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ORIGINAL RESEARCH article
Changing students minds and achievement in mathematics: the impact of a free online student course.
 1 Graduate School of Education, Stanford University, Stanford, CA, United States
 2 School of Education and Counseling Psychology, Santa Clara University, Santa Clara, CA, United States
This study reports on the impact of a “massive, open, online course” (MOOC) designed to change students' ideas about mathematics and their own potential and improve their mathematics achievement. Many students hold damaging fixed mindsets, believing that their intelligence is unchangeable. When students shift to a growth mindset (believing that their intelligence is malleable), their achievement increases. This study of a MOOC intervention differs from previous mindset research in three ways (1) the intervention was delivered through a free online course with the advantage of being scalable nationwide (2) the intervention infused mindset messages into mathematics, specifically targeting students' beliefs about mathematics (3) the research was conducted with a teacher randomized controlled design to estimate its effects. Results show that the treatment group who took the MOOC reported more positive beliefs about math, engaged more deeply in math in class, and achieved at significantly higher levels on standardized mathematics assessments.
Introduction
There are a number of damaging and pervasive myths about mathematics learning in the US that are believed by millions of school children, their parents and their teachers. These different myths hold students back on a daily basis and reduce their learning and achievement significantly ( Boaler, 2016 ). One of the most damaging is the idea that some people are born with a “math brain” and some are not, and that high achievement is only available to some students. Two areas of research are important in challenging this myth, and improving student learning. First, recent neuroscience showing the plasticity of the brain, revealing that brains can grow and change ( Maguire et al., 2000 ). Second, research on mindset showing that when people change their ideas about the malleability of their potential, from “fixed” (my ability is not changeable) to “growth” (my ability changes as I learn) their learning and achievement improves ( Dweck, 2006 ). Different studies, pioneered by Carol Dweck, have shown that students with a growth mindset achieve at higher levels than those with a fixed mindset ( Blackwell et al., 2007 ; Claro et al., 2016 ) and that when students change their mindset their achievement changes ( Aronson et al., 2002 ; Good et al., 2003 ). A second damaging myth is the idea that mathematics learning is all about procedures and memorization, rather than ideas, concepts, and creativity. Research shows that students who approach mathematics as a subject of memorization are lower achieving than those who approach it as a subject of ideas that they can think deeply about ( Boaler and Zoido, 2016 ). A third myth that students believe is that good mathematics students have to be fast when some of the world's leading mathematicians are slow thinkers ( Boaler, 2016 ). This study examines the impact of a “massive open online course” (MOOC) for students centered on changing these ideas and teaching students how to learn mathematics well.
The MOOC includes six modules, each of which takes 15–20 min to complete. The teacher of the course is the lead author, Jo Boaler, professor of mathematics education at Stanford, accompanied by some of her undergraduate students. Some of the key ideas in the course are:
• Everyone can learn mathematics to high levels
• Mistakes, challenge and struggle are the best times for brain growth
• Depth of thinking is more important than speed
• Mathematics is a creative and beautiful subject
• Good strategies for learning mathematics including talking and drawing
• Mathematics is all around us in life and is important—this was shown by different undergraduates showing mathematics in soccer, nature, juggling, and dance.
The course includes a series of short videos interspersed with opportunities for students to reflect on the ideas, connect with other students in the course, and work on openended mathematics tasks designed to shape students' perceptions related to these core ideas. (see Supplemental Note for more information about the online course.)
This paper describes the results of a randomized controlled trial (RCT) which examined the impact of the course on middle school students' engagement in mathematics class, their beliefs and mindset, and their academic achievement on state tests—the Smarter Balanced Assessment Consortium (SBAC) Summative Assessment. The SBAC assessments determine students' progress toward college and career readiness in English language arts/literacy and mathematics. These are given at the end of the school year and consist of two parts: a computer adaptive test and a performance task.
Research into the impact of free online classes—or MOOCs—has shown disappointing results with the early promise of equitable access to education being replaced with a harsh reality of low finishing rates and a predominance of privileged learners ( Hansen and Reich, 2015 ). This study gives a very different result, showing that a strategically designed course, with careful considerations to access, significantly impacted students' mathematics learning pathways and subsequent achievement, regardless of students' gender, ethnicity, language learning level, or wealth.
California school districts were recruited through a variety of announcements at conferences and workshops. School districts that were willing to provide data on the impact of the course were admitted. This study was carried out in accordance with the recommendations of Stanford University Research Compliance Office. The protocol was approved by the Stanford Graduate School of Education Institutional Review Board. All subjects gave written informed consent in accordance with the Declaration of Helsinki. Our analysis shows results from four school districts in California, with 1,090 students enrolled in 10 different middle schools across four districts. There were 439 students who took the online class, and 651 students who were control students. There were 14 teachers in this sample. Tables 1 , 2 provide additional descriptive statistics about the sample.
Table 1 . Observations used in the study.
Table 2 . Baseline traits description.
Using a delayedtreatment research design that enabled randomization of students without the constraint of certain students getting access to a helpful course while their classmates did not, we recruited middle school teachers who taught at least 2 classes of 6th, 7th or 8th grade mathematics. For each teacher half of their classes were randomly assigned to the treatment group and half to the control group. Students in the treatment and control conditions were taught by the same teachers, thus controlling for teacher characteristics. Classes assigned to the treatment group took the online class in the first few months of the school year. Students who completed at least 4 of the modules were considered as having received the treatment.
Table 3 provides a project timeline of key activities. The students in the control group were given access to the course at the conclusion of the study.
Table 3 . Project timeline.
Using Ordinary Least Squares (OLS) regression controlling for baseline differences (gender, ethnicity, free and reduced lunch status, English language learner (ELL) status, and special education status) we tested for the effects of the intervention. Summary descriptive statistics for SBAC measures are provided in Table S1.
We found a treatment effect indicating that MOOC participants obtained higher scores in their SBAC math overall scale score, overall proficiency levels, and concepts and procedures ( Smarter Balanced Assessment Consortium, 2013 ). In fact, students who receive the treatment obtained 0.33 standard deviation gains in SBAC math overall scale score; i.e., the average student in the treatment group would score higher than 63% of the control group that was initially equivalent (see Tables 4 , 5 ). The subscales of the SBAC test were also significantly higher for the treatment group. More details on the model specifications is given in Table S2. In addition, further analyses show a positive and significant treatment effect for student subgroups defined by ethnicity, gender, economic disadvantage status, ELL status, special education status, and school grade (see Table S3).
Table 4 . Regression estimates, MOOC effect on SBAC math scores.
Table 5 . Standard deviations gains on SBAC math scores.
The strong design offered by the RCT performed in this study was partially offset by data access constraints. At different stages of the data collection, some schools that were part of the original study design provided incomplete data on their students. The first call for participants yielded 193 teachers who expressed interest and participated in the initial orientation to the project. On the first survey measure in August 2014, 73 teachers and 6,727 students responded in 27 school districts. By December 2014, the sample size had reduced to 31 teachers and 1,645 students in 10 school districts. The attrition in the study was largely due to technical difficulties at the school level–students needed an email address and password to take the online class, which many districts could not provide. Some schools also reported being unable to access the online course from their classrooms because of district firewall security settings that could not be resolved in the timeframe of the study. Further attrition occurred when some districts did not provide full data from state tests, usually because of staff capacity.
Importantly, the attrition was not systematic and was not linked to the outcome variables. In fact, attrition can introduce selection bias in randomized trials so this was investigated fully, as explained below. The most crucial internal validity concern when estimating causal effects is the assumption that students' assignment to treatment and control condition is random. Under this assumption, the estimates are valid if students' baseline traits are statistically similar for treatment and control students. Table 6 validates this assumption by examining whether students' traits vary with treatment/control condition (Table S4 includes complete regression information). Each point estimate is from a separate regression where each baseline student's covariate (i.e., gender, ethnicity, economic disadvantage status, limited English proficiency, and special education) is the dependent variable. The estimated effect of treatment status on these covariates is small and statistically insignificant, suggesting that students' baseline traits are statistically similar for both treated and control students. This validity check shows that the treatment and control groups are comparable and equivalent at baseline. In other words, treatment and comparison groups are statistically equivalent in every observable aspect except for the intervention, ruling out the threat of selection bias.
Table 6 . Auxiliary regressions of baseline covariate balance.
Changes in Student Classroom Engagement
To understand possible mechanisms for the improved academic achievement treatment effect, the study also examined student engagement and beliefs related to mathematics teaching and learning. Participating teachers were asked to evaluate students' engagement, before and after students took the online class, in both their treatment and control classrooms. Teachers observed students along four dimensions of engagement: (a) student participates in class discussions, (b) student works as hard s/he can, (c) student appears to be involved in classwork, and (d) student gives up quickly. Table 7 indicates the two practices which showed significant differences in how students participated in class between control and treatment classes.
Table 7 . Outcomes and baseline traits of students in the randomized controlled trial MOOC.
The study of student engagement demanded a lot of teacher time, and only 4 of 14 teachers returned full data on the students' engagement in math class. The data from this subset show significant effects for students who took the online course. The effect size of the treatment on student engagement was 0.47 SD (see last row of Table 5 ), meaning that the average student in the treatment group would score higher on the engagement scale than 68% of the control group accounting for baseline differences (see Pre/postgains in student engagement, the last column of Table 8 for the regression estimates). Students in the treatment group participated more in class discussions and did not give up on work as quickly as their counterparts in the control classes. These findings provide insight into the reasons that students in the treatment group achieved at significantly higher levels on state mathematics tests. One compromise in our design is that because we used a “withinteacher” design, teachers were aware of which of their classes were designated as control and treatment, thus posing a potential threat to the validity of this measure. We include the engagement gain measure as a mediator variable for the positive treatment effect on standardized test introduced earlier. Our aim in using this measure was to explore possible factors through which students' beliefs about math may have resulted in deeper forms of classroom engagement, helping to explain increases in student achievement. Future studies will include student engagement surveys that do not rely on teacher reports, which will strengthen our design.
Table 8 . Regression estimates, MOOC effect on student mindset surveys.
Changes in Student Mindset
A pre and postsurvey, measuring shifts in students' beliefs, was completed by 156 students and provides further insight into students' increased academic achievement. (These numbers are low as although 1,090 students took surveys, only 156 students took both the pre and the postsurvey). Despite the response rate of 14%, Table S5 shows that the subset of students who completed the pre and postsurvey were, in fact, representative of the larger sample group of 1,090 students.
There was a significant treatment effect on three student beliefs (see Table 8 for regression estimates and Figure 1 which compares treatment and control group survey responses). Students in the treatment had significantly higher reports of growth mindset (Mindset) and their perceptions of mathematics being an interesting and creative subject (Math Creative). They also reported feeling less fearful or easily deterred in math (Fear of Math). The specific survey items for each cluster and alpha levels are given in Table 9 .
Figure 1 . Student mindset survey scores and effect size by group.
Table 9 . Measures of student mindset survey (cluster items with alpha levels) 1 = Strongly disagree, 2 = Disagree, 3 = Somewhat disagree, 4 = Somewhat agree, 5 = Agree, 6 = Strongly agree.
The significant changes in engagement and beliefs that students showed is likely to explain, at least partly, the increase in students' mathematics achievement. This finding supports a growing body of work that shows a link between students' mindsets about their potential and their ideas about mathematics. It is difficult to maintain a growth mindset and the idea you can learn any mathematics when the subject is presented as a series of short, closed questions—with no space for growth or learning within them. Sun (2015) showed that students developed more growth mindsets when teachers presented mathematics as subject with more opportunities for growth and learning, as opposed to performing and answering questions. The finding that students in this study shifted in seeing mathematics as a more creative subject, as well as developing more growth mindsets, supports this important link (see also Boaler, 2016 ).
Consistent with previous research, this study finds a significant connection between students' mindset and their learning outcomes ( Mueller and Dweck, 1998 ). Students in the treatment group reported more growth mindset beliefs and more challengeseeking behaviors than those in the control group. What is distinctive about this study is the impact of an online class in changing students' mindsets toward mathematics, with subsequent changes in student achievement. Much of the research on mindset has focused on changing students' mindsets outside of any content teaching and learning; by contrast this study examines an intervention that combines mindset with changed views of mathematics and mathematical engagement. This study shows that an intervention addressing the intersection of mindset and mathematics can improve students' academic achievement, as well as students' behavior and beliefs about mathematics.
These findings are particularly important in light of continued concerns with US mathematics achievement. In the most recent international comparisons students in the US ranked 40th out of 72 countries ( OECD, 2016 ). This is an issue that has prevailed for decades despite a vast body of research that has shown the ways to teach mathematics well ( Schoenfeld, 2002 ; Boaler, 2015 , 2016 ). Low mathematics achievement is not the only problem that faces the US—math anxiety is widespread among school children and the general population ( Ashcraft and Krause, 2007 ; Foley et al., 2017 ). Most of the attention that is given to this issue considers the curriculum standards and textbooks used in classrooms. While these issues are important they may not be more important than a completely neglected issue—the fact that most students sit in mathematics classrooms, from kindergarten to University, thinking “I am not a math person.” In addition to this damaging belief, few students have learned to approach mathematics as a conceptual domain, rather than a set of procedures. The evidence from this randomized control trial shows the academic impact of changing these beliefs and approaches for students.
We acknowledge the limitations of our study. Our sample was drawn from middle school students in one state, and so additional studies with wider grade spans and in more varied geographical areas would be needed to generalize more broadly. In addition, for the student engagement measures, we relied on teachers' classroom observations of students. Our study would have been strengthened if we had also included student engagement surveys.
Most of the research on MOOCS has portrayed disappointing results—with online classes having low retention and perpetuating the inequities of open access that MOOCs were originally aimed to challenge ( 9 ). The online class that was the subject of this study had a different outcome of students continuing the course and significantly improving their beliefs and achievement, regardless of students' gender, ethnicity, language learning level, or wealth. The fact that this MOOC was used as part of an educational intervention and administered by teachers is part of the reason for the students continued participation. Another, we contend is the pedagogy of active engagement inside the course. Most MOOC's are lecture based, which would likely have been ineffective, even inside a classroom setting. In the “How to Learn Math” course students were invited to engage every few minutes, through answering questions, commenting on videos, and interacting with others. As MOOCs are developed and refined over the next few decades, it seems that an important advancement will be the inclusion of opportunities for more active engagement.
Many school students in the US and world are held back by damaging ideas about learning and their potential—particularly in mathematics. There is a widespread myth that students are either born with a math brain or they are not, and when students struggle they often decide they are just not a math person. The RCT that is the focus of this study has shown that students can be liberated from these damaging ideas and when they are it improves their participation and achievement. Online courses for teachers that also focus on mindset messages, and ideas for teaching mathematics actively, have also been shown to change students' achievement and beliefs (Anderson et al., under review). Together these studies reveal the importance of changing the mindsets of teachers and students, in order that students can learn mathematics without being held back by damaging beliefs. They also show the potential of online courses  which have great scalability and widescale access—as effective teaching opportunities, bringing some of the best teachers and the most cuttingedge research to the students who most need it.
Author Contributions
JB designed and directed the study, CW led recruitment and implementation, GPN and KS ran analyses, the full team interpreted results, JD and GPN contributed to and managed the writing and revising process among all team members.
This study was funded by National Science Foundation (NSF), Research on Education and Learning (REAL), Award Number 1443790. JB, Principal Investigator, Stanford University.
Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Supplementary Material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2018.00026/full#supplementarymaterial
Supplementary Note on Online Student Course
The online class that was the focus of this study has now been taken by over 160,000 participants—students of mathematics of all levels from elementary school to college. It has also been taken by tens of thousands of teachers and parents as both sets of adults are helped by knowledge of the latest research on ways to learn mathematics. In addition to individuals taking the course teachers of students as young as 5 have shared the videos with their students. The class is free and can be taken at any time and at any pace. Students can take the class in their school class, as students in the study did, or at home. The modular nature of the course has enabled teachers to use the course in a variety of ways: using the course in summer school, as a way to launch the school year, or infused throughout the year.
The class, which is also available with Spanish subtitles, is open to anyone with an internet connection. The ideas from the class are also disseminated in different forms including papers, videos and mathematics curriculum materials on youcubed.org, a Stanford center and accompanying website of almost entirely free resources. Accompanying teacher courses on ways to teach mathematics well are also available.
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Keywords: growth mindset, mathematical mindset, MOOC, math achievement, student beliefs, student engagement, randomized control trial
Citation: Boaler J, Dieckmann JA, PérezNúñez G, Sun KL and Williams C (2018) Changing Students Minds and Achievement in Mathematics: The Impact of a Free Online Student Course. Front. Educ . 3:26. doi: 10.3389/feduc.2018.00026
Received: 26 February 2018; Accepted: 11 April 2018; Published: 25 April 2018.
Reviewed by:
Copyright © 2018 Boaler, Dieckmann, PérezNúñez, Sun and Williams. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jo Boaler, [email protected] Jack A. Dieckmann, [email protected]
Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.
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Future themes of mathematics education research: an international survey before and during the pandemic
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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 futureoriented 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 COVID19 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 followup 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 1question 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 Geoscience of Utrecht University (Bèta L19247). 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 doublecheck 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, selfreflection (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 COVID19 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, antiracism, 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, problembased learning, crosscurricular 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 challengebased 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 highquality 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 datadriven 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 antiexpertise and postfact 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 K12 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 HeuvelPanhuizen, 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 pandemicspecific 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 crossingrelated 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 selfregulation. 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 (sociopolitical 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, preservice 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, outoffield 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 subjectmatter knowledge and particularly noted that many teachers seem illprepared 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 postsecondary 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 COVID19 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 crosscutting 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 highquality 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 highquality mathematics education, and unfair distribution of resources. A related future research theme is how the socalled 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 psychologicalsocial phenomena such as emotion, love, belief, attitudes, interest, curiosity, fun, engagement, joy, involvement, motivation, selfesteem, 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 socalled eassessment, 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. BiknerAhsbahs & 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 socalled 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 multifaceted 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 crosssectional research, in particular in the context of teacher professional development:
It is not enough to investigate what happens in preservice 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 crosssectional 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 selfcriticism 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 reflectioninviting 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 socalled didactical, topicspecific research is not less important today but perhaps less fashionable for funding schemes that promote innovative, groundbreaking research. On the other hand, over time it has become clear that mathematics education is a multifaceted sociocultural 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 [contentspecific vs sociopolitical] 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 longerterm 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 teacherresearchers (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 nonresearchers. 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. SuazoFlores 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 nonresearchers (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 makinginequitiesexplicit, 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 listwise 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 (paperfolded 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 inbetween 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 inbetween 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 mathematicsfocused 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’ problemposing skills. Students should be provided opportunities to formulate their own problems based on situations. However, there are few problemposing 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 nonresearchers (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 HeuvelPanhuizen, 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 researchinformed 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 K12 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 “unthought”: 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 lowtech 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 futureoriented 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 lowtech 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 WeChatbased minilessons has been developed and delivered through the WeChat video function during the pandemic. Even when the pandemic is under control, minilessons are still developed and circulated through WeChat. We therefore think it is important to study the use and influence of lowtech 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 testtaking 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 paperandpencil 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 highquality work related to mathematics education in fields such as educational psychology and the cognitive and learning sciences. Encouraging the reporting of highquality 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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Acknowledgments
We thank Anna Sfard for her advice on the survey, based on her own survey published in Sfard ( 2005 ). We are grateful for Stephen Hwang’s careful copyediting for an earlier version of the manuscript. Thanks also to Elisabeth Angerer, Elske de Waal, Paul Ernest, Vilma Mesa, Michelle Stephan, David Wagner, and anonymous reviewers for their feedback on earlier drafts.
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Appendix 1: Explanation of Fig. 1
We have divided Fig. 1 in 12 rectangles called A1 (bottom left) up to C4 (top right) to explain the details (for image annotation go to https://www.fisme.science.uu.nl/toepassingen/28937 )
4   Dark clouds: Negative affect  Parabola mountain  Rainbow: equity, diversity, inclusion Ships in the distance Bell curve volcano  Sun: positive affect, energy source 
3   Pyramids, one with Pascal’s triangle  Elliptic lake with triangle  Shinto temple resembling Pi  Platonic solids  Climbers: ambition, curiosity   Gherkin (London)  NEMO science museum (Amsterdam)  Cube houses (Rotterdam)  Hundertwasser waste incineration (Vienna)  Los Manantiales restaurant (Mexico City)  The sign post “this way” pointing two ways signifies the challenge for students to find their way in society  Series of prime numbers. 43*47 = 2021, the year in which Lizzy Angerer made this drawing  Students in the crow’s nest: interest, attention, anticipation, technology use  The picnic scene refers to the video (Eames & Eames, )   Bridge with graduates happy with their diplomas  Vienna University building representing higher education 
2   Fractal tree  Pythagoras’ theorem at the house wall   Lady with camera and man measuring, recording, and discussing: research and assessment  The drawing hand represents design (inspired by M. C. Escher’s 1948 drawing hands lithograph) 
1  Home setting:  Rodin’s thinker sitting on hyperboloid stool, pondering how to save the earth  Boy drawing the fractal tree; mother providing support with tablet showing fractal  Paperfolded boat  Möbius strips as scaffolds for the tree  Football (sphere)  Ripples on the water connecting the home scene with the teaching boat  School setting:  Child’s small toy boat in the river  Larger boat with students and a teacher  Technology: compass, laptop (distance education)  Magnifying glass represents research into online and offline learning  Students in a circle throwing dice (learning about probability)  Teacher with book: professional selfdevelopment  Sunflowers hinting at Fibonacci sequence and Fermat’s spiral, and culture/art (e.g., Van Gogh) 
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Bakker, A., Cai, J. & Zenger, L. Future themes of mathematics education research: an international survey before and during the pandemic. Educ Stud Math 107 , 1–24 (2021). https://doi.org/10.1007/s1064902110049w
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Enhancing achievement and interest in mathematics learning through MathIsland
 Charles Y. C. Yeh ORCID: orcid.org/0000000345816575 1 ,
 Hercy N. H. Cheng 2 ,
 ZhiHong Chen 3 ,
 Calvin C. Y. Liao 4 &
 TakWai Chan 5
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Conventional teacherled instruction remains dominant in most elementary mathematics classrooms in Taiwan. Under such instruction, the teacher can rarely take care of all students. Many students may then continue to fall behind the standard of mathematics achievement and lose their interest in mathematics; they eventually give up on learning mathematics. In fact, students in Taiwan generally have lower interest in learning mathematics compared to many other regions/countries. Thus, how to enhance students’ mathematics achievement and interest are two major problems, especially for those lowachieving students. This paper describes how we designed a gamebased learning environment, called MathIsland , by incorporating the mechanisms of a construction management game into the knowledge map of the elementary mathematics curriculum. We also report an experiment conducted with 215 elementary students for 2 years, from grade 2 to grade 3. In this experiment, in addition to teacherled instruction in the classroom, students were directed to learn with MathIsland by using their own tablets at school and at home. As a result of this experiment, we found that there is an increase in students’ mathematics achievement, especially in the calculation and word problems. Moreover, the achievements of lowachieving students in the experimental school outperformed the lowachieving students in the control school (a control group in another school) in word problems. Moreover, both the lowachieving students and the highachieving students in the experimental school maintained a rather high level of interest in mathematics and in the system.
Introduction
Mathematics has been regarded as a fundamental subject because arithmetic and logical reasoning are the basis of science and technology. For this reason, educational authorities emphasize students’ proficiency in computational skills and problemsolving. Recently, the results of the Program for International Student Assessment (PISA) and the Trends in Mathematics and Science Study (TIMSS) in 2015 (OECD 2016 ; Mullis et al. 2016 ) revealed a challenge for Taiwan. Although Taiwanese students had higher average performance in mathematics literacy compared to students in other countries, there was still a significant percentage of lowachieving students in Taiwan. Additionally, most Taiwanese students show low levels of interest and confidence in learning mathematics (Lee 2012 ).
The existence of a significant percentage of lowachieving students is probably due to teacherled instruction, which still dominates mathematics classrooms in most Asian countries. It should be noted that students in every classroom possess different abilities and hence demonstrate different achievements. Unfortunately, in teacherled instruction, all the students are required to learn from the teacher in the same way at the same pace (Hwang et al. 2012 ). Lowachieving students, without sufficient time, are forced to receive knowledge passively. Barr and Tagg ( 1995 ) pointed out that it is urgent for lowachieving students to have more opportunities to learn mathematics at their own pace. Researchers suggested onetoone technology (Chan et al. 2006 ) through which every student is equipped with a device to learn in school or at home seamlessly. Furthermore, they can receive immediate feedback from MathIsland, which supports their individualized learning actively and productively. Thus, this may provide more opportunities for helping lowachieving students improve their achievement.
The lowinterest problem for almost all students in Taiwan is usually accompanied by low motivation (Krapp 1999 ). Furthermore, students with continuously low performance in mathematics may eventually lose their interest and refuse to learn further (Schraw et al. 2001 ). This is a severe problem. To motivate students to learn, researchers design educational games to provide enjoyable and engaging learning experiences (Kiili and Ketamo 2007 ). Some of these researchers found that gamebased learning may facilitate students’ learning in terms of motivation and learning effects (Liu and Chu 2010 ), spatial abilities and attention (Barlett et al. 2009 ), situated learning, and problemsolving (Li and Tsai 2013 ). Given these positive results, we hope that our educational game can enhance and sustain the student’s interest in learning mathematics.
In fact, many researchers who endeavored to develop educational games for learning mathematics have shown that their games could facilitate mathematics performance, enjoyment, and selfefficacy (Ku et al. 2014 ; McLaren et al. 2017 ). Although some of the studies were conducted for as many as 4 months (e.g., Hanus and Fox 2015 ), one may still criticize them for the possibility that the students’ interest could be a novelty effect—meaning their interest will decrease as the feeling of novelty diminishes over time (Koivisto and Hamari 2014 ). Due to the limitations of either experimental time or sample sizes, most studies could not effectively exclude the novelty effect of games, unless they were conducted in a natural setting for a long time.
In this study, we collaborated with an experimental elementary school for more than 2 years. The mathematics teachers in the school adopted our online educational game, MathIsland . The students used their own tablet PCs to learn mathematics from the game in class or at home at their own pace. In particular, lowachieving students might have a chance to catch up with the other students and start to feel interested in learning mathematics. Most importantly, because the online educational game was a part of the mathematics curriculum, the students could treat the game as their ordinary learning materials like textbooks. In this paper, we reported a 2year study, in which 215 second graders in the school adopted the MathIsland game in their daily routine. More specifically, the purpose of this paper was to investigate the effect of the game on students’ mathematics achievement. Additionally, we were also concerned about how well the lowachieving students learned, whether they were interested in mathematics and the game, and how their interest in mathematics compared with that of highachieving students. In such a longterm study with a large sample size, it was expected that the novelty effect would be considerably reduced, allowing us to evaluate the effect of the educational game on students’ achievement and interest.
The paper is organized as follows. In the “ Related works ” section, we review related studies on computersupported mathematics learning and educational games. In the “ Design ” section, the game mechanism and the system design are presented. In the “ Method ” section, we describe the research method and the procedures of this study. In the “ Results ” section, the research results about students’ achievement and interest are presented. In the “ Discussion on some features of this study ” section, we discuss the longterm study, knowledge map design, and the two game mechanisms. Finally, the summary of the current situation and potential future work is described in the “ Conclusion and future work ” section.
Related works
Computersupported mathematics learning.
The mathematics curriculum in elementary schools basically includes conceptual understanding, procedural fluency, and strategic competence in terms of mathematical proficiency (see Kilpatrick et al. 2001 ). First, conceptual understanding refers to students’ comprehension of mathematical concepts and the relationships between concepts. Researchers have designed various computerbased scaffolds and feedback to build students’ concepts and clarify potential misconceptions. For example, for guiding students’ discovery of the patterns of concepts, Yang et al. ( 2012 ) adopted an inductive discovery learning approach to design online learning materials in which students were provided with similar examples with a critical attribute of the concept varied. McLaren et al. ( 2017 ) provided students with prompts to correct their common misconceptions about decimals. They conducted a study with the game adopted as a replacement for seven lessons of regular mathematics classes. Their results showed that the educational game could facilitate better learning performance and enjoyment than a conventional instructional approach.
Second, procedural fluency refers to the skill in carrying out calculations correctly and efficiently. For improving procedural fluency, students need to have knowledge of calculation rules (e.g., place values) and practice the procedure without mistakes. Researchers developed various digital games to overcome the boredom of practice. For example, Chen et al. ( 2012a , 2012b ) designed a Cross Number Puzzle game for practicing arithmetic expressions. In the game, students could individually or collaboratively solve a puzzle, which involved extensive calculation. Their study showed that the lowability students in the collaborative condition made the most improvement in calculation skills. Ku et al. ( 2014 ) developed minigames to train students’ mental calculation ability. They showed that the minigames could not only improve students’ calculation performance but also increase their confidence in mathematics.
Third, strategic competence refers to mathematical problemsolving ability, in particular, word problemsolving in elementary education. Some researchers developed multilevel computerbased scaffolds to help students translate word problems to equations step by step (e.g., GonzálezCalero et al. 2014 ), while other researchers noticed the problem of overscaffolding. Specifically, students could be too scaffolded and have little space to develop their abilities. To avoid this situation, many researchers proposed allowing students to seek help during word problemsolving (Chase and Abrahamson 2015 ; Roll et al. 2014 ). For example, Cheng et al. ( 2015 ) designed a Scaffolding Seeking system to encourage elementary students to solve word problems by themselves by expressing their thinking first, instead of receiving and potentially abusing scaffolds.
Digital educational games for mathematics learning
Because mathematics is an abstract subject, elementary students easily lose interest in it, especially lowachieving students. Some researchers tailored educational games for learning a specific set of mathematical knowledge (e.g., the Decimal Points game; McLaren et al. 2017 ), so that students could be motivated to learn mathematics. However, if our purpose was to support a complete mathematics curriculum for elementary schools, it seemed impractical to design various educational games for all kinds of knowledge. A feasible approach is to adopt a gamified content structure to reorganize all learning materials. For example, inspired by the design of most roleplaying games, Chen et al. ( 2012a , 2012b ) proposed a threetiered framework of gamebased learning—a game world, quests, and learning materials—for supporting elementary students’ enjoyment and goal setting in mathematics learning. Furthermore, while a game world may facilitate students’ exploration and participation, quests are the containers of learning materials with specific goals and rewards. In the game world, students receive quests from nonplayer virtual characters, who may enhance social commitments. To complete the quests, students have to make efforts to undertake learning materials. Today, quests have been widely adopted in the design of educational games (e.g., Azevedo et al. 2012 ; Hwang et al. 2015 ).
However, in educational games with quests, students still play the role of receivers rather than active learners. To facilitate elementary students’ initiative, Lao et al. ( 2017 ) designed digital learning contracts, which required students to set weekly learning goals at the beginning of a week and checked whether they achieved the goals at the end of the week. More specifically, when setting weekly goals, students had to decide on the quantity of learning materials that they wanted to undertake in the coming week. Furthermore, they also had to decide the average correctness of the tests that followed the learning materials. To help them set reasonable and feasible goals, the system provided statistics from the past 4 weeks. As a result, the students may reflect on how well they learned and then make appropriate decisions. After setting goals, students are provided with a series of learning materials for attempting to accomplish those goals. At the end of the week, they may reflect on whether they achieved their learning goals in the contracts. In a sense, learning contracts may not only strengthen the sense of commitment but also empower students to take more control of their learning.
In textbooks or classrooms, learning is usually predefined as a specific sequence, which students must follow to learn. Nevertheless, the structure of knowledge is not linear, but a network. If we could reorganize these learning materials according to the structure of knowledge, students could explore knowledge and discover the relationships among different pieces of knowledge when learning (Davenport and Prusak 2000 ). Knowledge mapping has the advantage of providing students concrete content through explicit knowledge graphics (Ebener et al. 2006 ). Previous studies have shown that the incorporation of knowledge structures into educational games could effectively enhance students’ achievement without affecting their motivation and selfefficacy (Chu et al. 2015 ). For this reason, this study attempted to visualize the structure of knowledge in an educational game. In other words, a knowledge map was visualized and gamified so that students could make decisions to construct their own knowledge map in games.
To enhance students’ mathematics achievement and interests, we designed the MathIsland online game by incorporating a gamified knowledge map of the elementary mathematics curriculum. More specifically, we adopt the mechanisms of a construction management game , in which every student owns a virtual island (a city) and plays the role of the mayor. The goal of the game is to build their cities on the islands by learning mathematics.
System architecture
The MathIsland game is a Web application, supporting crossdevice interactions among students, teachers, and the mathematics content structure. The system architecture of the MathIsland is shown in Fig. 1 . The pedagogical knowledge and learning materials are stored in the module of digital learning content, organized by a mathematical knowledge map. The students’ portfolios about interactions and works are stored in the portfolio database and the status database. When a student chooses a goal concept in the knowledge map, the corresponding digital learning content is arranged and delivered to his/her browser. Besides, when the student is learning in the MathIsland, the feedback module provides immediate feedback (e.g., hints or scaffolded solutions) for guidance and grants rewards for encouragement. The learning results can also be shared with other classmates by the interaction module. In addition to students, their teachers can also access the databases for the students’ learning information. Furthermore, the information consists of the students’ status (e.g., learning performance or virtual achievement in the game) and processes (e.g., their personal learning logs). In the MathIsland, it is expected that students can manage their learning and monitor the learning results by the construction management mechanism. In the meantime, teachers can also trace students’ learning logs, diagnose their weaknesses from portfolio analysis, and assign students with specific tasks to improve their mathematics learning.
The system architecture of MathIsland
 Knowledge map
To increase students’ mathematics achievement, the MathIsland game targets the complete mathematics curriculum of elementary schools in Taiwan, which mainly contains the four domains: numerical operation , quantity and measure , geometry , and statistics and probability (Ministry of Education of R.O.C. 2003 ). Furthermore, every domain consists of several subdomains with corresponding concepts. For instance, the domain of numerical operation contains four subdomains: numbers, addition, and subtraction for the first and second graders. In the subdomain of subtraction, there are a series of concepts, including the meaning of subtraction, onedigit subtraction, and twodigit subtraction. These concepts should be learned consecutively. In the MathIsland system, the curriculum is restructured as a knowledge map, so that they may preview the whole structure of knowledge, recall what they have learned, and realize what they will learn.
More specifically, the MathIsland system uses the representational metaphor of an “island,” where a virtual city is located and represents the knowledge map. Furthermore, the island comprises areas, roads, and buildings, which are the embodiments of domains, subdomains, and concepts in the curriculum, respectively. As shown in Fig. 2 , for example, in an area of numeral operation in MathIsland, there are many roads, such as an addition road and a subtraction road. On the addition road, the first building should be the meaning of addition, followed by the buildings of onedigit addition and then twodigit addition. Students can choose these buildings to learn mathematical concepts. In each building, the system provides a series of learning tasks for learning the specific concept. Currently, MathIsland provides elementary students with more than 1300 learning tasks from the first grade to the sixth grade, with more than 25,000 questions in the tasks.
The knowledge map
In MathIsland, a learning task is an interactive page turner, including video clips and interactive exercises for conceptual understanding, calculation, and word problemsolving. In each task, the learning procedure mainly consists of three steps: watching demonstrations, practicing examples, and getting rewards. First, students learn a mathematical concept by watching videos, in which a human tutor demonstrates examples, explains the rationale, and provides instructions. Second, students follow the instructions to answer a series of questions related to the examples in the videos. When answering questions, students are provided with immediate feedback. Furthermore, if students input wrong answers, the system provides multilevel hints so that they could figure out solutions by themselves. Finally, after completing learning tasks, students receive virtual money according to their accuracy rates in the tasks. The virtual money is used to purchase unique buildings to develop their islands in the game.
Game mechanisms
In the MathIsland game, there are two game mechanisms: construction and sightseeing (as shown in Fig. 3 ). The former is designed to help students manage their learning process, whereas the latter is designed to facilitate social interaction, which may further motivate students to better develop their cities. By doing so, the MathIsland can be regarded as one’s learning portfolio, which is a complete record that purposely collects information about one’s learning processes and outcomes (Arter and Spandel 2005 ). Furthermore, learning portfolios are a valuable research tool for gaining an understanding about personal accomplishments (Birgin and Baki 2007 ), because learning portfolios can display one’s learning process, attitude, and growth after learning (Lin and Tsai 2001 ). The appearance of the island reflects what students have learned and have not learned from the knowledge map. When students observe their learning status in an interesting way, they may be concerned about their learning status with the enhanced awareness of their learning portfolios. By keeping all activity processes, students can reflect on their efforts, growth, and achievements. In a sense, with the game mechanisms, the knowledge map can be regarded as a manipulatable open learner model, which not only represents students’ learning status but also invites students to improve it (Vélez et al. 2009 ).
Two game mechanisms for MathIsland
First, the construction mechanism allows students to plan and manage their cities by constructing and upgrading buildings. To do so, they have to decide which buildings they want to construct or upgrade. Then, they are required to complete corresponding learning tasks in the building to determine which levels of buildings they can construct. As shown in Fig. 4 , the levels of buildings depend on the completeness of a certain concept, compared with the thresholds. For example, when students complete one third of the learning tasks, the first level of a building is constructed. Later, when they complete two thirds of the tasks, the building is upgraded to the second level. After completing all the tasks in a building, they also complete the final level and are allowed to construct the next building on the road. Conversely, if students failed the lowest level of the threshold, they might need to watch the video and/or do the learning tasks again. By doing so, students can make their plans to construct the buildings at their own pace. When students manage their cities, they actually attempt to improve their learning status. In other words, the construction mechanism offers an alternative way to guide students to regulate their learning efforts.
Screenshots of construction and sightseeing mechanisms in MathIsland
Second, the sightseeing mechanism provides students with a social stage to show other students how well their MathIslands have been built. This mechanism is implemented as a public space, where other students play the role of tourists who visit MathIsland. In other words, this sightseeing mechanism harnesses social interaction to improve individual learning. As shown in Fig. 4 , because students can construct different areas or roads, their islands may have different appearances. When students visit a welldeveloped MathIsland, they might have a positive impression, which may facilitate their selfreflection. Accordingly, they may be willing to expend more effort to improve their island. On the other hand, the student who owns the island may also be encouraged to develop their island better. Furthermore, when students see that they have a completely constructed building on a road, they may perceive that they are good at these concepts. Conversely, if their buildings are small, the students may realize their weaknesses or difficulties in these concepts. Accordingly, they may be willing to make more effort for improvement. On the other hand, the student who owns the island may also be encouraged to develop their island better. In a word, the visualization may play the role of stimulators, so that students may be motivated to improve their learning status.
This paper reported a 2year study in which the MathIsland system was adopted in an elementary school. The study addressed the following two research questions: (1) Did the MathIsland system facilitate students’ mathematics achievement in terms of conceptual understanding, calculating, and word problemsolving? In particular, how was the mathematics achievement of the lowachieving students? (2) What was students’ levels of interest in mathematics and the system, particularly that of lowachieving students?
Participants
The study, conducted from June 2013 to June 2015, included 215 second graders (98 females and 117 males), whose average age was 8 years old, in an elementary school located in a suburban region of a northern city in Taiwan. The school had collaborated with our research team for more than 2 years and was thus chosen as an experimental school for this study. In this school, approximately one third of the students came from families with a low or middle level of socioeconomic status. It was expected that the lessons learned from this study could be applicable to other schools with similar student populations in the future. The parents were supportive of this program and willing to provide personal tablets for their children (Liao et al. 2017 ). By doing so, the students in the experimental school were able to use their tablets to access the MathIsland system as a learning tool at both school and home. To compare the students’ mathematics achievement with a baseline, this study also included 125 second graders (63 females and 62 males) from another school with similar socioeconomic backgrounds in the same region of the city as a control school. The students in the control school received only conventional mathematics instruction without using the MathIsland system during the 2year period.
Before the first semester, a 3week training workshop was conducted to familiarize the students with the basic operation of tablets and the MathIsland system. By doing so, it was ensured that all participants had similar prerequisite skills. The procedure of this study was illustrated in Table 1 . At the beginning of the first semester, a mathematics achievement assessment was conducted as a pretest in both the experimental and the control school to examine the students’ initial mathematics ability as second graders. From June 2013 to June 2015, while the students in the control school learned mathematics in a conventional way, the students in the experimental school learned mathematics not only in mathematics classes but also through the MathIsland system. Although the teachers in the experimental school mainly adopted lectures in mathematics classes, they used the MathIsland system as learning materials at school and for homework. At the same time, they allowed the students to explore the knowledge map at their own pace. During the 2 years, every student completed 286.78 learning tasks on average, and each task took them 8.86 min. Given that there were 344 tasks for the second and third graders, the students could finish 83.37% of tasks according to the standard progress. The data also showed that the average correctness rate of the students was 85.75%. At the end of the second year, another mathematics achievement assessment was administered as a posttest in both schools to evaluate students’ mathematics ability as third graders. Additionally, an interest questionnaire was employed in the experimental school to collect the students’ perceptions of mathematics and the MathIsland system. To understand the teachers’ opinions of how they feel about the students using the system, interviews with the teachers in the experimental school were also conducted.
Data collection
Mathematics achievement assessment.
To evaluate the students’ mathematics ability, this study adopted a standardized achievement assessment of mathematics ability (Lin et al. 2009 ), which was developed from a random sample of elementary students from different counties in Taiwan to serve as a norm with appropriate reliability (the internal consistency was 0.85, and the testretest reliability was 0.86) and validity (the correlation by domain experts in content validity was 0.92, and the concurrent validity was 0.75). As a pretest, the assessment of the second graders consisted of 50 items, including conceptual understanding (23 items), calculating (18 items), and word problemsolving (9 items). As a posttest, the assessment of the third graders consisted of 60 items, including conceptual understanding (18 items), calculating (27 items), and word problemsolving (15 items). The scores of the test ranged from 0 to 50 points. Because some students were absent during the test, this study obtained 209 valid tests from the experimental school and 125 tests from the control school.
Interest questionnaire
The interest questionnaire comprised two parts: students’ interest in mathematics and the MathIsland system. Regarding the first part, this study adopted items from a mathematics questionnaire of PISA and TIMSS 2012 (OECD 2013 ; Mullis et al. 2012 ), the reliability of which was sound. This part included three dimensions: attitude (14 items, Cronbach’s alpha = .83), initiative (17 items, Cronbach’s alpha = .82), and confidence (14 items Cronbach’s alpha = .72). Furthermore, the dimension of attitude was used to assess the tendency of students’ view on mathematics. For example, a sample item of attitudes was “I am interested in learning mathematics.” The dimension of initiatives was used to assess how students were willing to learn mathematics actively. A sample item of initiatives was “I keep studying until I understand mathematics materials.” The dimension of confidences was used to assess students’ perceived mathematics abilities. A sample item was “I am confident about calculating whole numbers such as 3 + 5 × 4.” These items were translated to Chinese for this study. Regarding the second part, this study adopted selfmade items to assess students’ motivations for using the MathIsland system. This part included two dimensions: attraction (8 items) and satisfaction (5 items). The dimension of attraction was used to assess how well the system could attract students’ attention. A sample item was “I feel Mathisland is very appealing to me.” The dimension of satisfaction was used to assess how the students felt after using the system. A sample item was “I felt that upgrading the buildings in my MathIsland brought me much happiness.” These items were assessed according to a 4point Likert scale, ranging from “strongly disagreed (1),” “disagreed (2),” “agreed (3),” and “strongly agreed (4)” in this questionnaire. Due to the absences of several students on the day the questionnaire was administered, there were only 207 valid questionnaires in this study.
Teacher interview
This study also included teachers’ perspectives on how the students used the MathIsland system to learn mathematics in the experimental school. This part of the study adopted semistructured interviews of eight teachers, which comprised the following three main questions: (a) Do you have any notable stories about students using the MathIsland system? (b) Regarding MathIsland, what are your teaching experiences that can be shared with other teachers? (c) Do you have any suggestions for the MathIsland system? The interview was recorded and transcribed verbatim. The transcripts were coded and categorized according to the five dimensions of the questionnaire (i.e., the attitude, initiative, and confidence about mathematics, as well as the attraction and satisfaction with the system) as additional evidence of the students’ interest in the experimental school.
Data analysis
For the first research question, this study conducted a multivariate analysis of variance (MANOVA) with the schools as a betweensubject variable and the students’ scores (conceptual understanding, calculating, and word problemsolving) in the pre/posttests as dependent variables. Moreover, this study also conducted a MANOVA to compare the lowachieving students from both schools. In addition, the tests were also carried out to compare achievements with the norm (Lin et al. 2009 ). For the second research question, several z tests were used to examine how the interests of the lowachieving students were distributed compared with the whole sample. Teachers’ interviews were also adopted to support the results of the questionnaire.
Mathematics achievement
To examine the homogeneity of the students in both schools in the first year, the MANOVA of the pretest was conducted. The results, as shown in Table 2 , indicated that there were no significant differences in their initial mathematics achievements in terms of conceptual understanding, calculating, and word problemsolving (Wilks’ λ = 0.982, F (3330) = 2.034, p > 0.05). In other words, the students of both schools had similar mathematics abilities at the time of the first mathematics achievement assessment and could be fairly compared.
At the end of the fourth grade, the students of both schools received the posttest, the results of which were examined by a MANOVA. As shown in Table 3 , the effect of the posttest on students’ mathematics achievement was significant (Wilks’ λ = 0.946, p < 0.05). The results suggested that the students who used MathIsland for 2 years had better mathematics abilities than those who did not. The analysis further revealed that the univariate effects on calculating and word problemsolving were significant, but the effect on conceptual understanding was insignificant. The results indicated that the students in the experimental school outperformed their counterparts in terms of the procedure and application of arithmetic. The reason may be that the system provided students with more opportunities to do calculation exercises and word problems, and the students were more willing to do these exercises in a gamebased environment. Furthermore, they were engaged in solving various exercises with the support of immediate feedback until they passed the requirements of every building in their MathIsland. However, the students learned mathematical concepts mainly by watching videos in the system, which provided only demonstrations like lectures in conventional classrooms. For this reason, the effect of the system on conceptual understanding was similar to that of teachers’ conventional instruction.
Furthermore, to examine the differences between the lowachieving students in both schools, another MANOVA was also conducted on the pretest and the posttest. The pretest results indicated that there were no significant differences in their initial mathematics achievement in terms of conceptual understanding, calculating, and word problemsolving (Wilks’ λ = 0.943, F (3110) = 2.210, p > 0.05).
The MANOVA analysis of the posttest is shown in Table 4 . The results showed that the effect of the system on the mathematics achievement of lowachieving students was significant (Wilks’ λ = 0.934, p < 0.05). The analysis further revealed that only the univariate effect on word problemsolving was significant. The results suggested that the lowachieving students who used MathIsland for 2 years had better word problemsolving ability than those students in the control school, but the effect on conceptual understanding and procedural fluency was insignificant. The results indicated that the MathIsland system could effectively enhance lowachieving students’ ability to solve word problems.
Because the mathematics achievement assessment was a standardized achievement assessment (Lin et al. 2009 ), the research team did a further analysis of the assessments by comparing the results with the norm. In the pretest, the average score of the control school was the percentile rank of a score (PR) 55, but their average score surprisingly decreased to PR 34 in the posttest. The results confirmed the fact that conventional mathematics teaching in Taiwan might result in an Mshape distribution, suggesting that lowachieving students required additional learning resources. Conversely, the average score of the experimental school was PR 48 in the pretest, and their score slightly decreased to PR 44 in the posttest. Overall, both PR values were decreasing, because the mathematics curriculum became more and more difficult from the second grade to the fourth grade. However, it should be noted that the experimental school has been less affected, resulting in a significant difference compared with the control school (see Table 5 ). Notably, the average score of word problemsolving in the posttest of the experimental school was PR 64, which was significantly higher than the nationwide norm ( z = 20.8, p < .05). The results were consistent with the univariate effect of the MANOVA on word problemsolving, suggesting that the MathIsland system could help students learn to complete word problems better. This may be because the learning tasks in MathIsland provided students with adequate explanations for various types of word problems and provided feedback for exercises.
To examine whether the lowachieving students had low levels of interest in mathematics and the MathIsland system, the study adopted z tests on the data of the interest questionnaire. Table 5 shows the descriptive statistics and the results of the z tests. Regarding the interest in mathematics, the analysis showed that the interest of the lowachieving students was similar to that of the whole sample in terms of attitude, initiative, and confidence. The results were different from previous studies asserting that lowachieving students tended to have lower levels of interest in mathematics (AlZoubi and Younes 2015 ). The reason was perhaps that the lowachieving students were comparably motivated to learn mathematics in the MathIsland system. As a result, a teacher ( #T301 ) said, “some students would like to go to MathIsland after school, and a handful of students could even complete up to forty tasks (in a day),” implying that the students had a positive attitude and initiative related to learning mathematics.
Another teacher ( T312 ) also indicated “some students who were frustrated with math could regain confidence when receiving the feedback for correct answers in the basic tasks. Thanks to this, they would not feel highpressure when moving on to current lessons.” In a sense, the immediate feedback provided the lowachieving students with sufficient support and may encourage them to persistently learn mathematics. Furthermore, by learning individually after class, they could effectively prepare themselves for future learning. The results suggested that the system could serve as a scaffolding on conventional instruction for lowachieving students. The students could benefit from such a blended learning environment and, thus, build confidence in mathematics by learning at their own paces.
The lowachieving students as a whole were also attracted to the system and felt satisfaction from it. Teacher ( #T307 ) said that, “There was a hyperactive and mischievous student in my class. However, when he was alone, he would go on to MathIsland, concentrating on the tasks quietly. He gradually came to enjoy learning mathematics. It seemed that MathIsland was more attractive to them than a lecture by a teacher. I believed that students could be encouraged, thus improve their ability and learn happily.” Another teacher ( #T304 ) further pointed out that, “For students, they did not only feel like they were learning mathematics because of the gamebased user interface. Conversely, they enjoyed the contentment when completing a task, as if they were going aboard to join a competition.” In teachers’ opinions, such a gamebased learning environment did not disturb their instruction. Instead, the system could help the teachers attract students’ attention and motivate them to learn mathematics actively because of its appealing game and joyful learning tasks. Furthermore, continuously overcoming the tasks might bring students a sense of achievement and satisfaction.
Discussion on some features of this study
In addition to the enhancement of achievement and interest, we noticed that there are some features in this study and our design worth some discussion.
The advantages of building a longterm study
Owing to the limitations of deployment time and sample sizes, it is hard for most researchers to conduct a longitudinal study. Fortunately, we had a chance to maintain a longterm collaboration with an experimental school for more than 2 years. From this experiment, we notice that there are two advantages to conducting a longterm study.
Obtaining substantial evidence from the gamebased learning environment
The research environment was a natural setting, which could not be entirely controlled and manipulated like most experiments in laboratories. However, this study could provide longterm evidence to investigate how students learned in a gamebased learning environment with their tablets. It should be noted that we did not aim to replace teachers in classrooms with the MathIsland game. Instead, we attempted to establish an ordinary learning scenario, in which the teachers and students regarded the game as one of the learning resources. For example, teachers may help lowachieving students to improve their understanding of a specific concept in the MathIsland system. When students are learning mathematics in the MathIsland game, teachers may take the game as a formative assessment and locate students’ difficulties in mathematics.
Supporting teachers’ instructions and facilitating students’ learning
The longterm study not only proved the effectiveness of MathIsland but also offered researchers an opportunity to determine teachers’ roles in such a computersupported learning environment. For example, teachers may encounter difficulties in dealing with the progress of both high and lowachieving students. How do they take care of all students with different abilities at the same time? Future teachers may require more teaching strategies in such a selfdirected learning environment. Digital technology has an advantage in helping teachers manage students’ learning portfolios. For example, the system can keep track of all the learning activities. Furthermore, the system should provide teachers with monitoring functions so that they know the average status of their class’s and individuals’ learning progress. Even so, it is still a challenge for researchers to develop a welldesigned visualization tool to support teachers’ understanding of students’ learning conditions and their choice of appropriate teaching strategies.
Incorporating a gamified knowledge map of the elementary mathematics curriculum
Providing choices of learning paths.
MathIsland uses a representational metaphor of an “island,” where a virtual city is located and represents the knowledge map. Furthermore, the island comprises areas, roads, and buildings, which are the embodiments of domains, subdomains, and concepts in the curriculum, respectively. Because the gamified knowledge map provides students with multiple virtual roads to learn in the system, every student may take different routes. For instance, some students may be more interested in geometry, while others may be confident in exploring the rules of arithmetic. In this study, we noticed that the lowachieving students needed more time to work on basic tasks, while highachieving students easily passed those tasks and moved on to the next ones. As a result, some of the highachieving students had already started to learn the materials for the next grade level. This was possibly because highachieving students were able to respond well to challenging assignments (Singh 2011 ). Therefore, we should provide highachieving students with more complex tasks to maintain their interest. For example, MathIsland should provide some authentic mathematical problems as advanced exercises.
Visualizing the learning portfolio
In this study, we demonstrated a longterm example of incorporating a gamified knowledge map in an elementary mathematical curriculum. In the MathIsland game, the curriculum is visualized as a knowledge map instead of a linear sequence, as in textbooks. By doing so, students are enabled to explore relationships in the mathematics curriculum represented by the knowledge map; that is, the structure of the different roads on MathIsland. Furthermore, before learning, students may preview what will be learned, and after learning, students may also reflect on how well they learned. Unlike traditional lectures or textbooks, in which students could only follow a predefined order to learn knowledge without thinking why they have to learn it, the knowledge map allows students to understand the structure of knowledge and plan how to achieve advanced knowledge. Although the order of knowledge still remains the same, students take primary control of their learning. In a sense, the knowledge map may liberate elementary students from passive learning.
Adopting the mechanisms of a construction management game
This 2year study showed that the adaptation of two game mechanisms, construction and sightseeing, into the elementary mathematical curriculum could effectively improve students’ learning achievement. The reason may be that students likely developed interests in using MathIsland to learn mathematics actively, regardless of whether they are high and lowachieving students.
Gaining a sense of achievement and ownership through the construction mechanism
Regardless of the construction mechanism, MathIsland allows students to plan and manage their cities by constructing and upgrading buildings. MathIsland took the advantages of construction management games to facilitate elementary students’ active participation in their mathematical learning. Furthermore, students may manage their knowledge by planning and constructing of buildings on their virtual islands. Like most construction management games, students set goals and make decisions so that they may accumulate their assets. These assets are not only external rewards but also visible achievements, which may bring a sense of ownership and confidence. In other words, the system gamified the process of selfdirected learning.
Demonstrating learning result to peers through the sightseeing mechanism
As for the sightseeing mechanism, in conventional instruction, elementary students usually lack the selfcontrol to learn knowledge actively (Duckworth et al. 2014 ) or require a social stage to show other students, resulting in low achievement and motivation. On the other hand, although previous researchers have already proposed various selfregulated learning strategies (such as Taub et al. 2014 ), it is still hard for children to keep adopting specific learning strategies for a long time. For these reasons, this study uses the sightseeing mechanism to engage elementary students in a social stage to show other students how well their MathIslands have been built. For example, in MathIsland, although the students think that they construct buildings in their islands, they plan the development of their knowledge maps. After learning, they may also reflect on their progress by observing the appearance of the buildings.
In brief, owing to the construction mechanism, the students are allowed to choose a place and build their unique islands by learning concepts. During the process, students have to do the learning task, get feedback, and get rewards, which are the three major functions of the construction functions. In the sightseeing mechanism, students’ unique islands (learning result) can be shared and visited by other classmates. The student’s MathIsland thus serves as a stage for showing off their learning results. The two mechanisms offer an incentive model connected to the game mechanism’s forming a positive cycle: the more the students learn, the more unique islands they can build, with more visitors.
Conclusion and future work
This study reported the results of a 2year experiment with the MathIsland system, in which a knowledge map with extensive mathematics content was provided to support the complete elementary mathematics curriculum. Each road in MathIsland represents a mathematical topic, such as addition. There are many buildings on each road, with each building representing a unit of the mathematics curriculum. Students may learn about the concept and practice it in each building while being provided with feedback by the system. In addition, the construction management online game mechanism is designed to enhance and sustain students’ interest in learning mathematics. The aim of this study was not only to examine whether the MathIsland system could improve students’ achievements but also to investigate how much the lowachieving students would be interested in learning mathematics after using the system for 2 years.
As for enhancing achievement, the result indicated that the MathIsland system could effectively improve the students’ ability to calculate expressions and solve word problems. In particular, the lowachieving students outperformed those of the norm in terms of word problemsolving. For enhancing interest, we found that both the lowachieving and the highachieving students in the experimental school, when using the MathIsland system, maintained a rather high level of interest in learning mathematics and using the system. The results of this study indicated some possibility that elementary students could be able to learn mathematics in a selfdirected learning fashion (Nilson 2014 ; Chen et al. 2012a , b ) under the MathIsland environment. This possibility is worthy of future exploration. For example, by analyzing student data, we can investigate how to support students in conducting selfdirected learning. Additionally, because we have already collected a considerable amount of student data, we are currently employing machine learning techniques to improve feedback to the students. Finally, to provide students appropriate challenges, the diversity, quantity, and difficulty of content may need to be increased in the MathIsland system.
Abbreviations
Program for International Student Assessment
The percentile rank of a score
Trends in Mathematics and Science Study
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Acknowledgements
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to thank the Ministry of Science and Technology of the Republic of China, Taiwan, for financial support (MOST 1062511S008003MY3), and Research Center for Science and Technology forLearning, National Central University, Taiwan.
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CYCY contributed to the study design, data acquisition and analysis, mainly drafted the manuscript and execution project. HNHC was involved in data acquisition, revision of the manuscript and data analysis.ZHC was contributed to the study idea and drafted the manuscript. CCYL of this research was involved in data acquisition and revision of the manuscript. TWC was project manager and revision of the manuscript. All authors read and approved the final manuscript.
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Charles Y.C. Yeh is currently an PhD student in Graduate Institute of Network Learning Technology at National Central University. The research interests include onetoone learning environments and gamebased learning.
Hercy N. H. Cheng is currently an associate professor and researcher in National Engineering Research Center for ELearning at Central China Normal University, China. His research interests include onetoone learning environments and gamebased learning.
ZhiHong Chen is an associate professor in Graduate Institute of Information and Computer Education at National Taiwan Normal University. His research interests focus on learning technology and interactive stories, technology intensive language learning and gamebased learning.
Calvin C. Y. Liao is currently an Assistant Professor and Dean’s Special Assistant in College of Nursing at National Taipei University of Nursing and Health Sciences in Taiwan. His research focuses on computerbased language learning for primary schools. His current research interests include a gamebased learning environment and smart technology for caregiving & wellbeing.
TakWai Chan is Chair Professor of the Graduate Institute of Network Learning Technology at National Central University in Taiwan. He has worked on various areas of digital technology supported learning, including artificial intelligence in education, computer supported collaborative learning, digital classrooms, online learning communities, mobile and ubiquitous learning, digital game based learning, and, most recently, technology supported mathematics and language arts learning.
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Yeh, C.Y.C., Cheng, H.N.H., Chen, ZH. et al. Enhancing achievement and interest in mathematics learning through MathIsland. RPTEL 14 , 5 (2019). https://doi.org/10.1186/s4103901901009
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“Teachers should teach math in a way that encourages students to engage in sensemaking and not merely to memorize or internalize exactly what the teacher says or does,” says Jon R. Star.
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There’s never just one way to solve a math problem, says Jon R. Star , a psychologist and professor of education at the Harvard Graduate School of Education. With researchers from Vanderbilt University, Star found that teaching students multiple ways to solve math problems instead of using a single method improves teaching and learning. In an interview with the Gazette, Star, a former math teacher, outlined the research and explained how anyone, with the right instruction, can develop a knack for numbers.
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STAR: Learning math should involve some sensemaking. It’s necessary that we listen to what our teacher tells us about the math and try to make sense of it in our minds. Math learning is not about pouring the words directly from the teacher’s mouth into the students’ ears and brains. That’s not the way it works. I think that’s how I learned math. But that’s not how I hope students learn math and that’s not how I hope teachers think about the teaching of math. Teachers should teach math in a way that encourages students to engage in sensemaking and not merely to memorize or internalize exactly what the teacher says or does.
GAZETTE: Tell us about the teaching method described in the research.
STAR: One of the strategies that some teachers may use when teaching math is to show students how to solve problems and expect that the student is going to end up using the same method that the teacher showed. But there are many ways to solve math problems; there’s never just one way.
The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.
GAZETTE: Why is this strategy an improvement over just learning a single method?
STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.
For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.
GAZETTE: What are the potential challenges for math teachers to put this in practice?
STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a wellintentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.
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Dynamics and stability analysis of nonlinear DNA molecules: Insights from the PeyrardBishop model
 Mostafa M. A. Khater 1,2,3 , , ,
 Mohammed Zakarya 4 ,
 Kottakkaran Sooppy Nisar 5 ,
 AbdelHaleem AbdelAty 6
 1. School of Medical Informatics and Engineering, Xuzhou Medical University, 209 Tongshan Road, 221004, Xuzhou, Jiangsu Province, PR China
 2. Institute of Digital Economy, Ugra State University, KhantyMansiysk, Russia
 3. Department of Basic Sciences, High Institute for Engineering and Technology Al Obour, Cairo 11828, Egypt
 4. Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia
 5. Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
 6. Department of Physics, College of Sciences, University of Bisha, Bisha 61922, Saudi Arabi
 Received: 30 January 2024 Revised: 18 June 2024 Accepted: 05 July 2024 Published: 05 August 2024
MSC : 35C08, 35Q05, 92C40, 70H06
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This study explores the nonlinear PeyrardBishop DNA dynamic model, a nonlinear evolution equation that describes the behavior of DNA molecules by considering hydrogen bonds between base pairs and stacking interactions between adjacent base pairs. The primary objective is to derive analytical solutions to this model using the Khater Ⅲ and improved Kudryashov methods. Subsequently, the stability of these solutions is analyzed through Hamiltonian system characterization. The PeyrardBishop model is pivotal in biophysics, offering insights into the dynamics of DNA molecules and their responses to external forces. By employing these analytical techniques and stability analysis, this research aims to enhance the understanding of DNA dynamics and its implications in fields such as drug design, gene therapy, and molecular biology. The novelty of this work lies in the application of the Khater Ⅲ and an enhanced Kudryashov methods to the PeyrardBishop model, along with a comprehensive stability investigation using Hamiltonian system characterization, providing new perspectives on DNA molecule dynamics within the scope of nonlinear dynamics and biophysics.
 PeyrardBishop model ,
 DNA dynamics ,
 nonlinear evolution equations ,
 analytical solutions
Citation: Mostafa M. A. Khater, Mohammed Zakarya, Kottakkaran Sooppy Nisar, AbdelHaleem AbdelAty. Dynamics and stability analysis of nonlinear DNA molecules: Insights from the PeyrardBishop model[J]. AIMS Mathematics, 2024, 9(9): 2344923467. doi: 10.3934/math.20241140
Related Papers:
[1]  , (2020), 2461–2483. http://doi.org/10.3934/math.2020163 > J. Manafian, O. A. Ilhan, S. A. Mohammed, Forming localized waves of the nonlinearity of the DNA dynamics arising in oscillatorchain of PeyrardBishop model, , (2020), 2461–2483. http://doi.org/10.3934/math.2020163 doi: 
[2]  , (2021), 035224. https://doi.org/10.1088/14024896/abdc57 > L. Ouahid, Plenty of soliton solutions to the DNA PeyrardBishop equation via two distinctive strategies, , (2021), 035224. https://doi.org/10.1088/14024896/abdc57 doi: 
[3]  , (2024), 102864. https://doi.org/10.1016/j.asej.2024.102864 > M. B. Riaz, M. Fayyaz, Rahman, R. U., Martinovic, J., O. Tunç, Analytical study of fractional DNA dynamics in the PeyrardBishop oscillatorchain model, , (2024), 102864. https://doi.org/10.1016/j.asej.2024.102864 doi: 
[4]  , (2022), 749–759. > K. K. Ali, M. I. Abdelrahman, K. R. Raslan, W. Adel, On analytical and numerical study for the peyrardbishop DNA dynamic model, , (2022), 749–759. 
[5]  , (2023), 232. https://doi.org/10.1007/s1108202204477y > M. I. Asjad, W. A. Faridi, S. E. Alhazmi, A. Hussanan, The modulation instability analysis and generalized fractional propagating patterns of the PeyrardBishop DNA dynamical equation, , (2023), 232. https://doi.org/10.1007/s1108202204477y doi: 
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 Figure 1. Numerical representations of the solitary wave solutions obtained through analytical methods are presented. Panels (af) display the bright solitary wave solutions computed using the Khater Ⅲ method (Eqs (2.1) and (2.2)). Panels (gi) illustrate the solitary wave solutions derived from an enhanced Kudryashov scheme (Eq (2.3)). These qualitative plots serve to validate the diversity of analytical solitary wave solutions and their localized wave profiles obtained from the two distinct techniques within the nonlinear framework of the PeyrardBishop DNA model
 Figure 2. Numerical representations of the solitary wave solutions obtained through analytical methods are depicted. Panels (ac) illustrate the bright solitary wave solutions computed utilizing the Khater Ⅱ method (Eq (2.4)). These qualitative plots serve to validate the diversity of analytical solitary wave solutions and their localized wave profiles acquired from the two distinct techniques within the nonlinear framework of the PeyrardBishop DNA model
 Figure 3. The Hamiltonian framework of the PeyrardBishop DNA model reveals conserved quantities, illustrated in Panels (a, b) and (c, d), depicting the momentum $\mathbb{M}$ as described by Eqs (2.5) and (2.6). These conserved dynamic properties arise from the intrinsic nonlinearity governing the system. Graphical representation of these conserved quantities offers valuable insights into the equilibrium conditions that uphold DNA strand integrity within the physicallyconsistent nonlinear dynamics of the PeyrardBishop model
 Figure 4. A stream plot of the bright solitary wave solution, expressed by Eq (2.1) and (2.2), is presented with specific parameter values denoted as (ai). Continuous lines depict the gradual variation in DNA strand displacement ($u$) concerning distance ($x$) and time ($t$), illustrating localized melting bubbles that smoothly rise and fall along the strand during the denaturation process. This visual representation qualitatively captures bubble breathing dynamics and nonlinear strand fluctuations associated with bubble nucleation and growth
 Figure 5. A stream plot of the dark solitary wave solution, formulated by Eqs (2.3) and (2.4), is presented with parameter values denoted as (jo). The oscillating contour pattern showcases periodic fluctuations in the strand displacement ($u$) across spatial and temporal coordinates, offering a visual representation of bubble oscillation phenomena within denaturing regions. Localized ripples propagate along the strand, retaining their shape and amplitude, effectively illustrating bubble propagation and coalescence through RippleLike soliton interactions during denaturationrenaturation transitions
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