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The impact of the COVID-19 pandemic on mathematics learning in higher education during learning from home (LFH): students' views for the new normal

R Y Tyaningsih 1 , Arjudin 1 , S Prayitno 1 , Jatmiko 2 and A D Handayani 2

Published under licence by IOP Publishing Ltd Journal of Physics: Conference Series , Volume 1806 , International Conference on Mathematics and Science Education (ICMScE) 2020 14-15 July 2020, Jawa Barat, Indonesia Citation R Y Tyaningsih et al 2021 J. Phys.: Conf. Ser. 1806 012119 DOI 10.1088/1742-6596/1806/1/012119

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1 Departemen Pendidikan Matematika, Universitas Mataram, Jl. Majapahit No. 62 Mataram, Indonesia

2 Departemen Pendidikan Matematika, Universitas Nusantara PGRI Kediri, Jl. KH. Achmad Dahlan No. 76 Kediri, Indonesia

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Covid-19 Pandemic has an impact on education that is bringing up new policies for learning from home. The purpose of this study was to determine students' views on learning mathematics in higher education while learning from home and its sustainability towards a new normal. This type of research is qualitative with data collection techniques using online surveys and interviews. Respondents in this study were 169 students of Mathematics Education in Higher Education. The results of this study indicate that 100% of lectures are conducted online with a composition of 34.32% in the form of Asynchronous Online Course, 19.53% Synchronous Online Course, and 46.15% using Hybrid Online Course. The effectiveness of online course that has been carried out was obtained 7.1% very effective, 20.1% effective, 53.8% sufficient, 17.8% less effective, and 1.2% ineffective. Open questions are provided in the questionnaire to allow students to convey constraints/difficulties experienced during Learning from Home (LFH), including network constraints, health problems, costs, the environment, course constraints, and time problems. Student responses regarding the application of the Blended Learning method in the new normal period showed 48% agreed, 31% were doubtful, and 21% disagreed.

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Article Contents

1 school and university transitions, 2 mathematics support, 3 mapping the effect of changing the medium of provision, 4 closing remarks, special issue editorial: restarting the new normal.

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Jonathan Gillard, Claire Ketnor, Ciarán Mac an Bhaird, Cathy Smith, Special issue editorial: restarting the new normal, Teaching Mathematics and its Applications: An International Journal of the IMA , Volume 40, Issue 4, December 2021, Pages 249–253, https://doi.org/10.1093/teamat/hrab026

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We are delighted to present this special issue of Teaching Mathematics and Its Applications with the title ‘Restarting the New Normal’. The title itself is an oxymoron since, at the time of conception, it was far from clear whether mathematics teaching would be restarting as before after a significant interruption or transforming to a new normal. The brief of this issue was to consider papers on research in one or more of the following areas: approaches to teaching post-16 mathematics to students during COVID-19 restrictions, the needs of mathematics learners in COVID-19-affected cohorts and general distance learning of mathematics.

Three guest editors, supported and led by an existing editor, began work on the special edition in September 2020. Through this edition we aimed to encourage research and reflection that directly informs future practice. We are appreciative of the effort of the authors, particularly because the papers were written at a time when we were all still affected by the impact of COVID-19.

Teaching during 2020 and 2021 involved quick professional development and learning for staff and students, even for the most technically competent among us. Many were suddenly teaching or supporting students in an environment we were not used to or prepared for. Internationally, there were numerous collaborative events, such as Teaching and Learning Mathematics Online ( http://talmo.uk/ ), which allowed for dialogue between innovators and those with existing experience of distance learning provision. Never before has there been a significant shift in such a short period of time within the teaching of post-16 mathematics.

The early stages of COVID-19 caused the closure of school, college and university campuses globally and led to an urgent requirement to instigate alternative ways of learning and teaching. The term ‘emergency remote teaching’ is often used to cover this initial response. Several of the authors in this special edition point to Hodges et al . (2020) who summarize emergency remote teaching as ‘a temporary shift of instructional delivery to an alternative delivery mode due to crisis circumstances’ that contrasts with ‘experiences that are planned from the beginning and designed to be online’. Remarkably, given the circumstances of that period, practitioners quickly started to investigate and evaluate these initial responses.

As it became apparent that online or remote teaching would continue in the latter parts of 2020 and beyond, the benefit of moving quickly to collate relevant research on best practices was clear. As a result, there are three types of paper within this issue: those focussed on the sudden shift that required instant changes to teaching practice, those considering more established approaches (including from the Open University who already taught online) and those that study iterations aiming to improve an initial design.

The first call for this special issue was in May 2020 and submissions were due in February 2021. Within a short window for submissions, there is naturally less time for planning and implementing research. However, the editors and reviewers were satisfied that the researchers have collected and analysed data to evidence the different effects of COVID-19 that they were investigating. In most cases, studies were undertaken at one institution, with the data analysed ranging from surveys and interviews to measures of student engagement and attainment. Covariables such as gender and ethnicity are also studied. A notable feature of the editing process for this special issue has been the collective refinement of claims and arguments. Reviewers and authors proved open to critiquing the relevance and generalizability of the results with reference to developing teaching practices and issues of professional and intellectual significance.

Time is a premium for everyone, especially during the pandemic, with colleagues juggling several commitments and high workloads. We were impressed in having received exceptional papers, in such exceptional times and in the quality of reviews. We enjoyed reading and looking after them during all the stages of the publication process. We hope you acknowledge, as we do, that these papers have captured incredible efforts that were made by teaching staff and students to provide an enriching learning environment. At time of writing, September 2021, it is clear that the papers not only offer the opportunity to look back at a unique and memorable time in history, but also start a process of recording the deliberations that underpin how we will navigate new teaching considerations and circumstances in mathematics.

Despite many difficulties within the COVID-19 pandemic, there may be some positive consequences. Initial research suggests that young people are now more interested in science careers as a result of COVID-19 ( British Science Association, 2020 ), and one may assume that this could translate to additional students of mathematics in the near future. As we identify below, positive aspects can be seen within this special edition, several of which would have possibly remained unknown if it had not been for the pandemic.

Before briefly describing the contents of the special issue, we make the following remarks, which should be kept in mind, since they may add context to the work described within:

Different countries were, and are, operating under different COVID-19 circumstances and restrictions at any given point in time.

Universities do not have the same timings in the academic year, and our colleagues in countries such as Australia were only part way into their academic year when they were first affected by COVID-19.

There is no one-size-fits-all approach to adjusting education in the time of a pandemic. Adjustments made are likely to be influenced by several factors, including staff expertise and the equipment available to them, the rules and regulations bestowed upon staff by their employer and government and the make-up of the student cohort.

The papers within this issue broadly concern topics such as school and university transitions, mathematics support and mapping the effect of changing the medium of provision. These headings are not exhaustive, and indeed it can be argued that several of the included papers fall under more than one of these classifications. The descriptions that follow are not an attempt to summarize the papers but to give an indication of how we classified them into each topic.

The paper by Hodds compares the entry competences of students arriving at a UK university in October 2020 with those who entered in previous years, by use of a common diagnostic test. Despite this paper offering some positive news, its results are in contrast with those of Golding who did note a decrease in mathematical preparedness, a profound impact upon the learning experience of the 16–18-year-old advanced mathematics students and a stark decline in their confidence. While these papers address mathematical preparedness within the curriculum, Lyakhova et al . point out the effect of COVID-19 on mathematics outreach events and report on student engagement with video materials offered to 16–18 year olds in Wales. The two papers focussing on school-age students both warn of disparate experiences, with a few students adapting successfully to a combination of asynchronous resources and interactive support while many struggled to learn mathematics at a distance despite their familiarity with technology in other guises. Papers in this section also raise equations about the purpose and effectiveness of current school-leaving examinations.

The paper by Gilbert et al . talks of the adjustments made to mathematics and statistics support services, describing results of a questionnaire distributed internationally early in the pandemic and follow-up interviews with practitioners 7 months later. They acknowledge some merits of support provision taking place online, but envisage that face-to-face support will return as the dominant mode of provision when circumstances allow. Crowley et al .’s evaluation of online mathematics support argues that such provision will become increasingly important and valuable in the future. Analysis of student interactions with their online mathematics support materials showed high engagement levels, but there could be some confounding variables partially explaining this success. Mullen et al . compare student and tutor perspectives on mathematics support in two institutions, one in Ireland and one in Australia. Responses in interviews mainly revolved around five key common themes. Finally, Mac an Bhaird et al . reflect on undergraduate experiences of online study groups and drop-in mathematics support, describing what they felt to be the positive and negative aspects of the online provision. The study groups were a new initiative as a reaction to low student engagement with drop-in sessions. The findings are used by the authors to suggest modes of future support.

Technology has a key role in underpinning the adjustments made to the delivery of teaching. Hilliam et al . offer their experiences from the Open University, who are well versed in online and distance learning. They summarize and evaluate their attempts before the pandemic to improve consistency of academic and pastoral support for all of their students. This paper discusses some issues and solutions that other practitioners are likely to meet as they develop their online provision.

Several papers focus on the early pedagogic issues and solutions trialled by mathematics lecturers, both in the emergency phase and as practice settled. Ní Fhloinn and Fitzmaurice describe the results of an international survey asking practitioners what hardware and software they have used and what training and support was made available to them. They also tried to understand the rationale of the choices made by those surveyed regarding the live and pre-recorded lectures. Lishchynska et al . compare face-to-face small-group tutorial delivery with a virtual alternative. Key issues were described, and factors to consider when developing future remote delivery are offered. Finally, Kempen and Liebendörfer investigated students’ reported use of a variety of teaching resources in a newly online linear algebra course in Germany. They offered the valuable comment, easily forgotten, that enabling social contact between students is important.

The impact of COVID-19 teaching on students should be considered over different timescales. Focussing on the immediate impact, Hyland and O’Shea research student perspectives on COVID-19 closures at Irish universities. The surveyed students also describe how they would like teaching to be delivered in the future. Büchele et al . examine measures of student participation, predictors of performance and student malpractice, comparing 2020 with two previous years. They identify an emerging gender gap negatively affecting women and comment on the effectiveness of their provision. Shaw and Tranter conduct a statistical analysis of awarding gaps in the student cohort, before and after an early period of lockdown. They provide evidence to suggest that the awarding gap for students with a lower-socioeconomic background had worsened, but the gap for Black and Minority Ethnic students had reduced.

This special issue represents a snapshot of the research conducted into the impact of COVID-19 upon post-16 mathematics education. We anticipate further innovations, continued evaluation and consideration of best practice, particularly as COVID-19 may have to be tolerated indefinitely ( Kissler et al ., 2020 ).

As we exit the pandemic hopefully and learn to live with COVID-19 as an endemic virus, it will be interesting to see which of our new practices remain and which return to as they were. We hope that the papers in this special issue, which have evaluated the relative merits and disadvantages of doing things differently, will help inform this dialogue.

The fact that the global pandemic is still very much on-going weighs heavily on all of us. We are keen to pay tribute to all students and staff working in education for rapidly adapting to the extreme circumstances that we have operated under, balancing the significant challenges of trying to live normally in a world that is currently anything but normal. A silver lining of the COVID-19 pandemic was the start of significant reflections on teaching practice in the discipline on a scale unseen before. It is safe to say that post-16 mathematics education will not be the same again.

Hodges , C. , Moore , S. , Lockee , B. , Trust , T. & Bond , A. ( 2020 ) The difference between emergency remote teaching and online learning . EDUCAUSE Review , 27 , 1 – 12 . https://er.educause.edu/articles/2020/3/the-difference-between-emergency-remote-teaching-and-online-learning   (accessed 30 September 2021) .

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British Science Association ( 2020 ) Young people are more interested in a scientific career as a result of COVID-19 . British Science Association . https://www.britishscienceassociation.org/blog/young-people-are-more-interested-in-a-scientific-career-as-a-result-of-covid-19   (accessed 30 September 2021) .

Kissler , S. M. , Tedijanto , C. , Goldstein , E. , Grad , Y. H. & Lipsitch , M. ( 2020 ) Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period . Science , 368 , 860 – 868 . https://www.science.org/doi/10.1126/science.abb5793   (accessed 30 September 2021) .

Dr Jonathan Gillard , Reader in Statistics, Cardiff University. Jonathan is a Reader in Statistics at the School of Mathematics at Cardiff University. He is an Editorial Board member of MSOR Connections. Jonathan is currently interested in the statistical analysis of the National Student Survey and the Teaching Excellence Framework. In the past he has published on the effective delivery of mathematics support services, diagnostic testing and new methods for giving student feedback.

Dr Claire Ketnor , Principal Lecturer, Sheffield Hallam University. Claire is the Teaching and Learning Portfolio Lead for the Department of Engineering and Mathematics at Sheffield Hallam University. In teaching mathematics, she develops innovative methods with the aim of improving inclusivity. Claire is an active researcher in teaching and learning pedagogy, having also published under her previous surname Cornock. Her current work includes looking into students’ views on making mistakes. https://orcid.org/0000-0002-6918-5918 .

Dr Ciarán Mac an Bhaird , Assistant Professor and MSC Director, Maynooth University. Ciarán was appointed to his roles in the Department of Mathematics and Statistics at Maynooth University in 2007. He has received multiple awards in recognition of his teaching and support of students. He was a founding committee member of the Irish Mathematics Learning Support Network and conducts research in algebraic number theory, mathematics education and the history of mathematics. https://orcid.org/0000-0001-5971-7709 .

Dr Cathy Smith , Senior Lecturer, Open University. Cathy leads the mathematics education team at The Open University, UK. Her current work involves mathematics teacher professional development, supervision of research students and research. She has a long-standing research interest in pedagogies of advanced mathematics education and in studying discourses of participation in mathematics.

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

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• Arthur Bakker   ORCID: orcid.org/0000-0002-9604-3448 1 ,
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Before the pandemic (2019), we asked: On what themes should research in mathematics education focus in the coming decade? The 229 responses from 44 countries led to eight themes plus considerations about mathematics education research itself. The themes can be summarized as teaching approaches, goals, relations to practices outside mathematics education, teacher professional development, technology, affect, equity, and assessment. During the pandemic (November 2020), we asked respondents: Has the pandemic changed your view on the themes of mathematics education research for the coming decade? If so, how? Many of the 108 respondents saw the importance of their original themes reinforced (45), specified their initial responses (43), and/or added themes (35) (these categories were not mutually exclusive). Overall, they seemed to agree that the pandemic functions as a magnifying glass on issues that were already known, and several respondents pointed to the need to think ahead on how to organize education when it does not need to be online anymore. We end with a list of research challenges that are informed by the themes and respondents’ reflections on mathematics education research.

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1 An international survey in two rounds

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

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

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

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

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

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

2 Methodological approach

2.1 themes for the coming decade (2019).

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

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

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

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

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

Artistic impression of the future themes

2.2 Has the pandemic changed your view? (2020)

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

3 The themes

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

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

3.1 Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

3.1.1 Teaching strategies

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

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

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

3.1.2 Curriculum

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

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

3.2 Goals of mathematics education

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

3.2.1 Societal goals

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

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

3.2.2 Educational goals

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

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

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

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

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

3.3 Relation of mathematics education to other practices

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

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

3.4 Teacher professional development

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

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

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

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

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

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

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

3.5 Technology

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

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

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

3.6 Equity, diversity, and inclusion

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

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

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

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

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

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

3.8 Assessment

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

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

4 Mathematics education research itself

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

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

4.2 Methodology

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

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

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

4.3 Reflection on our discipline

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

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

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

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

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

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

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

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

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

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

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

4.4 Interdisciplinarity and transdisciplinarity

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

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

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

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

5 The themes in their coherence: an artistic impression

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

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

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

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

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

6 Research challenges

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

6.1 Aligning new goals, curricula, and teaching approaches

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

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

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

6.2 Researching mathematics education across contexts

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

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

6.3 Focusing teacher professional development

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

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

6.4 Using low-tech resources

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

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

6.5 Staying in touch online

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

6.6 Studying and improving equity without perpetuating inequality

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

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

6.7 Assessing online

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

6.8 Doing and publishing interdisciplinary research

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

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

7 Concluding remarks

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

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

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

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Mathematics self-concept and challenges of learners in an online learning environment during COVID-19 pandemic

• Rex Bringula   ORCID: orcid.org/0000-0002-1789-9601 1 ,
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In this mixed-methods research, the relationship between four factors of individual online learners and their mathematics self-concept was explored. In addition, the challenges the students faced in learning mathematics online during the Coronavirus disease (COVID-19) pandemic were determined. The participant students were from two mathematics classes offered online during the summer of 2020. Pure online classes were first offered during this period because face-to-face learning sessions were suspended due to the COVID-19 pandemic. It was found that students owned the devices they were using for online classes. Internet connection and power interruption were the most problematic aspects of online learning. Students had positive as well as negative mathematics online learning self-concepts. Individual factors were partly related to mathematics self-concept. Qualitative data shows that students faced technological, personal, domestic, assessment, pedagogical, consultation, and test anxiety challenges. Implications and recommendations for teaching mathematics in an online environment are offered.

Introduction

Academic self-concept (ASC) is one's academic self-perceptions of one's general ability in school (Shavelson et al., 1976 ). It also refers to the interests of the students towards a particular course (Joyce & Yates, 2007 ). This idea has been widely investigated in the field of mathematics that eventually forms a field of study named mathematics self-concept (Lee & Kung, 2018 ; Pajares & Miller, 1994 ; Reyes, 1984 ). Mathematics self-concept has “to do with how sure a person is of being able to learn new topics in mathematics, perform well in mathematics class, and do well on mathematics tests” (p. 560, Reyes, 1984 ). It also refers to an individual's perception of his/her abilities related to mathematics as compared to others (Bong & Skaalvik, 2003 ). It is related to mathematics achievement (Lee & Kung, 2018 ). Thus, it is important to understand how students perceived their mathematics learning abilities.

However, these studies (e.g., Bong & Skaalvik, 2003 ; Lee & Kung, 2018 ; Pajares & Miller, 1994 ; Reyes, 1984 ) were investigated in the context of face-to-face learning where students can seek immediate learning interventions from teachers or their classmates. Moreover, while prior works (e.g., Baticulon et al., 2021 ; Fabito et al., 2021 ) outlined the barriers to online learning in the context of the COVID-19 pandemic, the mathematics self-concept in the context of online learning is not yet investigated. The sudden shift from face-to-face to online learning platforms posed challenges to students. Shifting to an online learning platform requires access to a device, Internet, and physical learning space, and a strong habit of learner’s autonomy. Due to the digital divide, device ownership is a persistent challenge in the effective implementation of online learning. For developing countries, intermittent Internet connection remains a barrier to effective online learning (Salac & Kim, 2016 ).

The shift from a classroom environment to a home learning environment raises another concern for students. Students that have no access to a personal physical online learning environment could be disrupted by noise and other distractions (Baticulon et al., 2021 ; Bringula et al., 2021 ). The pedagogical style also changed. In particular, there are parts of the content of the syllabus that students have to learn on their own (i.e., asynchronous session)—tasks that may not be a practice in a face-to-face setup. The ability of the students to learn and study course material during asynchronous sessions poses difficulties to online learners. These barriers are found to hinder the effective implementation of online learning. However, it is unclear whether these variables have a significant relationship with the students’ mathematics online learning self-concept (i.e., perceived abilities of the students to learn online mathematics online courses).

In other words, while technological, personal, institutional, and community barriers are identified in online learning in this time of the COVID-19 pandemic (Baticulon et al., 2021 ; Fabito et al., 2021 ), the relationship of these constraints to the mathematics self-concept of the students in a fully online class is still unknown. In addition, mathematics self-concept in an online learning setup is not considered in the existing guidelines on improving mathematics online education (e.g., Lee & Kung, 2018 ). Understanding the mathematics self-concept of the students at the early stages of the implementation of online learning could inform educators to improve or diversify their mathematics online teaching strategies. Hence, it is imperative to understand the relationship of these variables with the mathematics self-concept of the students in an online learning environment.

The findings of this study will benefit the teachers, the students, and the schools. Teachers may apply the recommended approach derived from the findings of the study in improving their online pedagogies. Schools may utilize the findings of the study in providing institutionalized strategies in addressing students’ online learning mathematics self-concept. In turn, students' online mathematics self-concept may be improved and students' mathematics learning may be achieved. Consequently, students' attrition to mathematics online learning courses may be lessened.

This study attempted to contribute to the existing threads of discussion of mathematics self-concept in the context of a fully online learning environment. The study investigated the mathematics self-concept of two classes of computing students during the summer of 2020. The results of the study served as a basis in the formulation of a checklist of suggestions for online mathematics teachers. Specifically, the study aims to answer the following research questions (RQ). (RQ) (1) What are the online learners-related factors in terms of device ownership, perceived Internet speed, personal physical learning space access, and mathematics learning autonomy? (2) What is the mathematics self-concept of online learners in terms of mathematics ability, mathematics interest, and perceived mathematics performance? (3) Is there a significant relationship between learner-related factors and mathematics self-concept? (4) Is there a significant difference in the mathematics self-concept of online learners between those with personal learning space and those without personal learning space?, and (5) What are the experiences of the respondents in terms of challenges in online learning?

Literature review

Online mathematics education.

Berge et al. ( 2000 ) set forth the following recommendations for teachers to ensure effective implementation of online learning courses: (1) state the hardware and software requirements of the course, (2) be available for consultation, (3) be creative in interacting with students online, (4) provide course performance feedback, (5) listen to student concerns and encourage class participation, (6) establish clear policies, goals, course objectives, and course expectations, (7) be knowledgeable with the online learning software, (8) use different pedagogical styles, (9) encourage collaboration among students, and (10) be proactive and solve problems to avoid escalation. Meanwhile, Herrington et al. ( 2004 ) suggested that online activities should be composed of relevant, ill-defined, varied, and complex tasks that could be integrated across different subject areas. Moreover, teachers may also provide activities supporting opportunities for collaboration and reflection.

Different studies reported the factors that influenced the achievement of online mathematics education students. The results of these studies served as focal points on improving online mathematics teaching. The study of Wadsworth et al. ( 2007 ) disclosed that four learning strategies (motivation, concentration, information processing, and self-testing) and self-efficacy predicted online mathematics grade achievement. They suggested that online mathematics educators provided real-world examples and conduct meetings with students regarding learning strategies. In a similar study, Glass and Sue ( 2008 ) showed that assignments were the most preferred learning object and had the most impact on learning. Thus, adequate practice drills and timely feedback were necessary for online mathematics education. The findings of Wadsworth et al. ( 2007 ) and Glass and Sue ( 2008 ) are consistent with the guidelines of Herrington et al. ( 2004 ).

Güzeller and Akin ( 2012 ) compared the mathematics achievement, attitudes, anxiety, and self-efficacy of students in the web-based mathematics instructions (WBMI) and traditional mathematics instructions (TMI). It was found that there was a significant difference between the WBMI and TMI in terms of achievement, attitudes, anxiety, and self-efficacy, having more favorable results on the WBMI. Students were advised to take the WBMI to familiarize themselves with this platform. Kim et al. ( 2014 ) extended the study of Güzeller and Akin (2012) by investigating the impact of anxiety and other forms of academic emotions (anger, shame, boredom, enjoyment, and pride) on online mathematics achievement. Kim et al. ( 2014 ) further hypothesized that motivation, self-efficacy, and cognitive processes (e.g., cognitive strategy, self-regulated learning) influenced online mathematics achievement. Academic emotions were accounted for 37% of the variation in student achievement.

Finally, Cho and Heron ( 2015 ) determined the impact of self-regulated learning (SRL), learning strategies, and emotions on satisfaction with online learning and online mathematics achievement. The study revealed that motivation influenced achievement, and both motivation and emotion were related to satisfaction. Based on these findings, the study formulated the following recommendations for teachers: enhance students' self-efficacy, design supporting tools in online courseware, provide course orientation, provide SRL support through social media, and restructure the format of the course.

Academic self-concept, academic self-efficacy, and mathematics self-concept

Arens et al. ( 2020 ) discussed the similarity and the difference between academic self-concept and academic self-efficacy. In general, both academic self-concept and academic self-efficacy address students' competence (Bong & Skaalvik, 2003 cited in Arens et al., 2020 ). However, academic self-concept is related to the self-perceived competence of a student in an academic domain in general (e.g., math; Marsh & Craven, 2006 ). Meanwhile, academic self-efficacy is self-perceived confidence to perform successfully a given task in a specific domain (e.g., Bandura, 2001 ; Zimmerman, 2000 ). In other words, self-concept is a domain-specific construct while self-efficacy is a domain- and task-specific construct (Arens et al., 2020 ).

Self-concept is investigated in the field of mathematics. Mathematics self-concept is attributed to students’ aspirations to pursue degree programs in STEM (science, technology, engineering, and mathematics) (Sax et al., 2015 ). Students with positive mathematics self-concept stay and finish their chosen degree program which, in turn, contributes to improving the school's student retention (Ackerman et al., 2013 ). This can be explained by the findings that mathematics self-concept is positively related to mathematics achievement (Kung, 2009 ; Lee & Kung, 2018 ). Moreover, mathematics self-concept and mathematics self-efficacy (i.e., a belief of student’s ability in solving mathematical problems or tasks related to mathematics; Masitoh & Firtriyani, 2018 ) predicted mathematics achievement (Kung, 2009 ). Thus, it is necessary to understand the mathematics self-concept of the students since it serves as a basis to cultivate the students’ learning interests in the program.

The measurement of mathematics self-concept is primarily based on the prior works of self-concept (e.g., Marsh & O'Neill, 1984 ; Marsh et al., 1983 , 2005 ). The questionnaire was composed of different areas including mathematics, verbal abilities, academic capabilities, problem-solving/creativity, physical abilities/sports, physical appearance, relations with same-sex peers, relations with opposite-sex peers, relations with parents, religion/spirituality, honesty/reliability, emotional stability/security, and general self-concept. The mathematics factor was composed of items such as “I find many mathematical problems interesting and challenging”, “Mathematics makes me feel inadequate”, “I am quite good at mathematics”, “I have trouble understanding anything that is based upon mathematics”, and “I have always done well in mathematics classes”. Recent studies on mathematics self-concept based their works on self-concept questionnaire developed by Marsh and colleagues (Marsh & O'Neill, 1984 ; Marsh et al., 1983 , 2005 ). For example, Lee and Kung ( 2018 ) investigated the mathematics self-concept and mathematics achievement of junior high school students in Taiwan. The authors devised a 13-item questionnaire measuring competence in mathematics, affection towards mathematics, and comparison of mathematics abilities.

Barriers to online learning

Cavanaugh et al. ( 2009 ) reported the barriers in the existing literature to online learning implementation as well as the benefits and challenges of online learning. Higher levels of motivation, expanding educational access, providing high-quality learning opportunities, improving student outcomes and skills, allowing educational choice, and administrative efficiency are notable benefits of online learning. Meanwhile, the high cost of start-ups, digital divide issues, governmental approval, and student readiness are the challenges raised in the implementation of online learning.

In a recent study, Binti Abd Aziz et al. ( 2020 ) investigated the barriers to online learning. According to the authors, addressing these barriers could lead to effective online learning practices. They identified the barriers in terms of attitudes, interruptions, technology skills, and personal skills. Attitudes towards online learning refer to the feelings of the people towards online learning. Computer, online, and computer application literacy skills are the components of technology skills. Interruptions to online learning are defined as the limits to technological access because students may be living in rural areas, being part of a minority group, having disabilities, or due to mature age. Personal skills are skills relating to prior experience of using online learning. Path analysis disclosed that attitudes toward online learning and technology skills are the main barriers to online learning.

The inability of the students to study at their own pace has also posed a barrier to students' online learning. Learner autonomy is the ability of learners to assume control or to take charge of their learning (Benson, 2001 ). Autonomous learners were able to understand the online learning process (Fotiadou et al., 2017 ), which enabled them to achieve high grades in online learning classes (Yen & Liu, 2009 ). In other words, mathematics learner autonomy is the ability of learners to understand and assume control learning of the online materials with little supervision (Benson, 2001 ; Fotiadou et al., 2017 ).

In the Philippines, Pena-Bandalaria ( 2009 ) reported that personal concerns (e.g., difficulty to interact and contact teachers, difficulty to seek help, difficulty in understanding the topics), technical concerns (e.g., problems accessing the course site), and the digital divide were barriers to online learning. This finding is consistent with the results of the study of Gledhill et al. ( 2017 ). Gledhill et al. ( 2017 ) also revealed that limited or poor access to the Internet, technology, and networks were the constraints of e-learning in less developed countries. Perceived Internet speed is the subjective evaluation of the speed of the Internet in supporting online learning sessions (Gledhill et al., 2017 ). Natividad ( 2021 ), and Salac and Kim ( 2016 ) explained the slow Internet connection in the Philippines. They agreed that Internet connection in the Philippines is slow due to limited Internet infrastructure which is brought by outdated laws and heavy bureaucratic processes for the development of Internet infrastructure. Furthermore, intermittent power supply was a major problem that hinders e-learning implementation in less developed countries (Bhuasiri et al., 2012 ).

In this time of COVID-19 pandemic, Philippine higher education institutions also implemented emergency online learning programs (Murphy, 2020 ). The emergency implementation may caught students unprepared (Aguilera-Hermida, 2020 ; Daniel, 2020 ). Baticulon et al. ( 2021 ) reported the major barriers of Filipino medical students to adopt online learning. These barriers can be classified as technological (lack of devices, issues with the online platform, Internet connectivity), individual (students’ learning style, physical and mental health), domestic (concerns at home, financial distress), institutional (school curriculum), and community barriers (lockdown restrictions, infrastructure challenges, and sociopolitical issues). Students found it difficult to understand the learning materials on their own. It was also reported that students had difficulty studying at home because of noise, distractions, and small space. Personal physical learning space refers to the space dedicated to online learning that is free from distraction or noise (Baticulon et al., 2021 ).

In a similar study of Fabito et al. ( 2021 ), they found that difficulty of clarifying topics or discussions with the professors, lack of study or working area dedicated for online activities, and lack of good Internet connection were the top three barriers and challenges that the 300 computing students (Computer Science and Information Technology) experienced. The study concluded that students and teachers were both not prepared to undergo full online learning. In a similar study, Bringula et al. ( 2021 ) found that the number of owned devices had a positive influence on the perceived academic online learning performance of computing students. Device ownership refers to the number of devices students own in accessing the LMS (Bringula et al., 2021 ). It was shown that students that own multiple devices are more likely to have positive dispositions towards their academic online learning performance than those students who have difficulty access to online learning devices.

Synthesis of literature review

There is a wealth of studies proposing the effective delivery of mathematics online education. These recommendations did not consider students' mathematic online self-concept. Moreover, the recommended teaching strategies are not set forth in the context of COVID-19 pandemic where students faced physical and psychological challenges (Baticulon et al., 2021 ; Bringula et al., 2021 ; Gledhill et al., 2017 ; Pena-Bandalaria, 2009 ). In particular, the continuous lockdowns in the Philippines exacerbate the existing phenomenon of digital divide, e.g., students cannot utilize public pay-for-access of computers and Internet in computer shops (Baticulon et al., 2021 ). Students may experience difficulty on engaging to mathematics online learning due to limited access to basic online resources. In turn, students may feel less capable of learning the mathematics online materials at their own pace. Determining the possible connection between barriers to online learning and students’ mathematics self-concept may help teachers, parents, and educational institutions to formulate pedagogical interventions to achieve desired online mathematics achievement.

Methodology

Research design, research setting, participants, sample, and data gathering procedure.

This mixed-method study was conducted in one department of a university in Manila. In the quantitative part, the participants of the study were second-year college students (subsequently referred to as online learners) enrolled in two classes in Numerical Analysis. There were 69 online learners enrolled in the said course. This was the only mathematics course offered at the time the study was conducted. The study was conducted after the first week of full implementation of pure online learning through a learning management system (LMS). Learning materials and lecture sessions were both delivered in synchronous and asynchronous methods, but mostly delivered in asynchronous methods (36 h out of 54 h). Online learners were informed at the beginning of the online class sessions about this setup. The survey form was distributed in the LMS. An online survey form was distributed to all online learners but only 54 students participated in the study. Online learners in the study are composed of male (59%) and female (41%) students with an average age of 20 years old.

Research instrument

The study utilized a content-validated survey form that served as a research instrument. The survey form consisted of two parts. The first part gathered information about the online learners profiles such as device ownership, perceived Internet speed, personal physical learning space ownership, and mathematics learning autonomy. They were also asked whether they have personal/private physical space for online learning (Baticulon et al., 2021 ). Perceived Internet speed can be answered using the responses “Very slow”, “Slow”, “Sometimes fast, sometimes slow”, “Fast”, and “Very fast”.

The mathematics learning autonomy intends to determine the level of independence to learn mathematics. It was measured using a 5-point scale in which the most negative response (i.e., total dependence to teachers or classmates) had an assigned value of 1 while the most positive response had a value of 5 (i.e., can independently learn the course content). These variables were selected because these were deemed relevant to the participants of the study and these were believed to influence engagement in online learning in the context of the COVID-19 pandemic (Pynos, 2016 ).

The second part solicited data on the mathematics self-concept of online learners. Mathematics self-concept consisted of mathematics ability (12 items), interest (2 items), and perceived mathematics performance (1 item). Online learners were asked about their perceived abilities and interest in mathematics learning when the course is delivered in an online setting. All items of ability and interest were preceded by the phrase “Considering that the course is delivered online,…”. The items could be answered using the responses “very untrue to me”, “untrue to me”, “unsure”, “true to me”, and “very true to me”. These verbal responses had assigned values from -2 to 2, where the most negative response has a value of -2 while the most positive response has a value of 2. Students were asked to complete the sentence “Considering that the course is delivered online, your mathematics grade will be…” to measure their perceived mathematics performance. This question could be answered using the responses “higher than face-to-face”, “about the same with face-to-face”, “lower than face-to-face”, and “not sure/cannot tell”.

The definition of academic self-concept relating to ability (Shavelson et al., 1976 ) and interest (Joyce & Yates, 2007 ) served as the basis in the construction of the research instrument. The items of the research instrument were adapted from Joyce and Yates ( 2007 ), Marsh et al. ( 1983 ), and Lee and Kung ( 2018 ). Only the ability component in the questionnaire of Lee and Kung ( 2018 ) was adopted in this study. The items applicable to online learning were retained. The adapted research instrument was then pilot-tested to 50 students who were not part of the study. Factor and Cronbach’s alpha analyses revealed that all items were found valid (factor loading ≥ 0.50) and reliable (Cronbach’s alpha α ≥ 0.70).

Statistical treatment of data

Frequency counts, means, and cross-tabulation were used to describe the data. Spearman Rank correlation was employed to determine the relationship between the profile of the online learners and mathematics self-concept. Mann–Whitney U test was used to determine whether there is a significant difference between the mathematics self-concept of online learners when categorized by personal physical learning space access. A 0.05 level of significance was used to determine the significance of the results.

Interview sessions, selection of interview participants, and participants

In the qualitative part, a series of separate interviews were conducted with three Information Technology (IT) students to further explain the findings of the study. The participants were selected based on their mathematics abilities and access to personal learning space. The authors sought the recommendations of the teachers who handled the course to identify the possible participants. The teachers identified and categorized the mathematics abilities of the students. These classifications are reliable because teachers know their students’ capabilities (Cheong et al., 2004 ; Lambert, 2002 ; Reeve, 2006 ). The participants consisted of 3 male, third-year students with an average age of 20 years old and they had varying degrees of mathematics abilities (i.e., struggling, average, and high performing). One of the informants (i.e., respondents) has no personal space dedicated to online learning. Two female participants were invited but they refused to participate in the study.

The interviews were conducted through Google Meet. Informants were interviewed one-by-one on different occasions to protect their identities. Students were asked about their challenges in online learning. The students were asked about their study practices in a face-to-face class (e.g., attending classes, review preparations for quizzes and exams, practicing solving math problems, reading materials, and taking lecture notes), challenges experienced in an online learning class, their perceived abilities in a face-to-face and online learning class, and their recommendations relating to the improvement of online pedagogy.

Qualitative data analysis

The interviews were transcribed and were tabulated in a word processor. The tabulated results were then presented for validation. The purpose of the validation process was to determine whether other students had the same experiences. The validation of interview results was done through a presentation to another set of students (i.e., the validators) with the same level of mathematics abilities and also enrolled in the mathematics class during the summer period. Throughout these procedures, all identities of the participants were kept confidential. The validators were asked whether they agree or disagree with the responses collected from the interviews. The validators were composed of IT students (3 males with an average age of 20 years old; 2 third-year students and 1 s-year student; one of the respondents has no personal learning space). After the validation process, the tabulated responses were analyzed through qualitative content analysis (Hsieh & Shannon, 2005 ; Mayring, 2014 ).

The responses were coded and categorized based on Baticulon et al.'s ( 2021 ) classifications of barriers to online learning. The codes were keywords or phrases that represented the challenges of learning mathematics in an online environment or their recommendations to their teachers about teaching mathematics in an online setup. The authors also made their classifications if an item did not fit on the Baticulon et al.'s ( 2021 ) classification. The codes were then assigned to themes (i.e., challenges of online learning and the recommendations to improve online learning). One of the authors coded the responses. When coding the text, the coder was guided by the themes. The process was repeated until all keywords and phrases were assigned to the themes. Afterward, the research team deliberated whether they agree (or disagree) with the themes. In case of disagreement, the deliberation process was repeated until a consensus was reached (Bringula et al., 2019 ).

Both the quantitative and qualitative results of the study were presented during the department general faculty meeting to elicit feedback and inputs on how mathematics online teaching practices be improved and to validate the themes proposed in this study. The attendees of the meeting served as the external validators. The external validators involved two mathematics teachers (both female with at least 25 years of teaching experience), one LMS administrator (female with 20 years of work experience; conducts LMS training and develops University-wide online course materials), and one academic administrator. All external validators agreed with the themes and recommendations of the study.

RQ1: Learners-related factors

It is found that 98% of the online learners own 1 or 2 devices (see Table 1 ). More than half of the respondents do not have personal learning space during online learning sessions. Seventy percent reported that intermittent Internet connection is the most problematic aspect of online learning. They perceived that mathematics delivered in an online platform is harder to learn than in a face-to-face setup. A large percentage of online learners (89%) mainly rely on lectures from teachers or from the help of their classmates to understand mathematical concepts. Fifty-six percent of the students rely on their teacher’s or classmates’ consultation to understand the lessons.

RQ2: Mathematics self-concept in an online learning setup

Online learners have negative notions about their capabilities in terms of understanding the lessons, solving problems easily, finishing the course, performing better relative to their classmates’ abilities, and performing better relative to their schoolmates’ abilities (Table 2 ). They feel it is more enjoyable to learn in a face-to-face setting than online. In terms of mathematics performance, 43% of the students feel that they will get lower grades in the online course than when it is done in a face-to-face session. Another 43% reported that they are unsure of what grade they will get at the end of the semester (Fig.  1 ). This perceived academic performance could be attributed to the unfamiliarity of the new learning environment. Students feel unprepared for online learning (Daniel, 2020 ; Fabito et al., 2021 ; Murphy, 2020 ) and they may feel it undesirable (Aguilera-Hermida, 2020 ).

Perceived mathematics performance in an online learning setup

Despite these negative notions, students have positive outlooks in terms of achieving good grades, doing well in the course, attending classes, doing assignments, helping their classmates in their assignments, recalling the lessons, and passing the course. They perceive that learning through the LMS is interesting. These findings suggest that they believe that their abilities can still meet the demands of the course. They are confident that they can still perform well despite the challenges and uncertainties they are facing.

RQ3: Relationship between learners-related factors and mathematics self-concept

It was also determined the relationship between the profile of online learners and their mathematics self-concept (Table 3 ). The number of devices they can use has positive relationship with understanding ( r  = 0.27, p  < 0.05) and recalling ( r  = 0.34, p  < 0.05) the lesson. Perceived Internet speed is positively related to the ease of attending class ( r  = 0.29, p  < 0.05). Statistical analyses found that correlations exist between mathematics learning autonomy and mathematics self-concept. Out of the 15 correlation analyses, six variables have significant correlations with mathematics learning autonomy. Mathematics learning autonomy have positive relationship with the abilities to get good grades ( r  = 0.42), solve problems easily ( r  = 0.40), do assignments easily ( r  = 0.31), recall the lessons ( r  = 0.34), and perform better than their classmates ( r  = 0.40) or schoolmates ( r  = 0.39).

RQ4: Difference in the mathematics self-concept of online learners between with and without personal learning spaces

Further analysis was conducted to determine if mathematics self-concept differs between online learners with and without personal learning spaces. Both online learners (with personal learning space, 30%; without learning space, 33%) agreed that mathematics learning in an online environment is harder than face-to-face. However, they have different opinions in terms of their mathematics grades. Online learners with no learning space perceived that they might have a lower grade than in a face-to-face course (30%) while those who have personal learning spaces are unsure of what grades they will get at the end of the semester (28%). The test of the difference between the means of self-concept of learners explains these results.

Mann Whitney U test revealed that the mathematics self-concept of the two groups of learners differ significantly in terms of achieving good grades ( U (52) = 252.0), solving problems easily ( U (52) = 243.5), doing well in the course ( U (52) = 249.5), answering assignments ( U (52) = 256.0), and recalling the lessons ( U (52) = 225.0) (Table 4 ). The results are unlikely to have arisen from sampling error ( p  < 0.05). Online learners with no personal learning space had lower mathematics concepts than privileged online learners.

RQ5: Challenges on online learning

Table 5 shows the challenges that the informants faced in learning mathematics in an online environment. All validators agreed that they experienced these challenges. All the informants and validators alike agreed that technological challenges are the most pressing concern in online learning. Only one of the informants reported power interruption. This informant is in the province and his province has been experiencing regular power interruptions. This statement confirms the study of Bhuasiri et al. ( 2012 ). The other informants and validators may not experience this because they are all in the National Capital Region where power interruption is rare.

The second challenge involves a problem that can only be solved by the students themselves. One of the informants said:

It is hard to focus on my studies. Unlike in a classroom setup, the environment is conducive to learning. All you need to do is to listen to the teacher. When you are at school, your mind is conditioned to study. When you are at home, you are in a comfort zone. I tend to do other things and delay doing my assignments. I admit: I become less productive and do not manage my time well when I am at home.

However, it must be noted that one of the validators did not agree that he procrastinates. This is the only item that the validators disagree with. Domestic challenges contribute to the distraction of students' online learning. Even while during class sessions, students were asked to run for errands or do simple household chores. Some situations are beyond the control of the students and their families (e.g., visitors). Noise and distractions contribute to domestic challenges. This finding confirms the study of Baticulon et al. ( 2021 ) and Fabito et al. ( 2021 ). One informant commented:

I have other responsibilities at home. Sometimes, I feel guilty because they are all busy doing household chores while I am on just on my computer throughout the day. Sometimes, I have to run errands. There are times that I have to respond to our neighbors' calls who are looking for my parents.

One validator agreed and said:

I agree with this. I want to also add that I only used the living room for my online learning sessions. Sometimes, the people in the house forgot that I am having an online learning session. They play music while walking around the living room. There was even an incident that they look at my laptop thinking that I was only watching movies.

Informants and validators reported that they experienced assessment challenges. Assessment challenges involve few practice drills, design of the online examination, clarity of instructions, and assessment feedback. One informant commented that teachers only provide about three questions and let them study and solve the other problems. This practice is construed as ineffective because, according to the informant, practice is an important activity that builds up their mathematics skills. The informant explained that this could be attributed to the desire of the teacher to cover the whole content of the course syllabus. When he was asked whether he prefers quality over quantity of the content, he chose the former. The validator stated: “There is no point in covering the whole syllabus when you did not understand any of them.”

The design of the examination refers to the way the questions are presented on an online platform. These include questions that require answers with long inputs of formula that are susceptible to typographical errors, multiple types of questions that are prone to guessing, time allotment, unclear instructions, and familiarity with the system itself. Informants and validators pointed out those problems that need to input their solutions in a text box entail a lot of time. They explained that they solve the problems on a piece of paper and then transfer them to the online submission system. They also raised their concerns on submission deadlines as they also have other courses with the same course requirements to satisfy. They also requested feedback on their activities and quiz results so that they would not commit the same mistakes.

The informants and validators further pointed out that they need more time as they familiarized themselves using the system. This is consistent with the study of Binti Abd Aziz et al. ( 2020 ). One informant said: “In an online exam, you are not only concerned with the correctness of your answers. You are also concerned about how you will input your answers correctly in the system. Teachers have to take into account that we also need time to familiarize ourselves with the system.”

The informants and validators also raised a pedagogical issue. The study of Baticulon et al. ( 2021 ) classified understanding the content of the course as a personal barrier. However, this is not the case based on the interviews with the informants and validators. According to informants and validators, it is difficult to understand the topics in online learning because of its delivery. When it is delivered appropriately, they can understand the topics and have better chances of passing the quizzes and exams. They prefer a combination of content delivery strategies including discussion of PowerPoint slides with step-by-step solutions through online meeting apps (e.g., Google Meet), and recorded videos of step-by-step solutions. One of the informants emphasized this comment: “Please do not let us study mathematics on our own. You do not just give the materials to us and let us understand the content.”

Before online learning, informants and validators seek the assistance of their teachers, classmates, or friends. The informants and validators understand that it is difficult to seek consultation because it is difficult to find a common time for consultation. Teachers and students have other responsibilities to attend to after online classroom sessions.

Students developed test anxiety because of the aforementioned challenges. According to the statement of one informant,

There is always a nervous factor when taking the quizzes or examinations since these are time-based activities. I am anxious since my Internet connection could suddenly become unstable or there might be a power interruption. Some teachers do not allow returning to the questions. Once you skipped the question, it will be given a zero mark. Unlike in a paper-based test, you can skip the questions and go back working on it if there is still time. I understand my teachers. They are thinking that a time-limited quiz/exam is a way to deter cheating.

This study investigated the classroom experience of online learners in a mathematics class during the summer of 2020. Toward this goal, the study attempted to determine the relationship between the online learners-related factors and their mathematics self-concept. Moreover, interviews were conducted to determine the challenges they faced in learning mathematics delivered on an online platform. The online-related factors in terms of device ownership revealed that they own 1 or 2 devices. Access to the device is not a problem to this set of participants relative to the general student population that may experience the digital divide (Cavanaugh et al., 2009 ; Pena-Bandalaria, 2009 ). This can be explained by the fact that the participants of this study are IT students, where learning activities, even before the pandemic, are highly dependent on devices.

The quantitative result shows that Internet connection is the most problematic aspect of online learning. An intermittent Internet connection can greatly affect the attendance of the students in online classes. This finding is consistent with the interview results in terms of technological challenges. This is a national problem since the Philippines has slow Internet connectivity (Chiu et al., 2017 ). According to Natividad ( 2021 ), and Salac and Kim ( 2016 ), the Philippines has a slow Internet connection because of the outdated Philippine law and red tape that hinders the quick installations of cell towers. This result confirms the findings of Bhuasiri et al. ( 2012 ) and Baticulon et al. ( 2021 ). Although only one of the informants reported an issue of power interruption, his concern is valid. His concern might not be similar to other informants or validators simply because the other informants and validators are all living in Metro Manila.

Almost half of the participants have no personal learning space during online learning sessions. Online learners with no personal learning space had lower mathematics concepts than privileged online learners. The lack of personal learning space during online sessions puts online learners in a disadvantaged position to attain an optimal learning experience. This is consistent with the findings of Baticulon et al. ( 2021 ), and Fabito et al. ( 2021 ). As explained in the interviews, students who lack personal learning space are more susceptible to distractions during, and even after online learning sessions. Noise and running errands are the most common forms of distraction. The interview results show that other members of the family may simply forget the students are in an online class. In short, as one student commented, access to personal physical learning space can create an environment conducive to online learning.

Online learners disclosed that they understood the content through lectures and constant consultation with teachers. This is consistent with the interview results that students dislike studying the course content on their own. This mathematics learning autonomy is the exact opposite of the nature of asynchronous learning. In asynchronous learning sessions, students have to study a lecture on their own. In other words, learners who are consultation-dependent will resist this educational shift.

As shown in Table 3 , students with low learning autonomy are expected to have lower dispositions of their mathematics abilities. Students who feel inferior about their mathematical abilities tend to have lower mathematics performance (Lee & Kung, 2018 ). In students’ point of view, asynchronous session activities (e.g., reading materials, assignments, practice drills, and quizzes) are challenging in the aspects of assessment, pedagogy, and consultation. These challenges explain why students experience test anxiety. Consequently, these difficulties contributed to their feeling of uncertain or low grade perceptions. The quantitative results provide insights to address this issue, i.e., students have to be gradually introduced into the concept of learners’ autonomy. Furthermore, family members may dedicate a place in the house that will serve as an online learning space.

The shift to an educational setting had a negative impact on the mathematics self-concept of learners. More than 80% of the respondents perceived that they will have a lower grade in mathematics. They also have negative notions of their mathematics self-concept in terms of understanding the lesson, solving problems, finishing the course, performing better relative to their classmates or schoolmates, and enjoying the online class. These negative notions on their capabilities and interest in online learning can be explained by the fact that full online learning is just implemented recently. While online learners have experience using the LMS before the COVID-19 pandemic, they are not yet fully familiar with a fully online learning setup. This is evident in one of the narratives of the informants. On one hand, the positive mathematics self-concept indicates that they are hopeful in the aspects of achieving good grades, attending classes, doing assignments, helping their classmates in their assignments, recalling lectures, passing the course, having the interest to learn, and doing well overall in the course. Teachers have to sustain these positive outlooks to achieve the course outcomes.

Device ownership has a positive relationship with understanding and recalling the lecture. Multiple devices such as laptops and mobile devices are dependable for students' online learning (Muyinda et al., 2010 ). Multiple device ownership allows online learners to view multiple screens and to store multiple copies of learning materials (Pynos, 2016 ). Multiple device usage in learning also creates seamless connectivity that enables the continuity of the learning experience (Milrad et al., 2013 ). This practice allows easy access to information that is useful for solving problems. For example, an online learner may be looking at his/her laptop screen for the given problem while he/she is looking into another device (e.g., cellphone) that displays the formula and the sample solved problems. Furthermore, multiple devices can address accessibility or installation issues.

Perceived Internet speed is positively related to the ease of attending class. This finding is expected. What is more interesting is that perceived Internet speed does not relate to the other items of mathematics self-concept. The results imply that a fast Internet connection is only necessary to attend the class but not necessarily related to the online learners' perceptions about their mathematics abilities. Their perceptions about their abilities and interest in mathematics are not related to the speed of Internet access. In other words, there is no link between the confidence of online learners in their mathematics abilities and their speed of Internet access.

Meanwhile, mathematics learning autonomy is correlated with most of the mathematics self-concept. This vivid finding denotes that mathematics self-concept is mostly related to the perceived abilities of online learners to study at their own pace, i.e., as students become more independent learners, they tend to have a higher mathematics self-concept. Teachers have to emphasize to the online learners that online learning is different from face-to-face where teachers can intervene when confusion or challenges arise in understanding the lessons. Teachers, at the onset of the course, are encouraged to orient online learners that they are expected to be independent learners. Problem sets and learning materials may be given in advance to develop the habit of independent learning.

Consistent with the literature, the respondents of this study experienced technological, personal, and domestic challenges. There is a challenge that students can be addressed by themselves (e.g., procrastination) but most are beyond their control. Domestic challenges require the support and understanding of family members. The students and their family members must have open communication. They should set house rules in terms of household chores and running errands.

The study of Baticulon et al. ( 2021 ) categorized the inability to understand the content of the course as a personal barrier to online learning. In this study, it was shown that this is a pedagogical challenge than a personal problem. Furthermore, it was disclosed that teachers have direct responsibilities on four out of the seven identified challenges. Challenges in the teaching and assessment had the most number of concerns. These results guide teachers to devise creative teaching and fair assessment strategies that could address these concerns. For teaching and learning activities, teachers are advised to provide ample time for lectures and deliver the contents through different forms of multimedia. At the end of each lecture, teachers may elicit feedback from students to assess if the students understood the lessons. It is advisable to gather feedback from struggling as well as high-performing students to understand the challenges of the students with diverse mathematical abilities. Group learning activities may be conducted using the Group Discussion function of the LMS. Teachers may provide practice drills that are not yet included for grade computation. Provide 2–3 days to allow students to do their assignments. The asynchronous sessions may also be utilized as consultation time. Teachers may use the randomized function of the LMS to pick random questions from its databank. Teachers may also request students to show their computer windows during the quiz (see “ Appendix A ”).

Another important role of the teachers is to sustain the positive and counter the negative mathematics self-concept of the students. Teachers at the onset of the course, the questionnaire here may be utilized to determine the mathematics self-concept of the students. Students should be oriented about the course expectations. Teachers should introduce independent learning gradually (“ Appendix A ”).

Conclusions, recommendations, limitations, and implications

This study investigated the profile of online learners and its influence on their mathematics self-concept. It is revealed that online learners in this study have access to devices. Physical learning space is one important aspect of an online learning environment. However, some online learners have physical learning space limitations which make online learning inconvenient. This limitation contributed to their low academic self-concept.

The majority had reported an intermittent Internet connection. Online learners have mixed notions about their mathematics capabilities and interest in learning mathematics in an online environment. They expressed uncertainties about the possible grades they will get at the end of the semester. The ability of online learners to study mathematics at their own pace is the most desired skill for online learners. Moreover, online learners with limited learning space are more likely to experience a lower mathematics self-concept because they cannot focus on the course. Thus, it can be concluded that the profile of online learners partly influences their mathematics self-concept.

Teachers play a significant role in improving and sustaining the mathematics self-concept of online learners. At the beginning of the class, teachers must inform online learners that having a habit of self-paced learning is a highly desirable discipline. Teachers have to sustain the positive mathematics self-concepts of online learners. They may assure online learners that online consultations are available when needed. Timely feedback on the works of online learners is highly encouraged to sustain their positive outlook about their capabilities. Individualized feedback can be provided to inform online learners that they are performing well (or not performing well) relative to his/her classmates.

The negative mathematics self-concepts of online learners serve as a basis for teachers to find ways to address these negative notions. Teachers have to be creative in delivering the content of the course (O’Doherty et al., 2018 ). For instance, PowerPoint slides with a voice recording or a previous video recording of the lesson may be utilized for lecture sessions. These materials may be accessed anytime and students with a slow Internet connection can still follow the phase of the course. Teachers may conduct synchronous learning sessions to answer questions or clarifications. An unwavering teacher’s dedication and understanding are suggested to assist online learners to finish the course.

The study is limited in terms of the participants and sample size. These limitations existed because of the timing of the shift of mode instructions in the university. There were only limited courses offered and a small number of students were enrolled when the study was conducted. Thus, the findings of the study may not be widely applicable beyond this population. Despite these limitations, this study provides clear insights into the students' mathematics self-concept and the challenges they faced in an online learning environment. The realities discovered in this study cannot be denied and deserves the attention of mathematics teachers. Nevertheless, a university-wide investigation of mathematics self-concept may be initiated to improve further the findings of the study.

There are issues raised in the study that cannot be solved by teachers. The members of the family must understand that online learners need physical learning space and minimal disruptions. To address this concern, teachers, or schools may send letters to parents about online learning to observe the online learning schedule of their children. Cooperation and understanding from family members are necessary for providing an environment conducive to online learning. It is strongly recommended that family members dedicate a physical learning space for online learners.

Educational institutions have to select an LMS that can support the demands of the course. The institution needs to understand the online learning requirements of the different degree programs. It is imperative to understand the strengths and limitations of the different LMS. A selection criteria committee may be instituted to select an LMS and to review its effectiveness relative to the needs of the students and faculty. Usability testing of the LMS may be done after the implementation. This will identify the ease of use and satisfaction of use of the LMS. The evaluation process may also evaluate whether the LMS supported the pedagogical requirements of the faculty (Pipan et al., 2008 ).

Another challenge the institution facing is the possibility of students that might be left behind because of inadequate access to devices. The formation of a technical support group is also desirable. Educational institutions may extend their help to online learners by lending laptops, tablets, and mobile Wi-Fi. Local government units may also offer assistance to underprivileged students. For example, a city local government unit in the Philippines provided online learning devices (e.g., laptops or tablets) to students (Casinas, 2020 ) and installed Internet centers to support online learning (Kabagani, 2020 ).

Lastly, the government may reinforce fast Internet connections through legislation. One of the possible legislations is to shorten the application of business process applications of constructing Internet facilities (Natividad, 2021 ; Salac & Kim, 2016 ). The government may allocate funds for the development of Internet infrastructures. These funds may be directed to rural areas. A government-private partnership may also be initiated. With this partnership, lengthy bureaucratic procedures will be avoided. Finally, the government may promote a market of competitiveness through the inclusions of other Internet providers (Salac & Kim, 2016 ).

Availability of data and materials

Data cannot be shared because of existing laws in the country of the authors.

Abbreviations

Academic self-concept

Coronavirus disease

Information technology

• Learning management system

Self-regulated learning

Traditional mathematics instructions

Mann–Whitney U test

Web-based mathematics instructions

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Acknowledgements

We would like to thank the participants of the study and those whom all helped us distribute the survey form. This paper is partly funded by the institutions of the authors.

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Appendix A: Checklist of suggestions for online mathematics teaching

Mathematics Self-concept in an Online Setting

Determine the mathematics self-concept of the students at the start of the class.

Inform the students of the setup of the course.

Introduce learners to the concept of independent learning.

Assist students in re-enforcing positive self-concepts.

Teaching Learning Activities

Provide ample time for lectures.

Immediately elicit feedback after the lecture.

Gather feedback from high-performing and struggling students about the phase and clarity of the lecture.

Use the combination of PowerPoint slides, online meetings, and videos for course content delivery.

Make the due dates reasonable (2–3 days).

Use the Group Discussion function of the LMS to encourage group study among students.

Assessment and Consultation

Provide individualized feedback on students’ activities.

Inform students about their class performance.

Dedicate synchronous sessions intended for consultation or feedback.

Provide more seatwork and practice drills.

Provide assessment activities that are not recorded.

Balance the types of the assessment – e.g., minimize multiple-choice, more on problems showing solutions, and discourage providing answers that are format-sensitive.

Balance the difficulty levels of the assessment.

Use the randomized function of the LMS to generate quizzes and exam questions.

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Bringula, R., Reguyal, J.J., Tan, D.D. et al. Mathematics self-concept and challenges of learners in an online learning environment during COVID-19 pandemic. Smart Learn. Environ. 8 , 22 (2021). https://doi.org/10.1186/s40561-021-00168-5

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COVID-19 and the use of digital technology in mathematics education

Mansour saleh alabdulaziz.

Department of Curriculum and Instruction, College of Education, Imam Abdulrahman Bin Faisal University, P.O. 1982, Dammam, Saudi Arabia

Once the COVID-19 crisis is over, will everything” return to normal” or will we instead witness an ongoing boom in online learning? A time of crisis is an opportunity for all education systems to look to the future; there is enormous potential for digital technology in mathematics education, regardless of the impact of COVID-19. In this paper, the researcher focuses on answering two research questions: (1) Is COVID-19 the gateway for digital learning in mathematics education? (2) What type of digital technology is being used in mathematics education during the COVID-19 pandemic? The study also provided a discussion on the implications that such digital technologies could have on research into the field of mathematics education and practice in addition to suggestions for future research directions on this topic. Interviews were chosen as techniques for the purpose of this research, which were undertaken with hundred and twenty mathematics teachers from different secondary schools in the Kingdom of Saudi Arabia. The researcher found that 98% of participants believed that COVID-19 is the gateway for digital learning in mathematics education. In addition, 97% claimed that the use of online education by schools had expanded greatly following the coronavirus outbreak. This has resulted in various forms of software being used to facilitate communicate between teachers and students included mobile technologies, touchscreens and pen tablets, digital library and designing learning objects in mathematics education, Massive Open Online Courses (MOOCs) in mathematics, and computer algebra systems (CAS) such as Mathematical, Maple, MuPAD, MathCAD, Derive and Maxima.

Introduction

The response of educational organisations across the globe to travel bans and quarantines has resulted in a shift towards learning online. This could lead to an upsurge in education – and better prepare us to deal with subsequent emergencies. The nature of global digital education is such that COVID-19 may fuel the development of strong capabilities in areas where there is sufficient connectivity, infrastructure, and resources.

In Saudi Arabia, for example the use of online education by universities and schools had expanded considerably as a result of the coronavirus outbreak. Currently, there is a dearth of research conducted on the use of digital platforms for learning mathematics (Mulenga & Marbán, 2020 ). However, one such a recent study indicates that students learn mathematics better with effective and appropriate technology (Perienen, 2020 ), while another previous study highlights that the adoption of technology in mathematics education improves learning (Niess, 2006 ). It is not yet known exactly “is COVID-19 the gateway for digital learning in mathematics education?” “What type of digital technology is being used in mathematics education during the COVID-19 closure period?” As they will be required to learn remotely in their respective homes. It is against this gap of knowledge that this study wishes to narrower.

The study also provides a discussion on the implications that such digital technologies could have on research into the field of mathematics education and practice in addition to suggestions for future research directions on this topic. This will help the reader to understand how recent developments in this area of research have evolved in the last few years.

Research importance

• Providing useful insights regarding the positive side-effects of COVID-19.
• The research keeps pace with global and local trends that advocate the need to benefit from the use of digital technology in mathematics education.
• Enriching educational libraries with a modern topic on COVID-19 and the use of digital technology in mathematics education.
• Instructing teachers to use of digital technology in mathematics education.
• Providing useful insights regarding the use or application of digital technology in mathematics education to those developing curricula in the Ministry of Education in the world.
• Contributing to opening up new prospects for further research in order to keep pace with technology and exploit its positive role in mathematics education.

Research questions

• Is COVID-19 the gateway for digital learning in mathematics education?
• What type of digital technology is being used in mathematics education during the COVID-19 pandemic?

Literature review

Theoretical framework.

The theoretical framework adopted to undertake this research include Technology Acceptance Model (TAM) (Davis et al., 1989 ) is the lense used to guide the data analysis and data interpretation to investigate the components that influence secondary students’ interests in online interactions through digital technology.

Davis’ ( 1993 ) Technology Acceptance Model (TAM) underpins this study as a theoretical framework. The TAM represents a good fit within a constructivist meta-theoretical paradigm, as it presents individual attitudes and subjective choice for using (or not using) ICT for teaching and learning. Two distinct attitude constructs namely, ‘perceived usefulness’ (PU) and ‘perceived ease of use’ (PEU) are used to frame the attitude of the academic towards engagement or indifference to the use of technology. These two behavioural constructs namely PU and PEU also directly influence whether actual engagement with the technology will occur.

Deployment of ICT innovations in mathematics education

The integration of technology within education is a highly complex process involving multiple factors and similar to all other innovative concepts, it is essential that it is not incorporated prior to testing the various different elements (Haddad & Draxler, 2002 ). It is important to substantiate innovations in terms of the level to which they are appropriate and suitable, their applicability in classrooms, their impact on the learning process and cost-effectiveness. With regard to mathematics education, numerous innovative concepts have been proposed, developed, piloted and implemented for usage with various different consequences. Particular fields in which they have verified to be successful are educational approaches based on ICT, application of open and distance learning (ODL), virtual educational platforms, distribution of open educational resources (OERs) and the propagation of research conclusions ( Iji & Abah,  2018 ).

Educational approaches based on ICT are teaching and learning methods in which ICT instruments are actively utilised to enhance the student learning (Agbo-Egwu et al., 2018 ). Schools around the world are already using a wide variety of extant digital technologies for mathematics teaching. According to Clark-Wilson et al. ( 2011 ), existing tools that are based on innovation include dynamic graphing tools, dynamic geometry tools, algorithmic programming languages, spreadsheets, data loggers (motion detectors and GPS), and computer algebra systems (CAS). Furthermore, CASs like Mathematica, Maple, MuPAD, MathCAD, Derive and Maxima are capable of facilitating active learning approaches, which enable students to become active participants in the process of discovering and consolidating their personal knowledge, thereby enhancing their theoretical and geometrical comprehension and providing a more on-depth learning strategy (Kumar & Kumaresan, 2008 ). Based on the observations of Abari ( 2014 ), student interest was maintained and their achievement levels increased subsequent to the enhancement of teaching in a higher level secondary school mathematics class through GeoGebra. The use of dynamic geometry systems (DGS) such as Cabri and Geometers Sketchpad (GSP), among others, appear to offer new perspectives on geometry in the school setting, in addition to more advanced levels by clearly facilitating the experimentation and exploration of geometrical formations and linkages (Iji et al., 2018 ). In addition to the ability to actively enhance teaching, new aspects of ICT innovations have emerged in mathematics education including the Class Learning Interactions – Observation (CLIO) tools, which allow all interactions that occur within the classroom to be systematically observed and monitored (Manny-Ikan et al., 2013 ).

An area of particular significance in terms of the implementation of ICT innovations within the field of mathematics teaching is Open and Distance Learning (ODL). The first usage of the name Massive Open Online Course was in relation to the 2008 version of the Connectivism and Connective Knowledge’ Course (Kady & Vadeboncoeur, 2013 ). Massive Open Online Courses (MOOCs) are considered to be the leading type of such courses. The design of MOOCs courses allows multiple learners to participate simultaneously, they provide students the ability to access courses at any time and from any location provided they are connected to the Internet, are publicly accessible with no entry criteria, and offer comprehensive course experiences via the Internet at no charge (Azevedo & Marques, 2017 ). Additionally, Open University frameworks offered in numerous countries are powered by a robust collection of technologies, providing high-quality competencies necessary for those who wish to be employed in long-tenured careers in the field of mathematics. ODL schemes are frequently developed for the purpose of establishing a social, cognitive and teaching presence through the Internet (Hanover Research Council, 2009 ). In this respect, synchronous online classrooms can even be more effective for educating younger children compared to conventional types of teaching as they allow visual, auditory and kinaesthetic processes to be integrated at the same time (Hastie et al., 2007 ). Likewise, crowd-based design approaches have been developed as a means of facilitating mathematical interactions between students and teachers in virtual environments (Hui et al., 2014 ).

Online teaching platforms are frequently used for augmenting discussion and cooperation among mathematicians. As suggested by Holzl ( 1999 ), the different tools utilised within virtual learning environments can include electronic mail, online forums, computer conferencing and chat groups. The development of different innovative technologies has enabled the replication of mathematical experiences based on technology both within and external to the classroom (Hofmann, 2014 ). Elluminate.com represents an effective example of an online classroom as it offers a basic user interface as well as a powerful participant window that displays the names of all session participants, along with a collection of interactivity tools including the ability to raise a hand when requesting to contribute to the debate. Messaging between users and the mathematics teacher is facilitated by the instant messaging functionality, while the whiteboard can be used by the teacher for projecting slides or by the students for writing or drawing with the text and drawing applications. Different examples of frequently utilised online learning platforms include Blackboard and Moodle ( Iji et al., 2018 ).

A different field in which ICT innovations have been deployed in mathematics teaching is mobile technologies. After the emergence of mobile technologies, one of the areas in which the fastest growth has been observed is educational applications, and it is anticipated that the expansion of these apps along with mobile technologies will continue going forward (Cherner et al., 2016 ). Hence, the following section will explain to readers how mobile technologies are used for mathematics teaching.

Use of Mobile Technologies in Mathematics Teaching and Learning

There has been increased focus among educational scholars and practitioners on the utilisation of mobile technologies (e.g., tablets and tablets) by teachers and learners in the field of mathematics. The particular attributes of mobile devices including the fact that they are portable, available, allow users to access the Internet, and are widely embraced by members of the younger generation and others mean that they are considered an emerging medium with the capability to expand the boundaries of mathematics teaching and learning outside the traditional classroom environment. White and Martin ( 2014 , p. 64) contended that the specific features of mobile devices (like the ability to capture and collect data, communicate and collaborate with different users, consume and critique media, build and generate individual forms of expression and representation) can be easily translated into the scientific, mathematical and engineering practices emphasised within the Common Core Math and Next Generation Standards (NGSS Lead States, 2013 ).

Researchers are increasingly focussing on the potential areas of application and possibilities of mobile technologies; however, this remains an under-researched subject with regard to mathematics education. Nonetheless, some studies have been conducted (e.g. Crompton & Traxler, 2015 ; Larkin & Calder, 2015 ) that have addressed the manner in which this type of technology could be utilised for mathematics teaching and learning.

The first studies into the application of mobile learning in mathematics can be traced back to the end of the 2000s (e.g. Franklin & Peng, 2008 ), and since that time, there has been considerable expansion in this kind of research in terms of both international conferences and sector-specific journals. The majority of studies analysed within this research can be categorised into three main groups: (a) research into the possible areas of application of mobile devices for mathematics teaching and learning; (b) affective studies on the utilisation of mobile devices; and (c) the utilisation of mobile devices for educating mathematics teachers.

Various researchers have concentrated on taking advantage of the features of mobile devices, including the benefits of being portable, mobile, and the ability to photograph and video actual phenomena that can subsequently be examined and discussed from a mathematical perspective. One such study was conducted by Wijers et al. ( 2010 ), who employed a game based on location named MobileMath for mobile devices with GPS technology that facilitated the creation and exploration of quadrilateral equations along with their properties in a real environment in an external location.

Other studies have concentrated on investigating the opinions and feelings experienced by mathematics teachers and students when teaching or studying mathematics via mobile devices. For instance, Holubz ( 2015 ) gathered feedback from teachers and students regarding a programme titled “Bring Your Own Device” (BYOD), which encouraged the utilisation of the Internet and mobile equipment when studying mathematics.

Lastly, it can be observed the design of inquiry tasks in mobile environments for preservice and inservice that several studies have analysed the usage of mobile devices for educating mathematics teachers. For example, Yerushalmy and Botzer ( 2011 ) presented a discussion on the theoretical aspects in addition to the problems and potential benefits underpinning teachers.

The study conducted by Crompton ( 2015 ) particular exemplifies the manner in which mobile devices can be utilised for promoting mathematical concept learning. As part of her work, Crompton proposed a research study based on design whereby iPads were utilised as a medium for supporting the learning of the notion of angles in primary school children.

Within this learning environment, mobile devices were employed by the children for the purpose of identifying and photographing forms that resembled angles that existed naturally in their environment (e.g., tree stumps, shoe patterns, or table corners). Subsequently, the photographed shapes were analysed by the students through dynamic geometry apps installed on their mobile devices. Consequently, this enabled the students to examine whether the naturally formed angles they observed in their physical surroundings in fact corresponded to the mathematic characteristics of an angle.

The usage of mobile technologies in the context of mathematics learning and teaching is a developing field of research that continues to enlarge at an exponential rate. Hence, the following part of this paper will provide an explanation on how touchscreens and pen tablets are used for Mathematics Teaching and Learning.

Use of touchscreens and pen tablets in mathematics teaching and learning

Researchers have contended that the attention spans of individuals could be impacted by the input devices utilised when performing activities or tasks supported by computers (Chen et al., 2017 ; Evans et al., 2011 ; Mangen, 2008 ; McLaughlin et al., 2009 ). For example, Chen et al. ( 2017 ) conducted a study in which they attempted to investigate and make a comparison between student’s attention span with regard to the time spent on a task and the amount of distractions when utilising touchscreens and pen tablets for problem solving tasks in the field of mathematics with virtual manipulatives. The findings revealed that those students who used touchscreens when performing the task had an increased attention span, meaning that the time spent on the task increased and they had less distractions compared to those who used pen tablets. Mangen ( 2008 ) argued that the action of clicking a mouse could distract the user from the information they are reading on the computer screen. Technologies that have emerged recently such as touchscreens, which offer intuitive and shared interfaces, introduce new methods of incorporating technology into educational practice, including the use of virtual manipulatives on touchscreen gadgets for supporting the learning of mathematics (e.g., Moyer-Packenham et al., 2016 ; Watts et al., 2016 ). Studies have indicated that when using touchscreens, there is a stronger association between the hand gestures of the user and the on-screen results compared with use of a mouse or physical keyboard (Romeo et al., 2003 ). Additionally, recent studies have shown that various teachers have tried to utilise pen-based technologies to promote student learning, specifically in the context of mathematics teaching (e.g., Cantu et al., 2008 ; Huang et al., 2017 ; Koile & Rubin, 2015 ), due to the fact that such technologies enable students to learn how to write equations or draw mathematical representations.

Digital library and designing learning objects in mathematics education

According to the definition provided by the Digital Library Manifesto (Candela et al., 2007 ), a digital library is a virtual entity that engages in a process of collecting, managing and preserving rich digital content of all types for the benefit of users. Clearly, such libraries require some form of digital storage. In the field of education, digital repositories utilise learning objects for the purpose of organising their content, which differentiates their organisational approach from those used for printed documents.

Learning objects (LO) suggested by IEEE Learning Technology Standards Committee ( 2002 ) are components of a novel kind of e-learning based on an object-focused approach in computer science. According to the definition, an LO is a digital object that one can use, reuse, and tag with metadata targeted at promoting learning.

The primary characteristics of learning objects are that they are accessible, interoperable and reusable (Polsani, 2003 ). Accessibility denotes the ability to tag learning objects with metadata, while interoperability is the technique via which learning objects are shared with other technology systems without the requirement to modify the objects, and reusability denotes the utilisation of learning objects in various different learning settings.

Widely used learning resources in virtual repositories include MERLOT (Multimedia Educational Resources for Learning and Online Teaching), Wisc-Online, DRI, Khan Academy, and EBA (Digital Repository of Turkey) (Borba et al., 2017 ).

In 1997, the Multimedia Educational Resource for Learning and Online Teaching (MERLOT) ( https://www.merlot.org/ ) was established. A resource developed by California State University, it has wide usage around the world. Users are not charged to use MERLOT and it is largely financed by higher education establishments in different countries.

The Khan Academy ( https://www.khanacademy.org ) is an individualised learning resource that caters to learners from different age groups; it provides practice tasks, educational videos and a tailored learning dashboard that allows learners to work at their own speed both within and out of the classroom environment. The mathematics missions provide guidance for early learners through to those studying calculus by using the latest adaptive technology, which is capable of identifying the learners’ strengths and learning deficiencies (Borba et al., 2017 ). Murphy et al. ( 2014 ) found also a connection between Khan Academy exercises and improved scores on basic mathematics.

What Will You Do In Math Today? ( http://researchideas.ca ) is an open repository of resources accessible on the Internet for teaching mathematics that was developed by George Gadanidis at Western University, Canada. This platform receives support from different organisations and incorporates a research-based mathematics text in which learning objects are categorised as numbers, patterns and algebra, measurements, geometry, data and probabilities (Borba et al., 2017 ).

Existing research into learning objects has largely focused on measures of quality, individualisation and mobile learning. Gadanidis et al. ( 2004 ) examined the pedagogy and the design of interfaces used in interactively visualising mathematical investigations. They reached the conclusion that a large proportion of interactive visualisations have poor designs in terms of both pedagogy and interface design. Research has demonstrated that an important aspect of the ability to predict the effectiveness of repositories is quality assurance of the LORs (Clements et al., 2015 ).

Similarities between the literature and this research

• This research is consistent with (White & Martin, 2014 ; NGSS Lead States, 2013 ) who explored the effect of mobile technologies in mathematics teaching and learning. It is also consistent with (Chen et al., 2017 ; Evans et al., 2011 ; Mangen, 2008 ; McLaughlin et al., 2009 ), who found the positive effect in touchscreens and pen tablets in mathematics teaching and learning. It is also consistent with (Polsani, 2003 ; Borba et al., 2017 ), who determine the effectiveness of using digital library and designing learning objects in mathematics education. This research is consistent with (Kumar & Kumaresan, 2008 ), who believe that emergence of such mathematical tools and its ability to deal with most of the secondary school cannot be ignored by mathematics educators. It is also consistent with (Azevedo & Marques, 2017 ), who found the advantage of using Massive Open Online Courses (MOOCs) in mathematics education. However, this research differs from all literature reviews in terms of handling the COVID-19 variable.
• The previous studies were implemented in non-Arab countries. This also represents the first study on this subject within Saudi Arabia.
• The researcher used semi-structured interviews to collect his data, but the tools used in previous studies varied due to differences in their objectives.
• The current study extended the recommendations of previous studies, such as that of Mulenga and Marbán ( 2020 ), the findings of his study motivate new areas of research. Other researchers could carry out studies on the effects of COVID-19 on Education. Others could investigate on some useful digital resources for students during the COVID-19 crisis and lockdown. It may also interest other researchers to examine if digital learning will eventually replace physical classroom in future. While digital learning is a life-long process for many students caught in the consequences of the spread of the deadly virus but may also be a way of coping with home confinement for all.
• At the time of data collection for this current study, schools were closed and there were confirmed cases of COVID-19 in Saudi. Health intervention measures had been put in place to restrict movements. The researcher interviewed the participants via Microsoft Teams or Zoom. Thus, this was very helpful to answer my research questions. In contrast to other studies, who during the time of data collection, schools were not yet closed and there were no confirmed cases of COVID-19. Health intervention measures had not yet been put in place to restrict movements. Thus, delivery mode was face-to-face in classroom settings and in the presence of the researcher.

What distinguishes this research from the existing literature

The researcher contend that this research is distinct because it is the only study to have explored COVID-19 and the use of digital technology in mathematics education in Saudi Arabia.

Aspects drawn from the literature reviews:

• Drawing on the pedagogical literature, literature reviews, and adopted scientific methodology to form the theoretical framework used in this research.
• Identification of the research methodology and tools appropriate for this research.
• Reviewing the statistical methods employed and adopting them as appropriate for this research.

Methodology

In this study, the researcher used a semi-structured interview, and the questions included in the interviews were discussed with ten academic faculty members of mathematics in universities in Saudi Arabia to determine the face validity and appropriateness of the content. The interview questions (please see Appendix 1 ) were written in Arabic and later translated into English, a process that was followed by asking a different person, to produce a translation to compare and en-sure accuracy.

Pilot interviews were then conducted to determine the relevance of the interview questions, as well as to assess the duration of the interview and to evaluate the ability to perform the task. The interview rehearsal was administered to two mathematics teachers.

The sample was selected randomly and consisted of 120 mathematics teachers whose teach third stage of secondary education, in the second semester, 2019-2020. These 120 teachers have various academic backgrounds. Some have between 3 to 10 years’ teaching experience and others between 11 and 25 years.

Sampling procedures

Emails and WhatsApp inviting teachers who were specializing in mathematics and other related areas to participate in the study. A reminder email was sent two weeks after the initial invitation to encourage participation. The message included an introductory letter and consent form that was requested be sent back to the researcher to indicate willingness to participate (the research’s principal topic, invitation paragraph, purpose of the study, why have I been chosen? Do I have to take part? Who will have the access to the research information (data)? Who do I speak to if problems arise? What will happen to the results of the research project? Ethical review of the study and contact for further information). Finally, the participants were thanked in advance for their participation. The researcher chose the first 120 participants that returned the letter to him to be part of his research since he was subject to time restrictions. The researcher interviewed the participants via Microsoft Teams or Zoom. Before the interview, in order to ensure a smooth interview process, the researcher copied the invitation and send it out to the participant. A 25 min interview was planned for each interviewee.

Data analysis

Thematic analysis, which is one of the tools of grounded theory was utilized in order to analyze the interview data. Initially, every interview was recorded and subsequently transcribed and the data were then read and re-read. The next stage involved the application of thematic coding (underlining the text in various colors) and then the data were matched to separate categories, thereby enabling reduction and synthesis of the large amount of data. Subsequent to this, every recognized commonality was divided into topics. It was necessary to supply the following three categories of the most significant with ethical issues. In the first category, all participants were informed that they were volunteers in this study and had the right to ask for any of the that responses they had given previously to be removed. In the second category, the confidentiality of participants’ identities and personal details was guaranteed, meaning that their names would not be included in the course of the translation procedure. The third category involved providing the participants comprehensive details regarding the purposes of the research.

Interviews were chosen as techniques for the purpose of this research, therefore, the researcher will discuss the findings concluded from answers to the interview questions and the literature review according to the research questions see Table  1 and Fig.  1 .

Summary of the answers to the research questions

The type of digital technology is being used in mathematics education during the COVID-19 pandemic?

The above table shows that 98% of participants believed that COVID-19 is the gateway for digital learning in mathematics education. 97% claimed that the use of online education by schools had expanded greatly following the coronavirus outbreak. In line with extreme changes worldwide, schools and universities have closed and thus interactions with colleagues and teaching through traditional lectures have transformed into an online, virtual experience. This has resulted in various forms of software being used to facilitate communicate between teachers and students. In the teaching and learning of mathematics, 40% of these used mobile technologies, whereas 30% used touchscreens and pen tablets please see Fig. ​ Fig.1. 1 . Furthermore, 3% concentrated on using digital library and designing learning objects in mathematics education, while 10% used computer algebra systems (CAS) such as Mathematica, Maple, MuPAD, MathCAD, Derive and Maxima. Additionally, 14% used Massive Open Online Courses (MOOCs) in mathematics education as follows:

One participant stated that:

“Umm…Yes, particularly the use of mobile technologies, touchscreens and pen tablets, and digital libraries in mathematical education.”

“Following the coronavirus outbreak, the nature of education may, in some ways, have fundamentally changed, potentially for the better.”

He provided the following example to illustrate this:

“Once the coronavirus outbreak has passed, the adoption of online education by schools will have expanded substantially. This is because the crisis will have provided a larger number of opportunities to develop online delivery. What remains uncertain is the extent to which the use of online teaching will remain, although it is often the case that once people become accustomed to a specific modality, they will be far more likely to use it.”

He also said:

“My students found many advantages when using mobile devices such as: cooperation and communication with various users, the capability of capturing and gathering data, constructing and generating individual types of representation and expression, and consuming and evaluating media. This can be easily translated into the scientific and mathematical.”

Another participant also contended that:

“Yes, COVID-19 is the gateway to digital learning in mathematical education. Students’ opinions of online learning may have become more positive as a result of the outbreak. Having previously viewed distance education as “very second rate”, some schools and students may now come to appreciate its potential.”

“I noticed that students’ attention spans were positively affected when touchscreens and pen tablets were used for problem-solving tasks in the field of mathematics. Therefore, as I have already mentioned, the opinions and attitudes of students towards online learning may have become more positive as a result of the outbreak.”

Another participant noted that:

“I think that COVID-19 is the gateway to digital learning in mathematical education. I did not use digital learning previously, but when COVID-19 arrived, I did use it, because these tools has become mandatory for all educational institutions, and I will continue to do so even after this pandemic is over.”

“I tried to use pen-based technologies to promote and support student learning of mathematics, such as using virtual manipulatives on touchscreen gadgets. However, I can say that such technologies enable students to learn how to draw mathematical representations easily.”

Similarly, another participant stated:

“I tried to use the devices with a touchscreen with my students. Consequently, I found that touchscreen gadgets involve virtual manipulatives that students can control in order to support the visualisation of mathematical concepts. Therefore, I can say that technology provides additional opportunities for learners to see and interact with mathematical concepts. Students can explore and make discoveries with digital tools.”

Another participant stated:

“I think that COVID-19 is the gateway to digital learning in mathematics education, as it solved the many problems that students face in the classroom. For example, several students who were previously reluctant to participate are now putting themselves forward. This is because quieter, more introverted students feel able to participate as they are not on display in front of their peers.”

Another participant also noted that:

“Teachers can see what every single student is doing, which is not how things usually work in the standard classroom.”

However, another contended that:

“Working online, it is difficult to establish whether a student is fully engaged and has sufficient understanding, a basic issue that has yet to be solved by technology.”

“Yes, prior to COVID-19, I did not encourage my students to use digital library in mathematics education, because I thought that digital technology is not easy to use. .However, I do try to use it now, and have found it to be an individualised learning resource which is accessible, interoperable and reusable.”

Furthermore, another participant noted that:

“Yes, I think that the positive side effects of COVID-19 enable me and my students to see the advantage of using digital library in mathematics education, such as MERLOT (Multimedia Educational Resources for Learning and Online Teaching), Wisc-Online, DRI, Khan Academy, and EBA (Digital Repository of Turkey).”

One participant wanted to talk about the positive side-effects of COVID-19:

“As expected in a period of crisis, every day we are inundated with negative information about COVID-19. However, although its negative effects are widely known, the outbreak of COVID-19 has also had a somewhat unexpected positive outcome.”

He went on to explain “the importance of using mathematics and understanding numeracy, which refers to the ability to use mathematics and numbers in everyday life. An individual’s level of numeracy in both their personal and work lives is extremely important. It also provides a tool to enhance critical thinking and the use of logic. These can facilitate decision making and the completion of both minor and major tasks. The greater a person’s ability to use numbers, the easier everyday chores will become. Indeed, it would not be inaccurate to state that a person’s aptitude with numbers will substantially determine their level success. Referring back to the previous question you asked me, I will now explain what I mean by the positive side-effect of COVID-19, and the use of digital technology in mathematics education.”

“All students are now studying from home, which means they have (an immense amount of) time in which to think and have to countenance the fact that everybody is buying far more than they need! I therefore asked the students to think about what they really need and what they really value. They should not buy products they will not use. Students thus have to budget on a monthly basis to avoid spending too much.”

“It is worthy of mention that I noticed that those students who used touchscreens when thinking about their real needs and values developed an increased attention span, compared with those who did not use them. In fact, before COVID-19 arrived in this country, I did not try to use touchscreens and pen tablets for problem-solving tasks in the field of mathematics, because I thought that digital technology is complicated and difficult to operate and use. However, I will now use them to support my students in mathematics.”

Another participant noted:

“Many students readily confess to a dislike of some basic mathematical concepts, and have misapprehensions about mathematics. These have a strong impact on their capacity to learn and understand mathematics and often cause a considerable amount of confusion. The most frequent misconceptions relate to the use of fractions. For instance, students may erroneously believe that 1/12 is smaller than 1/13 because 12 is less than 13.”

He then went on to add:

“For instance, when students were asked to multiply fractions by a whole number, some multiplied the numerator and denominator. This is a misconception as it shows students do not understand why you only multiply the numerator by a whole number. We should work to eradicate such misunderstandings as it is vital to apply knowledge about fractions to the real-world problems students encounter and must try and understand. Regarding health statistics, misunderstanding the size of numbers can have negative outcomes such as underestimating the risks of COVID-19.”

“Returning to your question, I can say that COVID-19 is the gateway to digital learning in mathematics education. As coronavirus grips the nation, digital technologies are increasingly being used to deliver lessons to students at home. In order to help my students with their misconceptions in mathematics, I tried to use Mathematica, Maple, MuPAD, MathCAD, Derive and Maxima, and noticed that they provide a more in-depth learning strategy.”

“Actually, I used these types of software because of the COVID-19 outbreak, but I do not think that I will use them if no COVID-19 is present. I think that this pandemic has given me an opportunity to start looking at the use of digital technology in mathematical education.”

Other participants sent a message to teachers who specialize in mathematics and other related areas across the world:

“All teachers are being provided with a unique opportunity to exploit students’ natural curiosity about the virus, the science underlying the mechanism of viral infections, and the mathematics elucidating pandemics.”

He continued:

“I do not think that we would have done this as teachers in the traditional classroom setting, but COVID-19 gave us an opportunity to use Massive Open Online Courses (MOOCs). This gave the students the ability to access courses at any time, and enabled multiple students to participate simultaneously. Increased access to digital technology for mathematics allows for a more customised learning experience. Because no two learners are exactly alike, technology can provide individual students with content and supports that are particularly helpful to their individual needs.”

“Students worldwide are coming to terms with the realities of life during a pandemic; this provided mathematics teachers with an abundance of opportunities to integrate current events into lessons in a way that helps students develop empathy, self-reflection, and personal growth.”

“During the pandemic, I used computer algebra systems (CAS) such as Mathematica, Maple, MuPAD, MathCAD, and I think that this digital technology facilitated active learning methods. It also gave the students an opportunity of becoming active participants in the process of discovering and consolidating their personal knowledge.”

He also stated:

“Unfortunately, before the coronavirus, I did not try to use any of this digital technology in mathematical education. However, the sudden and unprecedented closure of our nation’s school buildings, due to the COVID-19 pandemic, forced educators to face the most jarring and rapid change of perhaps any profession in history. Therefore, I can say that COVID-19 is the gateway to digital learning in mathematical education; therefore, I will never stop using digital technology in mathematical education, because digital technology brings mathematics education to life! We can bring videos, animations, and other into the learning process to help our students develop skills and understandings. And it can help to motivate and excite our students about their learning.”

“I think that when teachers’ anticipations towards the digital technology in mathematics education benefits are confirmed, these tools will enhance their satisfaction which ultimately achieves the perceived objectives.”

Discussion of results

The responses of the participants varied on the research questions. 98% contended that the use of digital technology in mathematics by schools had expanded considerably as a result of the coronavirus outbreak, and this was a positive aspect of the pandemic. The researcher think that due to Corona Virus Disease 2019 (COVID-19) crisis, e-learning has become a very urgent need and an imperative of education necessities in most countries all over the world. Its great importance manifested in solving the problem of quarantined students, reduce the effects of the corona-virus epidemics. According to the interviewees, teachers will perceive the digital technology as easy to use because these tools has become mandatory for all educational institutions all over the world. Another possible explanation for these findings is the fact that when teachers’ anticipations towards the digital technology in mathematics education benefits are confirmed, these tools will enhance their satisfaction and acceptance which ultimately achieves the perceived objectives. These findings could be explained by the reason that if teachers think or perceive that it is uncomplicated and simple to use the digital technology, then they are willing and intent to spend more effort and time to learn how to do so, which would undoubtedly improve their performance. In contrast, if the digital technology is complicated and difficult to operate and use, then teachers would be unwilling to try to use it.

Teachers are becoming familiar with its ‘ease of use’, and then found pedagogical purpose or ‘perceived usefulness’ (Davis, 1993 ). In this study, teachers’ ‘turn’ towards digital technology seemed to satisfy both TAM constructs of ‘ease of use’ and ‘perceived usefulness’ (Davis, 1993 ).Teachers’ beliefs and attitudes also changed with their practice as they experienced ‘ease of use’ and appreciated the ‘perceived usefulness’ of digital technology in mathematics education (Davis, 1993 ).

However, the question that arises is whether such a boom in online learning represents an enduring solution or a tool with which to respond to a crisis. The teachers’ responses indicated to the researcher that they will continue to use digital technology in mathematical education, because they have learned that technology can make mathematics easy. They provided the type of digital technology used in mathematics education during the COVID-19 pandemic? In addition, they gave us examples to show that digital technology in mathematics education encourages students to learn more than in a traditional classroom environment.

40% of them used mobile technologies in mathematics teaching and learning, one of them mentioned that “my students found many advantages when using mobile devices such as the ability to capture and collect data, communicate and collaborate with different users, consume and critique media, build and generate individual forms of expression and representation, and this is can be easily translated into the scientific and mathematical”. This is consistent with other researcher’ findings, such as (White & Martin, 2014 ; NGSS Lead States, 2013 ), who showed that the specific features of mobile devices (like the ability to capture and collect data, communicate and collaborate with different users, consume and critique media, build and generate individual forms of expression and representation) can be easily translated into the scientific, mathematical and engineering practices emphasised within the Common Core Math and Next Generation Standards. The participants mentioned that when the student finds it difficult to solve the task in mathematics, he can access to the mobile technologies and open the videos see the solutions, which allows students to learn at their own pace and in their own learning style. The researcher thinks that students and teachers are given new experiences through the application of mobile devices as instruments in mathematical education. Since the way in which we teach and learn is being quickly transformed by technology, it is important for teachers to be aware of conventional techniques of teaching mathematics and to be willing to support them. However, these methods need to be coordinated with efficient and relevant utilisation of technology, possibly involving applications and mobile devices.

With respect to the utilisation of mobile technologies for teaching and learning mathematics, the majority of learners have already determined that mobile phones constitute large parts of their lives both within and out of the classroom. While different technologies, such as pencils and paper in addition to computer software were also recognised as being part of this group, those currently studying in schools cannot perceive a world without mobile devices. Nonetheless, the availability of mobile technologies for students forms a relation between students and mathematics that is not broadly embraced by mathematics teachers, which interrupts the conventional transferal of mathematics knowledge between teachers and students, and has received minimal attention in the literature.

30% of participants used touchscreens and pen tablets in mathematics teaching and learning, one of them stated that “it is worth to mention that I noticed that those students who used touchscreens when thinking about what they really need and what they really value had an increased attention span, compared to those who did not use touchscreens. Actually before COVID-19 come to this country I did not try to use touchscreens and pen tablets for problem solving tasks in the field of mathematics, but now I will use them to support my students in mathematics”. This is consistent with (Chen et al., 2017 ; Evans et al., 2011 ; Mangen, 2008 ; McLaughlin et al., 2009 ), who indicated that the attention spans of individuals could be impacted by the input devices utilised when performing activities or tasks supported by computers. The participants analysed students’ attention when using touchscreens and pen tablets to solve equations problems with virtual manipulatives. They found that the students could maintain more attention, in terms of greater time-on-task and fewer distractors. This is consistent with recent studies have who shown that various teachers have tried to utilise pen-based technologies to promote student learning, specifically in the context of mathematics teaching (e.g., Cantu et al., 2008 ; Huang et al., 2017 ; Koile & Rubin, 2015 ), due to the fact that such technologies enable students to learn how to write equations or draw mathematical representations. Therefore, students have to pay attention to the learning process if effective learning is to occur, and this very important because if they do not do that the information they receive will quickly fade and rarely have a lasting impact. The researcher is of the opinion that teachers could utilise touchscreen in the mathematics classroom in order to support the activities which concentrate on student manipulations associated with the attention given to learning content.

3% of them concentrated on using digital library and designing learning objects in mathematics education, one of them mentioned that “yes, because before the COVID-19 I did not encourage my students to use Digital library in mathematics education, and now I tried to use it and I found that an individualised learning resource that they are accessible, interoperable and reusable.” This is consistent with (Polsani, 2003 ). Another participant also noted that: “yes, I think that the positive side-effects of COVID-19 that make me and my students to see the advantage of using Digital library in mathematics education such as MERLOT (Multimedia Educational Resources for Learning and Online Teaching), Wisc-Online, DRI, Khan Academy, and EBA (Digital Repository of Turkey)”. This is consistent with (Borba et al., 2017 ). For example, Khan Academy enabled students to move at a pace that is more appropriate to their learning needs. He reported that students are doing more mathematics problems than they would in a standard classroom. Therefore, we can say that it is not surprising that their skill level would also increase. One of the participants reported that many more of his students were getting 100% on their homework. This is consistent with Murphy et al., 2014 who found a connection between Khan Academy exercises and improved scores on basic mathematics. Khan Academy also gives teachers and students access to graphics and illustrations that are hard to replicate at the blackboard. At least two teachers in the study talked about the value of Khan Academy to teach students skills such as responsibility and self-discipline. The researcher think that the accessibility of mathematics education resources via the Internet (e.g., digital libraries and learning objects) has led numerous students to consult such resources prior to seeking assistance from teachers or textbooks, which leads one to question how such resources are structured to provide this type of access and the manner in which they are designed pedagogically to promote conceptual understanding.

A point worth mentioning is that there are useful insights regarding the positive side-effects of COVID-19. According to the participants, their opinion of digital technology in mathematics education has grown more positive as a result of the increased usage of it during the coronavirus school building closures. In addition, they plan to continue using those newfound skills even when school buildings reopen. We can see that this shift in practice has provided an opportunity to reconsider how technology use in mathematics can be utilised to improve student learning. On the other hand, one of participants mentioned that “In fact, before COVID-19 arrived in this country, I did not try to use touchscreens and pen tablets for problem-solving tasks in the field of mathematics, because I thought that digital technology is complicated and difficult to operate and use. However, I will now use them to support my students in mathematics.” Therefore, the researcher do not want to forget the IT help desks who also played a significant role in helping prevent those negative feelings from being much higher, actually the calls and emails were flooding in around the clock.

10% of the participants used computer algebra systems (CAS) such as Mathematica, Maple, MuPAD, MathCAD, Derive and Maxima, one of them stated that“ to help my students with their misconceptions in mathematics, I tried to use Mathematica, Maple, MuPAD, MathCAD, Derive and Maxima, and I noticed that they provided a more on depth learning strategy.” This is consistent with (Kumar & Kumaresan, 2008 ), who mentioned that these softwares can providing a more on-depth learning strategy. Another participant noted “ I used during the epidemic, computer algebra systems (CAS) such as Mathematica, Maple, MuPAD, MathCAD, and I think these digital technology facilitated active learning methods, which gave the students the chance to become active participants in the process of discovering and consolidating their personal knowledge”. This is consistent with also (Kumar & Kumaresan, 2008 ). The researcher believe that emergence of such mathematical tools and its ability to deal with most of the secondary school cannot be ignored by mathematics educators. Because what the researcher understanding from the participants that using a computer algebra system (CAS) during the pandemic crisis provided many opportunities for improving student learning. The students who were taught with CAS were more successful than students without CAS at three levels: basic computation, more advanced computation and complex symbolic problems. This is not surprising because eight of the participants reported for their students who were taught the concept of derivative with and without CAS. Four of the participants has also examined student motivation when used effectively during the pandemic crisis, they found that CAS can make mathematics more interesting and meaningful to students.

14% of them used Massive Open Online Courses (MOOCs) in mathematics education, one of the participants noted that “ I do not think we can done this as teachers in traditional classroom, but COVID-19 gave us the oppotinty to use Massive Open Online Courses (MOOCs), which gave the students the ability to access courses at any time, and allows multiple students to participate simultaneously”. This is consistent with (Azevedo & Marques, 2017 ). Further research is needed into the nature of the learning that participants engage in when participating in MOOCs and in what conditions before it will be possible to make definite conclusions. However, students are enabled, by digital learning, to study in the comfort of their own homes. If they have the required digital devices, they are able to occupy the front seats in the virtual mathematics classroom. It is implied by results that teachers hold the opinion that they will be enabled by digital learning to have a pedagogical mathematical move towards a less formalised teaching technique, such a method would be interesting and entertaining, instead of being traditional and rigorous.

To sum up, as we see from above that 40% of participants used mobile technologies in mathematics teaching and learning, and this is considered as high percentage compared with other digital technologies used. The main feature of mobile technologies that distinguishes it from other learning technologies is its mobility. The researcher think that mobile technologies are highly popular amongst secondary students due to their being easily carried, wireless, containing many apps making it easy for the student to do multiple tasks at one stand. As a result, commercial competitive industry has compelled manufacturers to present new creative features of competitive traits. In addition, it is only understandable why mobile phone companies have worked hard to develop the 5th generation mobile phones that enabled users not only to talk but actually do almost everything they now do with their PC. This mean that all other application or digital technologies ran in mobile technology at any environment, regardless of the OS, the Net or the type of cellular. Users can download any applications from many websites, whenever they want. They can run the application without being connected to the net. In addition, the mobile technologies is also Mobile phones are available and are part of the daily culture of almost every student. The researcher do not want to forget that coronavirus pandemic is a chance to see all these types and benefits of digital technologies in mathematics education, because 98% of participants above believed that COVID-19 is the gateway for digital learning in mathematics education. In addition, 97% claimed that the use of online education by schools, teachers and students had expanded greatly following the coronavirus outbreak.

Results show different types of digital technology used in mathematics education included (mobile technologies, touchscreens and pen tablets, digital library and designing learning objects in mathematics education, Massive Open Online Courses (MOOCs) in mathematics, and computer algebra systems (CAS) such as Mathematical, Maple, MuPAD, MathCAD, Derive and Maxima.), and the effects varied by the type of educational technology used. However, in view of the COVID-19 school closure period, it is apparent that digital learning in mathematics education is the instant positive response.

Implications for further studies

This study has showed that the adoption of digital learning as a response to COVID-19 stimulates the growth of digital learning in mathematics education in Saudi Arabia. The priviledge of the current situation for students engaged in digital learning is to position this transformation not just as a quick response but as a way of combating the spread of COVID-19 and the next transferable disease. The findings of this study motivate new areas of research. Other researchers could carry out studies on the effects of COVID-19 on other areas of the education field. Others could investigate on other useful digital resources for mathematics students during the COVID-19 crisis. It may also interest other researchers to answer the following questions: Will digital learning replace classroom education anytime soon? What the future holds for digitised education post-Covid-19. How will Covid-19 affect the future of igital mathematics education? While some believe that the unplanned and rapid move to digital technology – with no training, insufficient bandwidth, and little preparation will result in a poor user experience that is unconducive to sustained growth, others believe that a new hybrid model of education will emerge, with significant benefits. The researcher believed that a time of crisis is also an opportunity for all education systems to look into the future, adjust to possible threats, and build their capacity. Major world events are often an inflection point for rapid innovation – while we have yet to see whether this will apply to digital technology post-COVID-19. Finally, the researcher think that digital learning system designers and developers should pay further attention to these two essential factors (perceived usefulness and perceived ease of use).

Limitations of the study

Although this study was carefully prepared, it still faced a number of limitations:

• This study focused only on government secondary schools in the east of Saudi Arabia. However, the researcher believes that this city was a good place to conduct this study, because it has a big population which is drawn from different parts of the Kingdom of Saudi Arabia.
• The study sample focused on teachers only, because they are the first people who play a key role in educating students in the classroom. However, the study could have included students if there were no restrictions of time.

Recommendations

In view of the findings, the researchers recommend the following:

• These digital technologies must be included in mathematics curricula at various stages of education.
• The stakeholders should take advantage of the findings of this study to encourage teachers to continue using these technologies in mathematics education.
• Further research is needed to answer the questions that arose in the discussion section.

Acknowledgements

I would like to thank our almighty God who gave me the power and the will to finish this research. My heartfelt appreciation and gratitude go to my family, particularly my parents and my wife for their encouragement and cooperation.

• Did you use of digital technology in mathematics education before COVID-19 outbreak?
• Are you going to continue to use the digital technology in mathematics education after corona the COVID-19? Why? Do you think this is an enduring solution or a tool with which to respond to a crisis?
• How could digital technologies help during the pandemic?
• Regarding the use of technology in mathematics education, are there any particular technologies you feel excited about?
• How COVID-19 is changing mathematics teacher’s perception of using digital technology?
• Does “perceived usefulness” influence mathematics teachers’ acceptance of digital technology? How?
• Does personal experience influence mathematics teachers acceptance of digital technology? How?
• Does “perceived ease of use” influence mathematics teachers’ acceptance of digital technology? How?
• Is there anything else you would like to add?

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Glimpses of Teaching in the New Normal: Changes, Challenges, and Chances

The current context on virtual education has provided a plethora of studies investigating educational institutions’ response strategies to remote and online learning formats. However, to provide a much-grounded description of the realities in the field, this study explored the role of teachers in the virtual learning environment through their narratives reflective of their experiences. Furthermore, it employed a qualitative narrative and descriptive research method anchored on the tenets of Husserlian descriptive phenomenology. Six higher education professors from different colleges and universities in Central Visayas, Philippines served as the participants of the study. Data were collected from in-depth interviews done virtually via Zoom. Based on participant narratives, the following emerged as themes: changes, challenges, and chances, respectively, in all the teaching-learning phases, from preparation and implementation to assessment. These changes , challenges , and chances shared by the participants have shed light on teaching being a multifaceted profession, putting emphasis on teachers as innovators of change. Thus, it is recommended that colleges and universities should establish an institutional based framework for emergency remote teaching. The framework should highlight policies on virtual education, upscale and upskill teachers, address learning losses, and promote strategies to build resilience in students and teachers.  

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Blended learning: the new normal and emerging technologies

• Charles Dziuban 1 ,
• Charles R. Graham 2 ,
• Patsy D. Moskal   ORCID: orcid.org/0000-0001-6376-839X 1 ,
• Anders Norberg 3 &
• Nicole Sicilia 1  

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This study addressed several outcomes, implications, and possible future directions for blended learning (BL) in higher education in a world where information communication technologies (ICTs) increasingly communicate with each other. In considering effectiveness, the authors contend that BL coalesces around access, success, and students’ perception of their learning environments. Success and withdrawal rates for face-to-face and online courses are compared to those for BL as they interact with minority status. Investigation of student perception about course excellence revealed the existence of robust if-then decision rules for determining how students evaluate their educational experiences. Those rules were independent of course modality, perceived content relevance, and expected grade. The authors conclude that although blended learning preceded modern instructional technologies, its evolution will be inextricably bound to contemporary information communication technologies that are approximating some aspects of human thought processes.

Introduction

Blended learning and research issues.

Blended learning (BL), or the integration of face-to-face and online instruction (Graham 2013 ), is widely adopted across higher education with some scholars referring to it as the “new traditional model” (Ross and Gage 2006 , p. 167) or the “new normal” in course delivery (Norberg et al. 2011 , p. 207). However, tracking the accurate extent of its growth has been challenging because of definitional ambiguity (Oliver and Trigwell 2005 ), combined with institutions’ inability to track an innovative practice, that in many instances has emerged organically. One early nationwide study sponsored by the Sloan Consortium (now the Online Learning Consortium) found that 65.2% of participating institutions of higher education (IHEs) offered blended (also termed hybrid ) courses (Allen and Seaman 2003 ). A 2008 study, commissioned by the U.S. Department of Education to explore distance education in the U.S., defined BL as “a combination of online and in-class instruction with reduced in-class seat time for students ” (Lewis and Parsad 2008 , p. 1, emphasis added). Using this definition, the study found that 35% of higher education institutions offered blended courses, and that 12% of the 12.2 million documented distance education enrollments were in blended courses.

The 2017 New Media Consortium Horizon Report found that blended learning designs were one of the short term forces driving technology adoption in higher education in the next 1–2 years (Adams Becker et al. 2017 ). Also, blended learning is one of the key issues in teaching and learning in the EDUCAUSE Learning Initiative’s 2017 annual survey of higher education (EDUCAUSE 2017 ). As institutions begin to examine BL instruction, there is a growing research interest in exploring the implications for both faculty and students. This modality is creating a community of practice built on a singular and pervasive research question, “How is blended learning impacting the teaching and learning environment?” That question continues to gain traction as investigators study the complexities of how BL interacts with cognitive, affective, and behavioral components of student behavior, and examine its transformation potential for the academy. Those issues are so compelling that several volumes have been dedicated to assembling the research on how blended learning can be better understood (Dziuban et al. 2016 ; Picciano et al. 2014 ; Picciano and Dziuban 2007 ; Bonk and Graham 2007 ; Kitchenham 2011 ; Jean-François 2013 ; Garrison and Vaughan 2013 ) and at least one organization, the Online Learning Consortium, sponsored an annual conference solely dedicated to blended learning at all levels of education and training (2004–2015). These initiatives address blended learning in a wide variety of situations. For instance, the contexts range over K-12 education, industrial and military training, conceptual frameworks, transformational potential, authentic assessment, and new research models. Further, many of these resources address students’ access, success, withdrawal, and perception of the degree to which blended learning provides an effective learning environment.

Currently the United States faces a widening educational gap between our underserved student population and those communities with greater financial and technological resources (Williams 2016 ). Equal access to education is a critical need, one that is particularly important for those in our underserved communities. Can blended learning help increase access thereby alleviating some of the issues faced by our lower income students while resulting in improved educational equality? Although most indicators suggest “yes” (Dziuban et al. 2004 ), it seems that, at the moment, the answer is still “to be determined.” Quality education presents a challenge, evidenced by many definitions of what constitutes its fundamental components (Pirsig 1974 ; Arum et al. 2016 ). Although progress has been made by initiatives, such as, Quality Matters ( 2016 ), the OLC OSCQR Course Design Review Scorecard developed by Open SUNY (Open SUNY n.d. ), the Quality Scorecard for Blended Learning Programs (Online Learning Consortium n.d. ), and SERVQUAL (Alhabeeb 2015 ), the issue is by no means resolved. Generally, we still make quality education a perceptual phenomenon where we ascribe that attribute to a course, educational program, or idea, but struggle with precisely why we reached that decision. Searle ( 2015 ), summarizes the problem concisely arguing that quality does not exist independently, but is entirely observer dependent. Pirsig ( 1974 ) in his iconic volume on the nature of quality frames the context this way,

“There is such thing as Quality, but that as soon as you try to define it, something goes haywire. You can’t do it” (p. 91).

Therefore, attempting to formulate a semantic definition of quality education with syntax-based metrics results in what O’Neil (O'Neil 2017 ) terms surrogate models that are rough approximations and oversimplified. Further, the derived metrics tend to morph into goals or benchmarks, losing their original measurement properties (Goodhart 1975 ).

Information communication technologies in society and education

Blended learning forces us to consider the characteristics of digital technology, in general, and information communication technologies (ICTs), more specifically. Floridi ( 2014 ) suggests an answer proffered by Alan Turing: that digital ICTs can process information on their own, in some sense just as humans and other biological life. ICTs can also communicate information to each other, without human intervention, but as linked processes designed by humans. We have evolved to the point where humans are not always “in the loop” of technology, but should be “on the loop” (Floridi 2014 , p. 30), designing and adapting the process. We perceive our world more and more in informational terms, and not primarily as physical entities (Floridi 2008 ). Increasingly, the educational world is dominated by information and our economies rest primarily on that asset. So our world is also blended, and it is blended so much that we hardly see the individual components of the blend any longer. Floridi ( 2014 ) argues that the world has become an “infosphere” (like biosphere) where we live as “inforgs.” What is real for us is shifting from the physical and unchangeable to those things with which we can interact.

Floridi also helps us to identify the next blend in education, involving ICTs, or specialized artificial intelligence (Floridi 2014 , 25; Norberg 2017 , 65). Learning analytics, adaptive learning, calibrated peer review, and automated essay scoring (Balfour 2013 ) are advanced processes that, provided they are good interfaces, can work well with the teacher— allowing him or her to concentrate on human attributes such as being caring, creative, and engaging in problem-solving. This can, of course, as with all technical advancements, be used to save resources and augment the role of the teacher. For instance, if artificial intelligence can be used to work along with teachers, allowing them more time for personal feedback and mentoring with students, then, we will have made a transformational breakthrough. The Edinburg University manifesto for teaching online says bravely, “Automation need not impoverish education – we welcome our robot colleagues” (Bayne et al. 2016 ). If used wisely, they will teach us more about ourselves, and about what is truly human in education. This emerging blend will also affect curricular and policy questions, such as the what? and what for? The new normal for education will be in perpetual flux. Floridi’s ( 2014 ) philosophy offers us tools to understand and be in control and not just sit by and watch what happens. In many respects, he has addressed the new normal for blended learning.

Literature of blended learning

A number of investigators have assembled a comprehensive agenda of transformative and innovative research issues for blended learning that have the potential to enhance effectiveness (Garrison and Kanuka 2004 ; Picciano 2009 ). Generally, research has found that BL results in improvement in student success and satisfaction, (Dziuban and Moskal 2011 ; Dziuban et al. 2011 ; Means et al. 2013 ) as well as an improvement in students’ sense of community (Rovai and Jordan 2004 ) when compared with face-to-face courses. Those who have been most successful at blended learning initiatives stress the importance of institutional support for course redesign and planning (Moskal et al. 2013 ; Dringus and Seagull 2015 ; Picciano 2009 ; Tynan et al. 2015 ). The evolving research questions found in the literature are long and demanding, with varied definitions of what constitutes “blended learning,” facilitating the need for continued and in-depth research on instructional models and support needed to maximize achievement and success (Dringus and Seagull 2015 ; Bloemer and Swan 2015 ).

Educational access

The lack of access to educational technologies and innovations (sometimes termed the digital divide) continues to be a challenge with novel educational technologies (Fairlie 2004 ; Jones et al. 2009 ). One of the promises of online technologies is that they can increase access to nontraditional and underserved students by bringing a host of educational resources and experiences to those who may have limited access to on-campus-only higher education. A 2010 U.S. report shows that students with low socioeconomic status are less likely to obtain higher levels of postsecondary education (Aud et al. 2010 ). However, the increasing availability of distance education has provided educational opportunities to millions (Lewis and Parsad 2008 ; Allen et al. 2016 ). Additionally, an emphasis on open educational resources (OER) in recent years has resulted in significant cost reductions without diminishing student performance outcomes (Robinson et al. 2014 ; Fischer et al. 2015 ; Hilton et al. 2016 ).

Unfortunately, the benefits of access may not be experienced evenly across demographic groups. A 2015 study found that Hispanic and Black STEM majors were significantly less likely to take online courses even when controlling for academic preparation, socioeconomic status (SES), citizenship, and English as a second language (ESL) status (Wladis et al. 2015 ). Also, questions have been raised about whether the additional access afforded by online technologies has actually resulted in improved outcomes for underserved populations. A distance education report in California found that all ethnic minorities (except Asian/Pacific Islanders) completed distance education courses at a lower rate than the ethnic majority (California Community Colleges Chancellor’s Office 2013 ). Shea and Bidjerano ( 2014 , 2016 ) found that African American community college students who took distance education courses completed degrees at significantly lower rates than those who did not take distance education courses. On the other hand, a study of success factors in K-12 online learning found that for ethnic minorities, only 1 out of 15 courses had significant gaps in student test scores (Liu and Cavanaugh 2011 ). More research needs to be conducted, examining access and success rates for different populations, when it comes to learning in different modalities, including fully online and blended learning environments.

Framing a treatment effect

Over the last decade, there have been at least five meta-analyses that have addressed the impact of blended learning environments and its relationship to learning effectiveness (Zhao et al. 2005 ; Sitzmann et al. 2006 ; Bernard et al. 2009 ; Means et al. 2010 , 2013 ; Bernard et al. 2014 ). Each of these studies has found small to moderate positive effect sizes in favor of blended learning when compared to fully online or traditional face-to-face environments. However, there are several considerations inherent in these studies that impact our understanding the generalizability of outcomes.

Dziuban and colleagues (Dziuban et al. 2015 ) analyzed the meta-analyses conducted by Means and her colleagues (Means et al. 2013 ; Means et al. 2010 ), concluding that their methods were impressive as evidenced by exhaustive study inclusion criteria and the use of scale-free effect size indices. The conclusion, in both papers, was that there was a modest difference in multiple outcome measures for courses featuring online modalities—in particular, blended courses. However, with blended learning especially, there are some concerns with these kinds of studies. First, the effect sizes are based on the linear hypothesis testing model with the underlying assumption that the treatment and the error terms are uncorrelated, indicating that there is nothing else going on in the blending that might confound the results. Although the blended learning articles (Means et al. 2010 ) were carefully vetted, the assumption of independence is tenuous at best so that these meta-analysis studies must be interpreted with extreme caution.

There is an additional concern with blended learning as well. Blends are not equivalent because of the manner on which they are configured. For instance, a careful reading of the sources used in the Means, et al. papers will identify, at minimum, the following blending techniques: laboratory assessments, online instruction, e-mail, class web sites, computer laboratories, mapping and scaffolding tools, computer clusters, interactive presentations and e-mail, handwriting capture, evidence-based practice, electronic portfolios, learning management systems, and virtual apparatuses. These are not equivalent ways in which to configure courses, and such nonequivalence constitutes the confounding we describe. We argue here that, in actuality, blended learning is a general construct in the form of a boundary object (Star and Griesemer 1989 ) rather than a treatment effect in the statistical sense. That is, an idea or concept that can support a community of practice, but is weakly defined fostering disagreement in the general group. Conversely, it is stronger in individual constituencies. For instance, content disciplines (i.e. education, rhetoric, optics, mathematics, and philosophy) formulate a more precise definition because of commonly embraced teaching and learning principles. Quite simply, the situation is more complicated than that, as Leonard Smith ( 2007 ) says after Tolstoy,

“All linear models resemble each other, each non nonlinear system is unique in its own way” (p. 33).

This by no means invalidates these studies, but effect size associated with blended learning should be interpreted with caution where the impact is evaluated within a particular learning context.

Study objectives

This study addressed student access by examining success and withdrawal rates in the blended learning courses by comparing them to face-to-face and online modalities over an extended time period at the University of Central Florida. Further, the investigators sought to assess the differences in those success and withdrawal rates with the minority status of students. Secondly, the investigators examined the student end-of-course ratings of blended learning and other modalities by attempting to develop robust if-then decision rules about what characteristics of classes and instructors lead students to assign an “excellent” value to their educational experience. Because of the high stakes nature of these student ratings toward faculty promotion, awards, and tenure, they act as a surrogate measure for instructional quality. Next, the investigators determined the conditional probabilities for students conforming to the identified rule cross-referenced by expected grade, the degree to which they desired to take the course, and course modality.

Student grades by course modality were recoded into a binary variable with C or higher assigned a value of 1, and remaining values a 0. This was a declassification process that sacrificed some specificity but compensated for confirmation bias associated with disparate departmental policies regarding grade assignment. At the measurement level this was an “on track to graduation index” for students. Withdrawal was similarly coded by the presence or absence of its occurrence. In each case, the percentage of students succeeding or withdrawing from blended, online or face-to-face courses was calculated by minority and non-minority status for the fall 2014 through fall 2015 semesters.

Next, a classification and regression tree (CART) analysis (Brieman et al. 1984 ) was performed on the student end-of-course evaluation protocol ( Appendix 1 ). The dependent measure was a binary variable indicating whether or not a student assigned an overall rating of excellent to his or her course experience. The independent measures in the study were: the remaining eight rating items on the protocol, college membership, and course level (lower undergraduate, upper undergraduate, and graduate). Decision trees are efficient procedures for achieving effective solutions in studies such as this because with missing values imputation may be avoided with procedures such as floating methods and the surrogate formation (Brieman et al. 1984 , Olshen et al. 1995 ). For example, a logistic regression method cannot efficiently handle all variables under consideration. There are 10 independent variables involved here; one variable has three levels, another has nine, and eight have five levels each. This means the logistic regression model must incorporate more than 50 dummy variables and an excessively large number of two-way interactions. However, the decision-tree method can perform this analysis very efficiently, permitting the investigator to consider higher order interactions. Even more importantly, decision trees represent appropriate methods in this situation because many of the variables are ordinally scaled. Although numerical values can be assigned to each category, those values are not unique. However, decision trees incorporate the ordinal component of the variables to obtain a solution. The rules derived from decision trees have an if-then structure that is readily understandable. The accuracy of these rules can be assessed with percentages of correct classification or odds-ratios that are easily understood. The procedure produces tree-like rule structures that predict outcomes.

The model-building procedure for predicting overall instructor rating

For this study, the investigators used the CART method (Brieman et al. 1984 ) executed with SPSS 23 (IBM Corp 2015 ). Because of its strong variance-sharing tendencies with the other variables, the dependent measure for the analysis was the rating on the item Overall Rating of the Instructor , with the previously mentioned indicator variables (college, course level, and the remaining 8 questions) on the instrument. Tree methods are recursive, and bisect data into subgroups called nodes or leaves. CART analysis bases itself on: data splitting, pruning, and homogeneous assessment.

Splitting the data into two (binary) subsets comprises the first stage of the process. CART continues to split the data until the frequencies in each subset are either very small or all observations in a subset belong to one category (e.g., all observations in a subset have the same rating). Usually the growing stage results in too many terminate nodes for the model to be useful. CART solves this problem using pruning methods that reduce the dimensionality of the system.

The final stage of the analysis involves assessing homogeneousness in growing and pruning the tree. One way to accomplish this is to compute the misclassification rates. For example, a rule that produces a .95 probability that an instructor will receive an excellent rating has an associated error of 5.0%.

Implications for using decision trees

Although decision-tree techniques are effective for analyzing datasets such as this, the reader should be aware of certain limitations. For example, since trees use ranks to analyze both ordinal and interval variables, information can be lost. However, the most serious weakness of decision tree analysis is that the results can be unstable because small initial variations can lead to substantially different solutions.

For this study model, these problems were addressed with the k-fold cross-validation process. Initially the dataset was partitioned randomly into 10 subsets with an approximately equal number of records in each subset. Each cohort is used as a test partition, and the remaining subsets are combined to complete the function. This produces 10 models that are all trained on different subsets of the original dataset and where each has been used as the test partition one time only.

Although computationally dense, CART was selected as the analysis model for a number of reasons— primarily because it provides easily interpretable rules that readers will be able evaluate in their particular contexts. Unlike many other multivariate procedures that are even more sensitive to initial estimates and require a good deal of statistical sophistication for interpretation, CART has an intuitive resonance with researcher consumers. The overriding objective of our choice of analysis methods was to facilitate readers’ concentration on our outcomes rather than having to rely on our interpretation of the results.

Institution-level evaluation: Success and withdrawal

The University of Central Florida (UCF) began a longitudinal impact study of their online and blended courses at the start of the distributed learning initiative in 1996. The collection of similar data across multiple semesters and academic years has allowed UCF to monitor trends, assess any issues that may arise, and provide continual support for both faculty and students across varying demographics. Table  1 illustrates the overall success rates in blended, online and face-to-face courses, while also reporting their variability across minority and non-minority demographics.

While success (A, B, or C grade) is not a direct reflection of learning outcomes, this overview does provide an institutional level indication of progress and possible issues of concern. BL has a slight advantage when looking at overall success and withdrawal rates. This varies by discipline and course, but generally UCF’s blended modality has evolved to be the best of both worlds, providing an opportunity for optimizing face-to-face instruction through the effective use of online components. These gains hold true across minority status. Reducing on-ground time also addresses issues that impact both students and faculty such as parking and time to reach class. In addition, UCF requires faculty to go through faculty development tailored to teaching in either blended or online modalities. This 8-week faculty development course is designed to model blended learning, encouraging faculty to redesign their course and not merely consider blended learning as a means to move face-to-face instructional modules online (Cobb et al. 2012 ; Lowe 2013 ).

Withdrawal (Table  2 ) from classes impedes students’ success and retention and can result in delayed time to degree, incurred excess credit hour fees, or lost scholarships and financial aid. Although grades are only a surrogate measure for learning, they are a strong predictor of college completion. Therefore, the impact of any new innovation on students’ grades should be a component of any evaluation. Once again, the blended modality is competitive and in some cases results in lower overall withdrawal rates than either fully online or face-to-face courses.

The students’ perceptions of their learning environments

Other potentially high-stakes indicators can be measured to determine the impact of an innovation such as blended learning on the academy. For instance, student satisfaction and attitudes can be measured through data collection protocols, including common student ratings, or student perception of instruction instruments. Given that those ratings often impact faculty evaluation, any negative reflection can derail the successful implementation and scaling of an innovation by disenfranchised instructors. In fact, early online and blended courses created a request by the UCF faculty senate to investigate their impact on faculty ratings as compared to face-to-face sections. The UCF Student Perception of Instruction form is released automatically online through the campus web portal near the end of each semester. Students receive a splash page with a link to each course’s form. Faculty receive a scripted email that they can send to students indicating the time period that the ratings form will be available. The forms close at the beginning of finals week. Faculty receive a summary of their results following the semester end.

The instrument used for this study was developed over a ten year period by the faculty senate of the University of Central Florida, recognizing the evolution of multiple course modalities including blended learning. The process involved input from several constituencies on campus (students, faculty, administrators, instructional designers, and others), in attempt to provide useful formative and summative instructional information to the university community. The final instrument was approved by resolution of the senate and, currently, is used across the university. Students’ rating of their classes and instructors comes with considerable controversy and disagreement with researchers aligning themselves on both sides of the issue. Recently, there have been a number of studies criticizing the process (Uttl et al. 2016 ; Boring et al. 2016 ; & Stark and Freishtat 2014 ). In spite of this discussion, a viable alternative has yet to emerge in higher education. So in the foreseeable future, the process is likely to continue. Therefore, with an implied faculty senate mandate this study was initiated by this team of researchers.

Prior to any analysis of the item responses collected in this campus-wide student sample, the psychometric quality (domain sampling) of the information yielded by the instrument was assessed. Initially, the reliability (internal consistency) was derived using coefficient alpha (Cronbach 1951 ). In addition, Guttman ( 1953 ) developed a theorem about item properties that leads to evidence about the quality of one’s data, demonstrating that as the domain sampling properties of items improve, the inverse of the correlation matrix among items will approach a diagonal. Subsequently, Kaiser and Rice ( 1974 ) developed the measure of sampling adequacy (MSA) that is a function of the Guttman Theorem. The index has an upper bound of one with Kaiser offering some decision rules for interpreting the value of MSA. If the value of the index is in the .80 to .99 range, the investigator has evidence of an excellent domain sample. Values in the .70s signal an acceptable result, and those in the .60s indicate data that are unacceptable. Customarily, the MSA has been used for data assessment prior to the application of any dimensionality assessments. Computation of the MSA value gave the investigators a benchmark for the construct validity of the items in this study. This procedure has been recommended by Dziuban and Shirkey ( 1974 ) prior to any latent dimension analysis and was used with the data obtained for this study. The MSA for the current instrument was .98 suggesting excellent domain sampling properties with an associated alpha reliability coefficient of .97 suggesting superior internal consistency. The psychometric properties of the instrument were excellent with both measures.

The online student ratings form presents an electronic data set each semester. These can be merged across time to create a larger data set of completed ratings for every course across each semester. In addition, captured data includes course identification variables including prefix, number, section and semester, department, college, faculty, and class size. The overall rating of effectiveness is used most heavily by departments and faculty in comparing across courses and modalities (Table  3 ).

The finally derived tree (decision rules) included only three variables—survey items that asked students to rate the instructor’s effectiveness at:

Helping students achieve course objectives,

Creating an environment that helps students learn, and

Communicating ideas and information.

None of the demographic variables associated with the courses contributed to the final model. The final rule specifies that if a student assigns an excellent rating to those three items, irrespective of their status on any other condition, the probability is .99 that an instructor will receive an overall rating of excellent. The converse is true as well. A poor rating on all three of those items will lead to a 99% chance of an instructor receiving an overall rating of poor.

Tables  4 , 5 and 6 present a demonstration of the robustness of the CART rule for variables on which it was not developed: expected course grade, desire to take the course and modality.

In each case, irrespective of the marginal probabilities, those students conforming to the rule have a virtually 100% chance of seeing the course as excellent. For instance, 27% of all students expecting to fail assigned an excellent rating to their courses, but when they conformed to the rule the percentage rose to 97%. The same finding is true when students were asked about their desire to take the course with those who strongly disagreed assigning excellent ratings to their courses 26% of the time. However, for those conforming to the rule, that category rose to 92%. When course modality is considered in the marginal sense, blended learning is rated as the preferred choice. However, from Table  6 we can observe that the rule equates student assessment of their learning experiences. If they conform to the rule, they will see excellence.

This study addressed increasingly important issues of student success, withdrawal and perception of the learning environment across multiple course modalities. Arguably these components form the crux of how we will make more effective decisions about how blended learning configures itself in the new normal. The results reported here indicate that blending maintains or increases access for most student cohorts and produces improved success rates for minority and non-minority students alike. In addition, when students express their beliefs about the effectiveness of their learning environments, blended learning enjoys the number one rank. However, upon more thorough analysis of key elements students view as important in their learning, external and demographic variables have minimal impact on those decisions. For example college (i.e. discipline) membership, course level or modality, expected grade or desire to take a particular course have little to do with their course ratings. The characteristics they view as important relate to clear establishment and progress toward course objectives, creating an effective learning environment and the instructors’ effective communication. If in their view those three elements of a course are satisfied they are virtually guaranteed to evaluate their educational experience as excellent irrespective of most other considerations. While end of course rating protocols are summative the three components have clear formative characteristics in that each one is directly related to effective pedagogy and is responsive to faculty development through units such as the faculty center for teaching and learning. We view these results as encouraging because they offer potential for improving the teaching and learning process in an educational environment that increases the pressure to become more responsive to contemporary student lifestyles.

Clearly, in this study we are dealing with complex adaptive systems that feature the emergent property. That is, their primary agents and their interactions comprise an environment that is more than the linear combination of their individual elements. Blending learning, by interacting with almost every aspect of higher education, provides opportunities and challenges that we are not able to fully anticipate.

This pedagogy alters many assumptions about the most effective way to support the educational environment. For instance, blending, like its counterpart active learning, is a personal and individual phenomenon experienced by students. Therefore, it should not be surprising that much of what we have called blended learning is, in reality, blended teaching that reflects pedagogical arrangements. Actually, the best we can do for assessing impact is to use surrogate measures such as success, grades, results of assessment protocols, and student testimony about their learning experiences. Whether or not such devices are valid indicators remains to be determined. We may be well served, however, by changing our mode of inquiry to blended teaching.

Additionally, as Norberg ( 2017 ) points out, blended learning is not new. The modality dates back, at least, to the medieval period when the technology of textbooks was introduced into the classroom where, traditionally, the professor read to the students from the only existing manuscript. Certainly, like modern technologies, books were disruptive because they altered the teaching and learning paradigm. Blended learning might be considered what Johnson describes as a slow hunch (2010). That is, an idea that evolved over a long period of time, achieving what Kaufmann ( 2000 ) describes as the adjacent possible – a realistic next step occurring in many iterations.

The search for a definition for blended learning has been productive, challenging, and, at times, daunting. The definitional continuum is constrained by Oliver and Trigwell ( 2005 ) castigation of the concept for its imprecise vagueness to Sharpe et al.’s ( 2006 ) notion that its definitional latitude enhances contextual relevance. Both extremes alter boundaries such as time, place, presence, learning hierarchies, and space. The disagreement leads us to conclude that Lakoff’s ( 2012 ) idealized cognitive models i.e. arbitrarily derived concepts (of which blended learning might be one) are necessary if we are to function effectively. However, the strong possibility exists that blended learning, like quality, is observer dependent and may not exist outside of our perceptions of the concept. This, of course, circles back to the problem of assuming that blending is a treatment effect for point hypothesis testing and meta-analysis.

Ultimately, in this article, we have tried to consider theoretical concepts and empirical findings about blended learning and their relationship to the new normal as it evolves. Unfortunately, like unresolved chaotic solutions, we cannot be sure that there is an attractor or that it will be the new normal. That being said, it seems clear that blended learning is the harbinger of substantial change in higher education and will become equally impactful in K-12 schooling and industrial training. Blended learning, because of its flexibility, allows us to maximize many positive education functions. If Floridi ( 2014 ) is correct and we are about to live in an environment where we are on the communication loop rather than in it, our educational future is about to change. However, if our results are correct and not over fit to the University of Central Florida and our theoretical speculations have some validity, the future of blended learning should encourage us about the coming changes.

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Acknowledgements

The authors acknowledge the contributions of several investigators and course developers from the Center for Distributed Learning at the University of Central Florida, the McKay School of Education at Brigham Young University, and Scholars at Umea University, Sweden. These professionals contributed theoretical and practical ideas to this research project and carefully reviewed earlier versions of this manuscript. The Authors gratefully acknowledge their support and assistance.

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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

Frontiers in Educational Research , 2024, 7(2); doi: 10.25236/FER.2024.070204 .

Reconstruction of the Mathematics Classroom Teaching System under the New Curriculum Philosophy—Analysis Based on Compulsory Mathematics Curriculum Standards (2022)

Jin Liu 1 , Jun Zhang 1 , Qingdong Liu 2

1 College of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453007, China

2 College of Education, Anyang University, Xinxiang, Henan, 453500, China

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The introduction of a new curriculum concept for mathematics under compulsory education is characterized as a reconstruction of the mathematics teaching system. Curriculum philosophy is both the basic concept of a discipline and an interpretation of the value orientation of the curriculum. By reviewing and reflecting on the curriculum concept in the Mathematics Curriculum Standards for Compulsory Education (2022 Edition), the study concluded that in mathematics classroom teaching, the formulation of teaching goals should be based on core literacy. The selection of teaching content should be based on the core content and basic ideas. In addition, the design of teaching activities should rely on authentic learning situations, while teaching assessment should be embedded in the learning process. This integrated approach helps to build a more comprehensive and meaningful mathematics learning experience.

Compulsory education mathematics curriculum standard (2022 edition); Curriculum concept; Core literacy; Core content and basic ideas; Process evaluation

Cite This Paper

Jin Liu, Jun Zhang, Qingdong Liu. Reconstruction of the Mathematics Classroom Teaching System under the New Curriculum Philosophy—Analysis Based on Compulsory Mathematics Curriculum Standards (2022). Frontiers in Educational Research (2024) Vol. 7, Issue 2: 19-26. https://doi.org/10.25236/FER.2024.070204.

[1] Ministry of Education of the People's Republic of China. Mathematics Curriculum Standards for Compulsory Education (2022 Edition) [M]. Beijing: Beijing Normal University Press, 2022: 5-6.

[2] Fei LIU , Wei HUANG. Reconstruction of Language Classroom Teaching System under the Concept of New Curriculum -Analysis Based on the Compulsory Education Language Curriculum Standards (2022 Edition)[J]. Journal of Tianjin Normal University (Basic Education Edition),2022:2

[3] Dong Zhang. Example of problems and improvements in the presentation of teaching objectives in junior middle school mathematics lessons[J]. Mathematics Bulletin, 2020:29

[4] Wenjun Xu. Determining Classroom Teaching Objectives Based on Mathematics Curriculum Standards [J]. Education Frontier (Curriculum Education Research): 2020, 4

[5] Zuxi Liu. Interview with Prof. Ningzhong Shi: Talking about the basic ideas of mathematics and the core literacy of mathematics[J]. Mathematics Bulletin: 2017, 3-4

[6] UNESCO International Commission for the Development of Education. Learning to Live: Educating the World Today and Tomorrow [M]. Beijing: Educational Science Press, 1996. 

[7] Wei Huang. Language classroom practice based on the consistency of teaching, learning, and assessment: essentials and operations[J]. Middle School Language Teaching, 2021(6):10.

[8] Yuhua Liu. Research on the evaluation of mathematics teaching based on core literacy [J]. Journal of Corps College of Education, 2019(66):77.

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Math discovery provides new method to study cell activity, aging, MSU research shows

Contact: Meg Henderson

STARKVILLE, Miss.—New mathematical tools revealing how quickly cell proteins break down are poised to uncover deeper insights into how we age, according to a recently published paper co-authored by a Mississippi State researcher and his colleagues from Harvard Medical School and the University of Cambridge.

Galen Collins, assistant professor in MSU’s Department of Biochemistry, Molecular Biology, Entomology and Plant Pathology, co-authored the groundbreaking paper published in the Proceedings of the National Academy of Sciences, or PNAS, in April.

“We already understand how quickly proteins are made, which can happen in a matter of minutes,” said Collins, who is also a scientist in the Mississippi Agricultural and Forestry Experiment Station. “Until now, we’ve had a very poor understanding of how much time it takes them to break down.”

The paper in applied mathematics, “ Maximum entropy determination of mammalian proteome dynamics ,” presents the new tools that quantify the degradation rates of cell proteins—how quickly they break down—helping us understand how cells grow and die and how we age. Proteins—complex molecules made from various combinations of amino acids—carry the bulk of the workload within a cell, providing its structure, responding to messages from outside the cell and removing waste. 

The results proved that not all proteins degrade at the same pace but instead fall into one of three categories, breaking down over the course of minutes, hours or days. While previous research has examined cell protein breakdown, this study was the first to quantify mathematically the degradation rates of all cell protein molecules, using a technique called maximum entropy.

“For certain kinds of scientific questions, experiments can often reveal infinitely many possible answers; however, they are not all equally plausible,” said lead author Alexander Dear, research fellow in applied mathematics at Harvard University. “The principle of maximum entropy is a mathematical law that shows us how to precisely calculate the plausibility of each answer—its ‘entropy’—so that we can choose the one that is the most likely.”

“This kind of math is sort of like a camera that zooms in on your license plate from far away and figures out what the numbers should be,” Collins said. “Maximum entropy gives us a clear and precise picture of how protein degradation occurs in cells.”

In addition, the team used these tools to study some specific implications of protein degradation for humans and animals. For one, they examined how those rates change as muscles develop and adapt to starvation.

“We found that starvation had the greatest impact on the intermediate group of proteins in muscular cells, which have a half-life of a few hours, causing the breakdown to shift and accelerate,” Collins said. “This discovery could have implications for cancer patients who experience cachexia, or muscle wasting due to the disease and its treatments.”

They also explored how a shift in the breakdown of certain cell proteins contributes to neurodegenerative disease.

“These diseases occur when waste proteins, which usually break down quickly, live longer than they should,” Collins said. “The brain becomes like a teenager’s bedroom, accumulating trash, and when you don’t clean it up, it becomes uninhabitable.”

Dear affirmed the study’s value lies not only in what it revealed about cell protein degeneration, but also in giving scientists a new method to investigate cell activity with precision.

“Our work provides a powerful new experimental method for quantifying protein metabolism in cells,” he said. “Its simplicity and rapidity make it particularly well-suited for studying metabolic changes.”

Collins’s post-doctoral advisor at Harvard and a co-author of the article, the late Alfred Goldberg, was a pioneer in studying the life and death of proteins. Collins noted this study was built on nearly five decades of Goldberg’s research and his late-career collaboration with mathematicians from the University of Cambridge. After coming to MSU a year ago, Collins continued collaborating with his colleagues to complete the paper.

“It’s an incredible honor to be published in PNAS, but it was also a lot of fun being part of this team,” Collins said. “And it’s very meaningful to see my former mentor’s body of work wrapped up and published.”

Since 1914, PNAS has been one of the most authoritative publications of high-impact research in the biological, physical and social sciences. More information and past issues can be found at www.pnas.org . The Mississippi Agricultural and Forestry Experiment Station conducts research that improves human health and well-being. Learn more at www.mafes.msstate.edu .  

Mississippi State University is taking care of what matters. Learn more at  www.msstate.edu .

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