3 ).

Distribution of publications from 1996 to 2023

The analysis showed that the Springer Group is the primary publisher of review articles in mathematics education, followed by Taylor & Francis, Elsevier, Sage, Frontiers, Wiley, MDPI, and the American Psychological Association (APA) (see Table  4 ). Other publishers published the remaining review articles ( n  = 43). This result may be attributed to the predominance of mathematics education journals published by Springer within the WoS database.

To explore the prevailing types of review studies in mathematics education, we scrutinised the review methodologies of the included studies, considering the review types presented earlier in Table  1 . The findings revealed that researchers conducted (according to their own classification) 10 different types of reviews in mathematics education as outlined in Fig.  4 .

Types of review studies Note: *systematic reviews and meta-analyses ( n  = 6), systematic reviews and bibliometric analyses ( n  = 3), meta-analyses and narrative reviews ( n  = 2), and meta-analysis and critical review ( n  = 1)

Our analysis did not yield further review types in mathematics education. Time-related analysis showed that recent studies were systematic reviews, meta-analyses, literature reviews, and scoping reviews, whereas early examples of review studies in mathematics education were primarily narrative or critical reviews or were not explicitly classified according to review type by their authors. Figure  4 shows that some researchers ( n  = 18) described their studies as literature reviews using Grant and Booth’s ( 2009 ) generic term, without providing further details about the type of review.

To comprehend the methodologies employed by researchers to conduct reviews and select eligible studies, we conducted an analysis of the guidelines and protocols the researchers used. The findings revealed that the PRISMA guidelines were the most frequently employed ( n  = 121), aligning with the distribution of review types—PRISMA guidelines are basically recommended for systematic reviews and meta-analyses (Page et al., 2021 ). For scoping reviews, the guidelines developed by Arksey and O’Malley ( 2005 ) were the most prevalent and were used in seven studies. In six instances, researchers applied various guidelines (e.g. PICO or SALSA guidelines) sourced from the literature. Almost half of the studies ( n  = 125) did not specify the use of guidelines for conducting literature searches and selecting eligible studies. Additionally, three studies aimed to provide protocols for conducting review studies. Furthermore, seven studies were preregistered as review studies, following the Open Science Framework (OSF) and/or the International Prospective Register of Systematic Reviews (PROSPERO) protocol.

A prevalent discourse among researchers in review studies revolved around determining the most suitable number of studies to include in reviews. Our results revealed that the sample sizes of the included studies (i.e. the number of primary studies) in the field of mathematics education ranged from 8 to 3,485. Unfortunately, this information was not reported in 19 review articles. In the remaining 240 review articles, the average was 99 included studies, with an overall total of 23,761. Most of the studies ( n  = 202) had sample sizes of less than 100, with an average of 34 (see Table  5 ). Although we harboured concerns that the review studies identified in this investigation might not have been aptly named and conceptualised by their authors, we deliberately refrained from addressing this issue because it fell outside the scope of our study. While including a substantial number of studies is common and potentially suitable for bibliometric analyses and meta-analyses, conducting a systematic review, scoping review, or narrative review that critically analyses exceptionally high volumes of studies may pose challenges. In this meta-review, for example, we observed that five articles included more than 1,000 studies in the review process. Two studies, enriched by bibliometric analysis, took this approach, while another study was identified by the authors as a scoping review with a sample size of 2,433. Additionally, two studies were labelled as systematic reviews with sample sizes of 1,968, and 3,485, respectively.

Finally, we conducted a content analysis to scrutinise the theoretical and conceptual frameworks underpinning the included review studies in mathematics education. The findings revealed that out of 259 review studies, only 61 incorporated any theoretical or conceptual framework. Notably, a subset of studies ( n  = 14) was based on technology-related conceptual frameworks, such as Technological Pedagogical Content Knowledge (TPACK), frameworks pertaining to augmented and virtual reality, embodied design, artificial intelligence, big data, and the European Framework for the Digital Competence for Educators (DigCompEdu). Another prevalent category ( n  = 10) relied on frameworks related to the knowledge and competence of individuals (e.g. teachers and/or students), encompassing models such as the competence as continuum framework, TPACK, the didactic-mathematical knowledge and competencies model, mathematical content knowledge, pedagogical content knowledge, mathematical knowledge for teaching, teacher noticing competence, and an integrative model for the study of developmental competencies in minority children. Bronfenbrenner’s ecological theories (e.g. ecological theory of human development, bioecological model of human development, ecological systems theory, and ecological dynamics—a blend of dynamic-systems theory and ecological psychology) were employed by researchers in five review studies in mathematics education. In a limited subset of the studies, social and cultural theories (e.g. sociocultural theory, social learning theory, and cultural activity theory ( n  = 3)), cognitive theories (e.g. cognitive developmental theory ( n  = 2)), affective theories (e.g. self-determination theory and expectancy-value theory ( n  = 2)), linguistic theories ( n  = 2), and constructivist theories ( n  = 2) were used as frameworks. Additionally, researchers used conceptual frameworks concerning computational thinking ( n  = 2) and engagement ( n  = 3) alongside a few less frequently reported frameworks.

4.2 Results of the bibliometric analysis (RQ2–RQ5)

To identify productive and most cited authors, important journals, and countries of origin of the authors, along with the underlying research collaborations between researchers and countries, as well as research trends and key topics of review studies in mathematics education, we conducted a bibliometric analysis based on co-authorship, co-occurrence, and citations.

4.2.1 Co-authorship analysis

We conducted a co-authorship analysis according to authors and countries within the units of analysis.

Co-authorship and author analysis

The bibliometric analysis, using VOSviewer, revealed that 761 authors contributed to mathematics education, each of whom conducted at least one review study. The review papers were predominantly authored through collaboration, with most being written by two authors (30,2%), followed by three authors (20,2%), four authors (19,4%), a single author (10,1%), five authors (8,9%), six authors (6,2%), seven authors (3,5%), eight authors (1,6%), and nine authors (0,4%). These results showed that researchers primarily collaborate with their colleagues in conducting review studies—a practice vital for reducing workload and enhancing the quality of analyses—with the advantage of incorporating the various perspectives of different authors.

Table  6 highlights the top 17 authors who published a minimum of three review papers each. Notably, Lieven Verschaffel is the only scholar present in both lists of prolific and highly cited authors. The researchers listed in Table  7 , except Lieven Verschaffel, contributed to the field with a single review study. Consequently, while these researchers rank among most cited authors, the low total link strength (TLS) values indicate their limited collaboration with other scholars. The TLS was automatically calculated by VOSviewer and represents the overall intensity of co-authorship connections between a particular researcher and others. According to the co-authorship analysis, it is also noteworthy that many of the highly cited authors’ review studies typically date back over ten years, which is expected as citations tend to accumulate gradually over time. The results from the detailed citation analyses provided in Sect. 4.2.3.

Upon examining the research domains of prolific and highly cited authors, we found a diverse range of topics spanning mathematics education, psychology, educational psychology, special education, and neuroscience. This diversity highlights the interdisciplinary nature of research in mathematics education, with contributions to the literature review studies from psychologists and special education and neuroscience scholars alongside mathematics educators.

Figure  5 shows a co-authorship network map for the authors of the included review studies based on the TLS. We set the minimum number of documents for an author as one, which encompassed 761 authors who contributed to review papers in mathematics education. This bibliometric co-authorship analysis yielded 51 clusters, each containing a minimum of five items (researchers). The prominent co-authorship clusters included a green cluster (led by Lieven Verschaffel), a blue cluster (led by Gabriele Kaiser and Mustafa Cevikbas), a red cluster (led by Nelson Gena), and a yellow cluster (led by Diane P. Bryant). Nelson Gena had the highest number of collaboration links, with a TLS of 26, followed by Lieven Verschaffel (TLS = 22), Gabriele Kaiser (TLS = 16), Soyoung Park (TLS = 16), Tassia Bradford (TLS = 13), Diane P. Bryant (TLS = 12), Johannes König (TLS = 12), Mikyung Shin (TLS = 12), Min Wook Ok (TLS = 12), Bert de Smedt (TLS = 10), Fred Spooner (TLS = 10), Jihyun Lee (TLS = 10), Mustafa Cevikbas (TLS = 10), Rosella Santagata (TLS = 10), Sarah R. Powell (TLS = 10), and Thorsten Scheiner (TLS = 10).

Co-authorship and author networks

Co-authorship and country analysis

We conducted a co-authorship–country analysis, setting the minimum number of documents for a country as one, and identified 50 countries. This selection resulted in five clusters, each containing a minimum of five items (countries).

The most prominent cluster was the green cluster, encompassing eight countries from various global regions: the United States (US; TLS = 30), Germany (TLS = 23), Australia (TLS = 21), China (TLS = 11), South Korea (TLS = 6), Sweden (TLS = 4), New Zealand (TLS = 2), and Jordan (TLS = 1). The US dominated research collaborations both within this cluster and overall.

The red cluster included nine countries, predominantly Nordic and European countries: Norway (TLS = 13), Finland (TLS = 7), Belgium (TLS = 6), the Netherlands (TLS = 6), Lithuania (TLS = 1), Portugal (TLS = 1), Luxembourg (TLS = 1), Scotland (TLS = 1), and Israel (TLS = 1).

The yellow cluster contained seven countries: Canada (TLS = 7), Malaysia (TLS = 7), Denmark (TLS = 3), Libya (TLS = 2), Singapore (TLS = 2), Indonesia (TLS = 1), and the United Arab Emirates (TLS = 1).

The blue cluster primarily highlighted European collaborations and included seven countries: England (TLS = 22), Switzerland (TLS = 4), Italy (TLS = 3), France (TLS = 3), Greece (TLS = 1), Chile (TLS = 1), and Saudi Arabia (TLS = 1).

Lastly, the purple cluster represented a network of predominantly South and North American countries featuring, among others, Brazil (TLS = 6), Ireland (TLS = 5), Mexico (TLS = 4), Ecuador (TLS = 2), and Cuba (TLS = 2)(See Fig. 6 ).

Co-authorship and country networks

4.2.2 Co-occurrence analysis

To explore the research hotspots within mathematics education, we ran a keyword co-occurrence analysis using Author Keywords.

Co-occurrence analysis based on author keywords

The author keyword co-occurrence analysis indicated that our repository contained 691 keywords (see Fig.  7 , left side), of which 23 met the minimum occurrence threshold of five occurrences ( n  = 5) (see Fig.  7 , right side). In the figure, the size of a node corresponds to the frequency of a keyword co-selected in review studies in mathematics education. The distance between any two keywords reflects their relative strength and topic similarity. Nodes within the same colour cluster indicate similar topics among these publications.

The red cluster comprises 11 closely related items, including ‘mathematics, meta-analysis, mathematics achievement, intervention, scoping review, bibliometric analysis, review, technology, learning disabilities, children, and math anxiety’. The green cluster emerges as the second prominent cluster, featuring 8 interrelated items such as ‘mathematics education, systematic review, systematic literature review, literature review, teacher education, education, teaching, and flipped classroom’. Lastly, the blue cluster consists of 4 items, namely ‘math, science, early childhood, and identity’.

Co-occurrence analysis of author keywords

Notably, the most frequently cited author keyword was ‘mathematics education’ ( n  = 55), followed by ‘systematic review’ ( n  = 44), ‘mathematics’ ( n  = 41), ‘meta-analysis’ ( n  = 34), ‘systematic literature review’ ( n  = 14), ‘literature review’ ( n  = 11), ‘teacher education’ ( n  = 9), ‘mathematics achievement’ ( n  = 8), ‘intervention’ ( n  = 6), ‘education’ ( n  = 6), ‘teaching’ ( n  = 6), ‘science’ ( n  = 6), ‘scoping review’ ( n  = 5), ‘bibliometric analysis’ ( n  = 5), ‘review’ ( n  = 5), ‘math’ ( n  = 5), ‘technology’ ( n  = 5), ‘flipped classroom’ ( n  = 5), ‘early childhood’ ( n  = 5), ‘children’ ( n  = 5), ‘identity’ ( n  = 5), ‘learning disabilities’ ( n  = 5), and ‘math anxiety’ ( n  = 6).

The keywords chosen by the authors highlighted the focus areas of reviews in mathematics education, emphasising themes such as mathematics achievement, teacher education, interventions, technology, and technology-enhanced approaches (e.g. flipped classrooms), special education, and early childhood education. Furthermore, the author keywords reflected the prevalent review types in mathematics education, specifically systematic reviews and meta-analyses. Additionally, they highlighted the interdisciplinary nature of reviews in mathematics education, encompassing both mathematics education and science education.

Furthermore, we conducted distinct author keyword co-occurrence analyses for review studies published within the periods of 2019 to 2023 and those preceding 2019, aiming to discern temporal trends in author keywords, particularly in recent years. The analysis yielded 606 keywords for the 2019–2023 period and 144 keywords for the period before 2019 (see Table  8 for the most popular 15 author keywords). A noteworthy disparity in prevalent keywords was observed between the two temporal segments. While predominant keyword regarding the review types prior to 2019 was meta-analysis, followed by literature review and systematic review, over the past five years, additional keywords such as scoping review and bibliometric analysis emerged, signalling an augmentation in the diversity of review types and methodologies. The findings indicated a notable increase in the popularity of systematic reviews over the past five years.

4.2.3 Citation analysis

To explore the most cited publications and journals in mathematics education, we conducted a citation analysis based on the units of analysis in documents and sources.

Citation and document analysis

The analysis of the 259 review papers in mathematics education included in this study indicated that they received a total of 7,050 citations between 1996 and 2023, averaging 251.79 citations per year and 27.22 citations per paper. Notably, 67% of these citations were received in the last five years (2019–2023).

The threshold for the minimum number of citations of documents was set at one, which 221 review studies out of 259 met. Figure  8 visualises the network between these review papers with the largest citation links and Table  9 shows the most cited documents. Not all the studies listed in Table  9 are among the top 10 studies with the highest TLS. Among them, only Gersten et al. ( 2009 ), Cheung and Slavin ( 2008 ), and Slavin and Lake ( 2008 ) are within the top 10 review studies in mathematics education with the highest TLS. While highly cited documents are influential in terms of direct references, the TLS metric provides additional insights into the collaborative relationships and connections between researchers and their work, which may not always correlate perfectly with citation counts as seen in our findings.

Our results showed that the largest number of citation links were for meta-analyses and systematic review studies. The most prominent review type among the most cited studies listed in Table  9 is meta-analysis ( n  = 6), followed by literature review ( n  = 2), systematic review ( n  = 1), and narrative review ( n  = 1). This result indicates the potential of meta-analysis studies in terms of citation performance. Most of these review studies were primarily published in high-ranking educational review journals ( n  = 6). Other review papers published in teacher education ( n  = 2), psychology ( n  = 1), and behavioural science and neuroscience journals ( n  = 1). These ten most cited review articles were all published in SSCI journals over a decade ago. Regarding research topics in the most cited papers, the dominant topics were mathematics achievement, content knowledge, working memory, learning disabilities, and educational technologies.

Specifically, we analysed the citation trends of the most cited 10 review papers over time and separately for the first five years after publication and the past five years (2019–2023). The results indicate a significant increase in the citations review studies have received in the last five years. We found that eight out of the ten most cited papers received more citations in the past five years (2019–2023) than in the first five years after their publication. The analysis revealed that the average annual citations for each paper ranged from 7 to 30. While the majority of these review studies ( n  = 8) received the least citations in the year of their publication, they received the most citations on average approximately 12 years after publication. This indicates that the peak citation period for review articles in mathematics education extends beyond the first decade following their publication.

Additionally, we investigated the ‘Enriched Cited References’ feature, which provides insight into why an author cited a particular reference; this beta enhancement is only available in selected journals (Clarivate, 2024 ). These references are presented to aid readers in quickly assessing sections of a review paper, allowing them to identify the most closely related or impactful references and infer their purpose. Articles containing enriched cited references are marked with the following labels (Clarivate, 2024 ):

Previously published research that contextualizes the current study within an academic domain.

References that supply the datasets, methodologies, concepts, and ideas directly utilized by the author or upon which the author’s work relies.

References introduced because the current study engages in a more thorough discussion.

References cited by the current study as yielding similar results. This may encompass methodological similarities or, in certain instances, replication of findings.

References noted by the current study as presenting contrasting results. This may also involve disparities in methodology or sample differences, influencing the outcomes.

The results, displayed in Table  10 , pertain to the classification of references based on the Enriched Cited References analysis conducted automatically by WoS. These results suggest that the most cited review studies in mathematics education were predominantly utilized by researchers to establish the background for their own research. Furthermore, these reviews also frequently employed to shape the discussion within the papers. In addition, some researchers utilize the mentioned most cited review studies to establish a conceptual, theoretical, or methodological basis. While the limited number of the studies cited these reviews to support their findings, they were not used to present opposing evidence. This suggests a reliance on existing literature review studies to inform, validate, or potentially challenge new research within the field.

Citation and source analysis

We conducted a citation source analysis and present the citation network map for the journals in Fig.  9 , listing the top 15 journals in Table  11 based on the citation and TLS metrics to represent the frequency of citations between articles in any two journals. The threshold for the minimum number of documents citing a source was one, and 103 records met the minimum number of citations of a source, also set at one. The network map shown in Fig.  9 indicates prominent clusters. The red cluster included 23 items (mostly special education, educational psychology, and educational review journals). The blue cluster included 16 items (predominantly educational psychology, educational technology, and educational review journals). The green cluster comprised 17 items (including mathematics and mathematics education journals, educational technology journals, and educational psychology journals).

The number of articles and the distribution of journals across various research fields were as follows: 25 educational sciences journals (43 papers), 20 psychology and educational psychology journals (41 papers), 15 special education journals (32 papers), 12 mathematics education journals (52 papers), 10 educational review journals (41 papers), 9 educational technology journals (28 papers), 3 mathematics journals (14 papers), and 9 other journals (8 articles).

Our findings indicate that ZDM– Mathematics Education ( n  = 16) has, so far, published the most review studies focusing on mathematics education, which is not unexpected due to the origin of the journal as a review journal publishing only special issues, for which a review article is compulsory in each issue. This was followed by Frontiers in Psychology ( n  = 14), Educational Research Review ( n  = 13), and Mathematics ( n  = 10) (see Table  11 for the top 15 journals).

The results highlighted that the most frequently cited papers were often published in specific educational review journals (e.g. Review of Educational Research , Educational Research Review , and Educational Psychology Review ), psychology and educational psychology journals (e.g. Frontiers in Psychology , Educational Psychology Review , European Journal of Cognitive Psychology , and Psychological Bulletin ), special education journals (e.g. Exceptional Children , Learning Disabilities Research & Practice , Learning Disability Quarterly , and Remedial and Special Educati on), educational technology journals (e.g. Computers & Education , Journal of Computer Assisted Learning , and Education and Information Technologies ), and mathematics and mathematics education journals (e.g. ZDM– Mathematics Education , Educational Studies in Mathematics , and Mathematics ).

Although the most visible mathematics education journals in citation network map were ZDM– Mathematics Education and Educational Studies in Mathematics (see Fig.  9 ), as mentioned earlier, twelve mathematics education journals provided platforms for review studies. These were ZDM– Mathematics Education ( n  = 16), Educational Studies in Mathematics ( n  = 5), International Journal of Science and Mathematics Education ( n  = 5), International Journal of Mathematical Education in Science and Technology ( n  = 5), International Electronic Journal of Mathematics Education ( n  = 3), Mathematics Education Research Journal ( n  = 3), International Journal for Technology in Mathematics Education ( n  = 3), International Journal of Education in Mathematics, Science and Technology ( n  = 3), Journal for Research in Mathematics Education ( n  = 2), Canadian Journal of Science, Mathematics and Technology Education ( n  = 1), Journal für Mathematik-Didaktik ( n  = 1), and Research in Mathematics Education ( n  = 1).

5 Discussion, conclusions, and limitations

In this study, we conducted a meta-review of literature review studies in mathematics education, enriched by a comprehensive bibliometric analysis. This paper significantly contributes to scholarly discourse by unravelling nuanced research trends, the most common review methodologies, and prevalent theoretical approaches in review studies in mathematics education. Based on content and bibliometric analysis, it delves into the research foci, providing an understanding of the relevant academic landscape. Additionally, it illuminates intricate connections among researchers, countries, and journals, elucidating collaborative networks in mathematics education research.

5.1 Insights from the meta-review and implications

The findings revealed a significant increase in the number of literature reviews in mathematics education, particularly in the past five years; 79% of the reviews we examined were published during this period. Multiple factors may have contributed to this surge, including researchers’ increased publication output during the pandemic (Cevikbas & Kaiser, 2023 ; Nane et al., 2023 ), challenges in collecting empirical data during the pandemic crisis (Uleanya & Yu, 2023 ), the relatively high citation rates associated with literature review studies, the growing prestige of educational review journals based on their increased impact factors (Miranda & Garcia-Carpintero, 2018 ), and the publication of review-oriented special issues in mathematics education journals.

Our findings revealed a prevalence of systematic reviews and meta-analyses; however, researchers also conducted diverse types of reviews, including scoping reviews, critical reviews, narrative reviews, theoretical reviews, and tutorial reviews. This methodological diversity is important as the advantages of one method can potentially overcome the disadvantages of another and combining different approaches can mitigate disadvantages (Taherdoost, 2023 ). Furthermore, our study revealed that rapid reviews, meta-reviews, umbrella reviews, mapping reviews, mixed-methods reviews, integrative reviews, interpretative reviews, iterative reviews, meta-syntheses, descriptive reviews, realist reviews, selective reviews, algorithmic reviews, and computational reviews indexed in WoS were not represented in mathematics education. The well-established PRISMA guidelines offer a defined framework for systematic reviews and meta-analyses to assist researchers in conducting reviews while adhering to quality and transparency criteria (Moher et al., 2009 ; Page et al., 2021 ). This adherence may have encouraged researchers to undertake such reviews, and future advancements in the development of specific guidelines and methodologies for each review type may further motivate researchers to conduct other types of reviews in mathematics education more frequently.

There were nuanced overlaps between the review types, leading to ambiguous distinctions. For instance, the structural similarity between systematic reviews and scoping reviews has led to misunderstandings. Munn et al. ( 2018 ) confirm inconsistency and confusion regarding the differentiation between scoping reviews and systematic reviews and offered guidelines for this decision-making process: a systematic review is preferable when addressing specific questions regarding the feasibility, appropriateness, significance, or efficacy of a specific treatment or practice. However, if the authors intend to demarcate the research field and explore its potential size and scope, a scoping review is more appropriate. Grant and Booth ( 2009 ) and Munn et al. ( 2018 ) clarified that a scoping review is preparation for a systematic literature review, not a deep study for a systematic literature review. The diverse taxonomies proposed by researchers have contributed to this complexity, with some employing various terms for similar review characteristics, and others applying the same terms to studies with distinct review attributes. Consequently, a consensus regarding the categorisation of review studies, both in a broad context and specifically in mathematics education, remains elusive. We also observed instances of researchers labelling their reviews inaccurately. However, we refrained from judging the appropriateness of these terminologies as they fall outside the scope of our study and may be difficult to justify due to the ambiguity of the current typologies. Borges Migliavaca et al. ( 2020 ) expressed a similar concern, highlighting substantial disparities in review studies concerning their conceptualisation, conduct, reporting, risk of bias assessment, and data synthesis. They called for the evidence synthesis community to promptly develop guidance and reporting standards for review studies. Future researchers could potentially examine inconsistencies in the conducting of review studies and their categorisation in mathematics education. In this study, we distilled the various existing types of review studies to provide clear explanations of the main review types and to help researchers and readers understand the key characteristics of various review studies (see Chap. 2).

An additional noteworthy consideration pertains to the sample sizes of review studies. A prevalent discourse considers the appropriate number of studies to be included in a review, but establishing such a minimum or maximum number may be challenging and not appropriate because this depends on various contextual factors, such as the research area, topic, inclusion/exclusion criteria, and applied protocols. For example, in technical terms, a systematic review can be conducted with as few as two studies or as many as a thousand. A review study with a small sample (e.g. two or three studies) may be due to the literature search methods used or insufficient number of existing studies in a particular field, suggesting a limited demand for such a review. As previously noted, the primary function of review studies is to inform readers in the relevant field about published studies to address the challenge posed by an increasing number of studies and to identify trends and research gaps (Fusar-Poli & Radua, 2018 ). Conversely, although it is technically feasible to include a substantial number of studies in a review (e.g. 1,000 or 2,000), conducting a comprehensive analysis (e.g. content analysis) of such a large dataset can present major time, cost, storage, memory, bias, and security challenges (Cohen et al., 2015 ). Nevertheless, the findings of our study provide insight into this issue. Notably, the sample size of the studies we analysed varied from 8 to 3,485, with an average of 99. Notably, most of these studies (78%) had sample sizes of less than 100, with an average of 34. Although this observation does not serve as a prescriptive recommendation, it offers valuable insights into the typical sample sizes with which mathematics education researchers have tended to work in the past.

Furthermore, as evidenced by our findings, literature reviews may serve various purposes, such as assessing the use of theoretical models or conceptual and methodological approaches, or advancing new theories, concepts, or research models through critical appraisal of previous research within a specific subject area (Cooper, 1988 ). However, our findings also indicate that it is not common in practice to use or develop a theoretical or conceptual framework in mathematics education review studies. Only 24% of the reviewed studies explicitly reported employing a specific framework, and very few sought to formulate a framework based on the literature under scrutiny. The results highlighted the researchers’ interest in frameworks related to technology, knowledge, and competence models. A few studies incorporated grand theories, such as constructivism, sociocultural theory, and cognitive development theory.

It is remarkable that despite focusing on mathematics education, there is a notable scarcity of review studies employing content-specific frameworks in mathematics education, such as those centred on problem-solving, reasoning, and mathematical thinking. Only a minority of the studies used frameworks related to mathematical modelling and mathematical content knowledge. This observation may reflect a gap in the literature, suggesting a need for greater integration of domain-specific frameworks into review studies in mathematics education to enhance the depth and specificity of the studies. Moreover, this trend prompts a critical examination of potential underlying factors. One plausible explanation lies in the interdisciplinary nature of review studies in mathematics education, which draws contributions from diverse fields including psychology, educational technology, special education, and neuroscience. The diverse disciplinary backgrounds of the researchers may influence their preferences for frameworks that are not necessarily specific to mathematics education but rather draw from broader fields.

5.2 Insights from the bibliometric analyses and implications

The bibliometric analysis revealed contributions to mathematics education, with 761 authors from 50 countries conducting review studies. In future studies, researchers may consider conducting detailed analyses of how these initiatives have influenced the landscape of mathematics education, examining their specific impacts on various subfields, and assessing their overall influence.

Our findings reveal a notable participation in literature review studies within mathematics education by scholars from diverse backgrounds, including educational psychologists, mathematics educators, and specialists in special education and neuroscience. This multidisciplinary engagement underscores the broader interest of researchers beyond the field of mathematics education. Notably, co-authorship connections within US institutions were the most extensive. The leading countries that published review studies included the US, Germany, China, Australia, and England. A robust network emerged among researchers in North America, Europe, Asia, and Australia, emphasising collaboration opportunities that warrant exploration by African and South American researchers.

Systematic reviews and meta-analyses stood out as the predominant review types in mathematics education, both in terms of the number of publications and citation counts. Systematic reviews offer rigorous and comprehensive syntheses of existing literature on specific research questions, providing valuable insights, identifying gaps in knowledge, and informing evidence-based decision-making in various fields. Moreover, meta-analyses enhance statistical power, resolve conflicting findings, and offer more precise estimates of effect sizes by combining data from various sources. However, there is a discernible need to diversify the types of reviews conducted in mathematics education.

The findings underscore a significant surge in both the quantity of review studies and their citation counts within mathematics education especially over the recent five-year period (2019–2023). This trend suggests a prevalent practice among authors to draw upon previously published reviews to contextualize their own studies, frequently engaging in discussions and citing references to corroborate or challenge existing findings. Such reliance on established literature highlights the discipline’s emphasis on leveraging prior knowledge to inform and substantiate new research endeavours.

The most cited review papers were associated with specific educational review journals, educational psychology journals, special education journals, educational technology journals, and mathematics education journals, further highlighting the interdisciplinary nature of impactful research in the field. The results revealed that ZDM– Mathematics Education , Educational Studies in Mathematics , International Journal of Science and Mathematics Education , and International Journal of Mathematical Education in Science and Technology were the key mathematics education journals committed to publishing review studies. The performance of these journals, particularly in recent years, reflects the escalating significance of review studies in mathematics education. Nevertheless, the limited visibility of some mathematics education journals in publishing review studies could be attributed, among other factors, to their restricted representation in the WoS database or to the overall small number of studies published yearly in particular mathematics education journals.

Prominent research topics in mathematics education review studies are digital technologies, technology-enhanced approaches (e.g. flipped classrooms), teacher education, mathematics achievement, early childhood education, and learning disabilities. Recent technological advances, including artificial intelligence and augmented/virtual reality, may soon attract mathematics education researchers’ attention to emerging technologies (Cevikbas, Bulut et al., 2023 ; Cevikbas, Greefrath et al., 2023 ). In addition to technology-enhanced mathematics education and special education, researchers have also explored the cognitive and affective aspects of learning and teaching mathematics.

In short, the absence of high-quality research syntheses may impede theoretical and conceptual advances within mathematics education (Webster & Watson, 2002 ). Therefore, future researchers may endeavour to develop discipline-specific standards and guidelines for conducting various types of review studies in mathematics education. Moreover, they could focus on expanding the content of mathematics education journals to accommodate a greater number of review studies. The scientific influence of review journals may also provide an opportunity to establish a dedicated review journal with a pronounced focus on mathematics education.

5.3 Limitations and conclusion

Finally, we want to point out that in this comprehensive meta-review, enriched by bibliometric analysis, we meticulously compiled and scrutinised the largest dataset of reviews in mathematics education available within the WoS database. Although this was a substantial sample ( n  = 259) that was reasonably representative of published review studies in mathematics education, it is important to acknowledge certain limitations. Our search was confined to WoS, and we specifically focused on review articles published in English. It is worth noting that the characteristics of review studies published in journals, international handbooks, or conference proceedings not indexed in WoS or published in a language other than English could potentially differ from those we examined. In addition, despite studies indexed in WoS theoretically being of high quality, we identified inconsistencies and variability in the review studies we examined, and it is possible that a more extensive search would have yielded different results.

In conclusion, we advocate producing high-quality review papers that adeptly synthesise available knowledge to improve professional practice (Templier & Paré, 2015 ). Such efforts may further advance mathematics education and contribute to the continuous improvement of teaching and learning activities, despite the demanding nature of comprehensive review studies.

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• 14 May 2024

Why mathematics is set to be revolutionized by AI

• Thomas Fink 0

Thomas Fink is the director of the London Institute for Mathematical Sciences, UK.

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Giving birth to a conjecture — a proposition that is suspected to be true, but needs definitive proof — can feel to a mathematician like a moment of divine inspiration. Mathematical conjectures are not merely educated guesses. Formulating them requires a combination of genius, intuition and experience. Even a mathematician can struggle to explain their own discovery process. Yet, counter-intuitively, I think that this is the realm in which machine intelligence will initially be most transformative.

In 2017, researchers at the London Institute for Mathematical Sciences, of which I am director, began applying machine learning to mathematical data as a hobby. During the COVID-19 pandemic, they discovered that simple artificial intelligence (AI) classifiers can predict an elliptic curve’s rank 1 — a measure of its complexity. Elliptic curves are fundamental to number theory, and understanding their underlying statistics is a crucial step towards solving one of the seven Millennium Problems, which are selected by the Clay Mathematics Institute in Providence, Rhode Island, and carry a prize of US$1 million each. Few expected AI to make a dent in this high-stakes arena.

AI now beats humans at basic tasks — new benchmarks are needed, says major report

AI has made inroads in other areas, too. A few years ago, a computer program called the Ramanujan Machine produced new formulae for fundamental constants 2 , such as π and e . It did so by exhaustively searching through families of continued fractions — a fraction whose denominator is a number plus a fraction whose denominator is also a number plus a fraction and so on. Some of these conjectures have since been proved, whereas others remain open problems.

Another example pertains to knot theory, a branch of topology in which a hypothetical piece of string is tangled up before the ends are glued together. Researchers at Google DeepMind, based in London, trained a neural network on data for many different knots and discovered an unexpected relationship between their algebraic and geometric structures 3 .

How has AI made a difference in areas of mathematics in which human creativity was thought to be essential?

First, there are no coincidences in maths. In real-world experiments, false negatives and false positives abound. But in maths, a single counterexample leaves a conjecture dead in the water. For example, the Pólya conjecture states that most integers below any given integer have an odd number of prime factors. But in 1960, it was found that the conjecture does not hold for the number 906,180,359. In one fell swoop, the conjecture was falsified.

Second, mathematical data — on which AI can be trained — are cheap. Primes, knots and many other types of mathematical object are abundant. The On-Line Encyclopedia of Integer Sequences (OEIS) contains almost 375,000 sequences — from the familiar Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) to the formidable Busy Beaver sequence (0, 1, 4, 6, 13, …), which grows faster than any computable function. Scientists are already using machine-learning tools to search the OEIS database to find unanticipated relationships.

DeepMind AI outdoes human mathematicians on unsolved problem

AI can help us to spot patterns and form conjectures. But not all conjectures are created equal. They also need to advance our understanding of mathematics. In his 1940 essay A Mathematician’s Apology , G. H. Hardy explains that a good theorem “should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds”. In other words, the best theorems increase the likelihood of discovering new theorems. Conjectures that help us to reach new mathematical frontiers are better than those that yield fewer insights. But distinguishing between them requires an intuition for how the field itself will evolve. This grasp of the broader context will remain out of AI’s reach for a long time — so the technology will struggle to spot important conjectures.

But despite the caveats, there are many upsides to wider adoption of AI tools in the maths community. AI can provide a decisive edge and open up new avenues for research.

Mainstream mathematics journals should also publish more conjectures. Some of the most significant problems in maths — such as Fermat’s Last Theorem, the Riemann hypothesis, Hilbert’s 23 problems and Ramanujan’s many identities — and countless less-famous conjectures have shaped the course of the field. Conjectures speed up research by pointing us in the right direction. Journal articles about conjectures, backed up by data or heuristic arguments, will accelerate discovery.

Last year, researchers at Google DeepMind predicted 2.2 million new crystal structures 4 . But it remains to be seen how many of these potential new materials are stable, can be synthesized and have practical applications. For now, this is largely a task for human researchers, who have a grasp of the broad context of materials science.

Similarly, the imagination and intuition of mathematicians will be required to make sense of the output of AI tools. Thus, AI will act only as a catalyst of human ingenuity, rather than a substitute for it.

Nature 629 , 505 (2024)

doi: https://doi.org/10.1038/d41586-024-01413-w

He, Y.-H., Lee, K.-H., Oliver, T. & Pozdnyakov, A. Preprint at arXiv https://doi.org/10.48550/arXiv.2204.10140 (2024).

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Writing math research papers: a guide for students and instructors.

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Robert Gerver

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Writing Math Research Papers  is primarily a guide for high school students that describes how to write aand present mathematics research papers. But it’s really much more than that: it’s a systematic presentation of a philosophy that writing about math helps many students to understand it, and a practical method to move students from the relatively passive role of someone doing what is assigned to them, to creative thinkers and published writers who contribute to the mathematical literature.

As experienced writers know, the actual writing is not the half of it. William Zinsser once taught a writing class at the New School for Social Research which involved no writing at all: students talked through their ideas in class and through that process discovered the real story which could be written from their tangle of experiences, hopes and dreams. The actual writing was secondary, once they understood how to find the story and organize it.

Gerver, an experienced high school mathematics teacher, takes a similar approach. The primary audience is high school students who want to prepare formal papers or presentations, for contests or for a “math day” at their high school. But the discovery, research and organizational processes involved in writing an original paper, as opposed to rehashing information from a reference book, can help any student learn and understand math, and the experience will be useful even if the paper is never written.

Gerver leads students through a discovery process beginning with examining their own knowledge of mathematics and reviewing the basics of problem solving. The “math annotation” project follows next, in which students organize their class notes for one topic for presentation to their peers, resulting in a product similar to a section of a textbook or handbook, complete with illustrations and the necessary background and review material. Practical advice about finding a topic, developing it by keeping a research journal, and creating a final product, either a research paper or oral presentation, follows.

Writing Math Research Papers  is directed primarily to students, and could be assigned as a supplementary textbook for high school mathematics classes. It will also be useful to teachers who incorporate writing into their classes or who serve as mentors to the math club, and for student teachers in similar situations. An appendix for teachers includes practical advice about helping students through the research and writing process, organizing consultations, and grading the student papers and presentations. Excerpts from student research papers are included as well, and more materials are available from the web site www.keypress.com/wmrp .

Robert Gerver, PhD, is a mathematics instructor at North Shore High School in New York. He received the Presidential Award for Excellence in Mathematical Teaching in 1988 and the Tandy Prize and Chevron Best Practices Award in Education in 1997. He has been publishing mathematics. Dr. Gerver has written eleven mathematics textbooks and numerous articles, and holds two U.S. patents for educational devices.

Sarah Boslaugh, ( [email protected] ) is a Performance Review Analyst for BJC HealthCare and an Adjunct Instructor in the Washington University School of Medicine, both in St. Louis, MO. Her books include An Intermediate Guide to SPSS Programming: Using Syntax for Data Management  (Sage, 2004), Secondary Data Sources for Public Health: A Practical Guide (Cambridge, 2007), and Statistics in a Nutshell (O'Reilly, forthcoming), and she is Editor-in-Chief of The Encyclopedia of Epidemiology (Sage, forthcoming).

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• 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

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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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Bifurcation and chaos in simple discontinuous systems separated by a hypersurface

• Hany A. Hosham 1 ,  ,  , 
• Thoraya N. Alharthi 2
• 1. Department of Mathematics, Faculty of Science, Taibah University, Yanbu, 41911, Saudi Arabia
• 2. Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia
• Received: 13 March 2024 Revised: 22 April 2024 Accepted: 09 May 2024 Published: 16 May 2024

MSC : 34A36, 34D23, 37G15, 34H10

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This research focuses on a mathematical examination of a path to sliding period doubling and chaotic behaviour for a novel limited discontinuous systems of dimension three separated by a nonlinear hypersurface. The switching system is composed of dissipative subsystems, one of which is a linear systems, and the other is not linked with equilibria. The non-linear sliding surface is designed to improve transient response for these subsystems. A Poincaré return map is created that accounts for the existence of the hypersurface, completely describing each individual sliding period-doubling orbits that route to the sliding chaotic attractor. Through a rigorous analysis, we show that the presence of a nonlinear sliding surface and a set of such hidden trajectories leads to novel bifurcation scenarios. The proposed system exhibits period-$ m $ orbits as well as chaos, including partially hidden and sliding trajectories. The results are numerically verified through path-following techniques for discontinuous dynamical systems.

• discontinuous systems ,
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• period-doubling ,
• sliding mode ,

Citation: Hany A. Hosham, Thoraya N. Alharthi. Bifurcation and chaos in simple discontinuous systems separated by a hypersurface[J]. AIMS Mathematics, 2024, 9(7): 17025-17038. doi: 10.3934/math.2024826

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• Figure 1. Family of flat periodic orbits located in the $ (x, y)- $ plane with constant time $ T = 2\pi/\beta $
• Figure 2. Numerical simulations of subsystems (3.4) and (3.6) for varying values of $ \alpha_3 $. Figure (a), (b) at $ \alpha_3 = -0.2 $, demonstrating single periodic orbit. Figure (c), (d) at $ \alpha_3 = -0.11 $, demonstrating period-doubling orbits. Figure (e), (f) at $ \alpha_3 = -0.1 $ demonstrating period-4 orbits. Figure (g), (h) at $ \alpha_3 = -0.011 $ demonstrating chaotic behavior
• Figure 3. The system establishes a sliding period doubling orbit at $ \alpha_1 = 0.402 $ (Figure 3(a), (b)), presenting a path to chaos at $ \alpha_1 = 0.385 $ (Figure 3(c), (d))
• Figure 4. A shifting hypersurface results in (a), (b) a unique periodic orbit at $ \gamma_3 = 0.811 $. (c), (d) Period-doubling orbit at $ \gamma_3 = 0.611 $. (e), (f) Period-4 orbit at $ \gamma_3 = 0.511 $
• Figure 5. Varying the parameter $ \alpha_2 $ results in (a) a unique periodic orbit at $ \alpha_2 = -1.0 $. (b) Period-doubling orbit at $ \alpha_2 = -1.1 $. (c), Period-3 orbits at $ \alpha_2 = -1.15 $. (d) chaotic attractor involving a sliding mode at $ \alpha_2 = -1.17 $
• Figure 6. Period-$ m $ orbit and chaotic attractor of the discontinuous system when the subsystem (3.6) has infinite equilibria. (a), (b) a period-6 orbit involving a sliding mode at $ \alpha_4 = 0.67 $. (c), (d) chaotic attractor involving a sliding mode at $ \alpha_4 = 0.7 $

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Title: orca-math: unlocking the potential of slms in grade school math.

Abstract: Mathematical word problem-solving has long been recognized as a complex task for small language models (SLMs). A recent study hypothesized that the smallest model size, needed to achieve over 80% accuracy on the GSM8K benchmark, is 34 billion parameters. To reach this level of performance with smaller models, researcher often train SLMs to generate Python code or use tools to help avoid calculation errors. Additionally, they employ ensembling, where outputs of up to 100 model runs are combined to arrive at a more accurate result. Result selection is done using consensus, majority vote or a separate a verifier model used in conjunction with the SLM. Ensembling provides a substantial boost in accuracy but at a significant cost increase with multiple calls to the model (e.g., Phi-GSM uses top-48 to boost the performance from 68.2 to 81.5). In this work, we present Orca-Math, a 7-billion-parameter SLM based on the Mistral-7B, which achieves 86.81% on GSM8k without the need for multiple model calls or the use of verifiers, code execution or any other external tools. Our approach has the following key elements: (1) A high quality synthetic dataset of 200K math problems created using a multi-agent setup where agents collaborate to create the data, (2) An iterative learning techniques that enables the SLM to practice solving problems, receive feedback on its solutions and learn from preference pairs incorporating the SLM solutions and the feedback. When trained with Supervised Fine-Tuning alone, Orca-Math achieves 81.50% on GSM8k pass@1 metric. With iterative preference learning, Orca-Math achieves 86.81% pass@1. Orca-Math surpasses the performance of significantly larger models such as LLAMA-2-70B, WizardMath-70B, Gemini-Pro, ChatGPT-3.5. It also significantly outperforms other smaller models while using much smaller data (hundreds of thousands vs. millions of problems).

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Century of statistical ecology reviewed

Crunching numbers isn't exactly how Neil Gilbert, a postdoctoral researcher at Michigan State University, envisioned a career in ecology.

"I think it's a little funny that I'm doing this statistical ecology work because I was always OK at math, but never particularly enjoyed it," he explained. "As an undergrad, I thought, I'll be an ecologist -- that means that I can be outside, looking at birds, that sort of thing."

As it turns out," he chuckled, "ecology is a very quantitative discipline."

Now, working in the Zipkin Quantitative Ecology lab, Gilbert is the lead author on a new article in a special collection of the journal Ecology that reviews the past century of statistical ecology .

Statistical ecology, or the study of ecological systems using mathematical equations, probability and empirical data, has grown over the last century. As increasingly large datasets and complex questions took center stage in ecological research, new tools and approaches were needed to properly address them.

To better understand how statistical ecology changed over the last century, Gilbert and his fellow authors examined a selection of 36 highly cited papers on statistical ecology -- all published in Ecology since its inception in 1920.

The team's paper examines work on statistical models across a range of ecological scales from individuals to populations, communities, ecosystems and beyond. The team also reviewed publications providing practical guidance on applying models. Gilbert noted that because, "many practicing ecologists lack extensive quantitative training," such publications are key to shaping studies.

Ecology is an advantageous place for such papers, because it is one of, "the first internationally important journals in the field. It has played an outsized role in publishing important work," said lab leader Elise Zipkin, a Red Cedar Distinguished Associate Professor in the Department of Integrative Biology.

"It has a reputation of publishing some of the most influential papers on the development and application of analytical techniques from the very beginning of modern ecological research."

The team found a persistent evolution of models and concepts in the field, especially over the past few decades, driven by refinements in techniques and exponential increases in computational power.

"Statistical ecology has exploded in the last 20 to 30 years because of advances in both data availability and the continued improvement of high-performance computing clusters," Gilbert explained.

Included among the 36 reviewed papers were a landmark 1945 study by Lee R. Dice on predicting the co-occurrence of species in space -- Ecology's most highly cited paper of all time -- and an influential 2002 paper led by Darryl MacKenzie on occupancy models. Ecologists use these models to identify the range and distribution of species in an environment.

Mackenzie's work on species detection and sampling, "played an outsized role in the study of species distributions," says Zipkin. MacKenzie's paper, which was cited more than 5,400 times, spawned various software packages that are now widely used by ecologists, she explained.

• Environmental Issues
• Computer Modeling
• Mathematical Modeling
• Origin of Life
• Early Climate
• Artificial intelligence
• Computational genomics
• Albert Einstein
• Bioinformatics
• Mathematical model
• Water turbine
• Numerical weather prediction

Story Source:

Materials provided by Michigan State University . Original written by Caleb Hess. Note: Content may be edited for style and length.

Journal Reference :

• Neil A. Gilbert, Bruna R. Amaral, Olivia M. Smith, Peter J. Williams, Sydney Ceyzyk, Samuel Ayebare, Kayla L. Davis, Wendy Leuenberger, Jeffrey W. Doser, Elise F. Zipkin. A century of statistical Ecology . Ecology , 2024; DOI: 10.1002/ecy.4283

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