research plan mathematics

Research plan

The research plan is a document in which the PhD student develops a plan for his/her thesis. It should include an introduction, a brief state of the art to put in context the plan, the motivation and objectives of the work, the work done so far, a plan, detailing tasks and time planning, and a list of conferences and indexed journals were the work can be presented or submitted to.  A typical table of content is

• Cover page including the text “Research plan”, tentative title of the thesis, PhD program, name of the PhD student, name of the advisor(s) and date.
• Brief summary
• Introduction (state of the art, motivation, objectives)
• Work done so far
• Work plan (work to be done, tasks and time planning, tentative list of conferences and indexed journals, stays, etc). This must include a 1--page schedule of the forthcoming work until the end of the Thesis .

A short document of 10-30 pages is recommended, with annexes for details.

The panel is composed by 3 doctors with relevant expertise in the field, as president, secretary (from UPC) and vocal. As a general rule, the panel is composed by 2 researchers associated to the DMA program and an external one. In any case, the panel must include at least 1 researcher associated to the DMA program, and three of them cannot belong to the same research group. Finally any of the members of the panel can be a coauthor of the PhD student. 

[These rules have been approved by the Board of Studies of DMA at a meeting 05/07/2018, and included the condition of coauthor at the CADMAMAMME meeting 08/04/2021]

Presentation

The presentation should last no more than 30 min, followed by comments and questions from the panel.

• the pdf document of the research plan
• the form detailing the panel proposal (president, secretary and vocal), the date and time for the presentation, and the information for each one of the members of the panel.
• A scan of the registration form , previously signed by the advisors and the student.
• An updated version of the Document d'activitats del Doctorand (DAD)

Please, contact the PhD program admin or the coordinator for further information.

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

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

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

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

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1 An international survey in two rounds

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

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

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

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

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

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

2 Methodological approach

2.1 themes for the coming decade (2019).

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

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

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

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

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

Artistic impression of the future themes

2.2 Has the pandemic changed your view? (2020)

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

3 The themes

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

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

3.1 Approaches to teaching

We distinguish specific teaching strategies from broader curricular topics.

3.1.1 Teaching strategies

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

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

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

3.1.2 Curriculum

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

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

3.2 Goals of mathematics education

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

3.2.1 Societal goals

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

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

3.2.2 Educational goals

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

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

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

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

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

3.3 Relation of mathematics education to other practices

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

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

3.4 Teacher professional development

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

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

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

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

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

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

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

3.5 Technology

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

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

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

3.6 Equity, diversity, and inclusion

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

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

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

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

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

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

3.8 Assessment

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

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

4 Mathematics education research itself

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

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

4.2 Methodology

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

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

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

4.3 Reflection on our discipline

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

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

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

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

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

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

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

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

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

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

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

4.4 Interdisciplinarity and transdisciplinarity

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

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

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

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

5 The themes in their coherence: an artistic impression

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

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

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

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

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

6 Research challenges

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

6.1 Aligning new goals, curricula, and teaching approaches

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

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

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

6.2 Researching mathematics education across contexts

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

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

6.3 Focusing teacher professional development

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

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

6.4 Using low-tech resources

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

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

6.5 Staying in touch online

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

6.6 Studying and improving equity without perpetuating inequality

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

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

6.7 Assessing online

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

6.8 Doing and publishing interdisciplinary research

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

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

7 Concluding remarks

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

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

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Acknowledgments

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

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Appendix 1: Explanation of Fig. 1

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

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

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How to write a research plan: Step-by-step guide

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Today’s businesses and institutions rely on data and analytics to inform their product and service decisions. These metrics influence how organizations stay competitive and inspire innovation. However, gathering data and insights requires carefully constructed research, and every research project needs a roadmap. This is where a research plan comes into play.

There’s general research planning; then there’s an official, well-executed research plan. Whatever data-driven research project you’re gearing up for, the research plan will be your framework for execution. The plan should also be detailed and thorough, with a diligent set of criteria to formulate your research efforts. Not including these key elements in your plan can be just as harmful as having no plan at all.

Read this step-by-step guide for writing a detailed research plan that can apply to any project, whether it’s scientific, educational, or business-related.

• What is a research plan?

A research plan is a documented overview of a project in its entirety, from end to end. It details the research efforts, participants, and methods needed, along with any anticipated results. It also outlines the project’s goals and mission, creating layers of steps to achieve those goals within a specified timeline.

Without a research plan, you and your team are flying blind, potentially wasting time and resources to pursue research without structured guidance.

The principal investigator, or PI, is responsible for facilitating the research oversight. They will create the research plan and inform team members and stakeholders of every detail relating to the project. The PI will also use the research plan to inform decision-making throughout the project.

• Why do you need a research plan?

Create a research plan before starting any official research to maximize every effort in pursuing and collecting the research data. Crucially, the plan will model the activities needed at each phase of the research project.

Like any roadmap, a research plan serves as a valuable tool providing direction for those involved in the project—both internally and externally. It will keep you and your immediate team organized and task-focused while also providing necessary definitions and timelines so you can execute your project initiatives with full understanding and transparency.

External stakeholders appreciate a working research plan because it’s a great communication tool, documenting progress and changing dynamics as they arise. Any participants of your planned research sessions will be informed about the purpose of your study, while the exercises will be based on the key messaging outlined in the official plan.

Here are some of the benefits of creating a research plan document for every project:

Project organization and structure

Well-informed participants

All stakeholders and teams align in support of the project

Clearly defined project definitions and purposes

Distractions are eliminated, prioritizing task focus

Timely management of individual task schedules and roles

Costly reworks are avoided

• What should a research plan include?

The different aspects of your research plan will depend on the nature of the project. However, most official research plan documents will include the core elements below. Each aims to define the problem statement, devising an official plan for seeking a solution.

Specific project goals and individual objectives

Ideal strategies or methods for reaching those goals

Required resources

Descriptions of the target audience, sample sizes, demographics, and scopes

Key performance indicators (KPIs)

Project background

Research and testing support

Preliminary studies and progress reporting mechanisms

Cost estimates and change order processes

Depending on the research project’s size and scope, your research plan could be brief—perhaps only a few pages of documented plans. Alternatively, it could be a fully comprehensive report. Either way, it’s an essential first step in dictating your project’s facilitation in the most efficient and effective way.

• How to write a research plan for your project

When you start writing your research plan, aim to be detailed about each step, requirement, and idea. The more time you spend curating your research plan, the more precise your research execution efforts will be.

Account for every potential scenario, and be sure to address each and every aspect of the research.

Consider following this flow to develop a great research plan for your project:

Define your project’s purpose

Start by defining your project’s purpose. Identify what your project aims to accomplish and what you are researching. Remember to use clear language.

Thinking about the project’s purpose will help you set realistic goals and inform how you divide tasks and assign responsibilities. These individual tasks will be your stepping stones to reach your overarching goal.

Additionally, you’ll want to identify the specific problem, the usability metrics needed, and the intended solutions.

Know the following three things about your project’s purpose before you outline anything else:

What you’re doing

Why you’re doing it

What you expect from it

Identify individual objectives

With your overarching project objectives in place, you can identify any individual goals or steps needed to reach those objectives. Break them down into phases or steps. You can work backward from the project goal and identify every process required to facilitate it.

Be mindful to identify each unique task so that you can assign responsibilities to various team members. At this point in your research plan development, you’ll also want to assign priority to those smaller, more manageable steps and phases that require more immediate or dedicated attention.

Select research methods

Research methods might include any of the following:

User interviews: this is a qualitative research method where researchers engage with participants in one-on-one or group conversations. The aim is to gather insights into their experiences, preferences, and opinions to uncover patterns, trends, and data.

Field studies: this approach allows for a contextual understanding of behaviors, interactions, and processes in real-world settings. It involves the researcher immersing themselves in the field, conducting observations, interviews, or experiments to gather in-depth insights.

Card sorting: participants categorize information by sorting content cards into groups based on their perceived similarities. You might use this process to gain insights into participants’ mental models and preferences when navigating or organizing information on websites, apps, or other systems.

Focus groups: use organized discussions among select groups of participants to provide relevant views and experiences about a particular topic.

Diary studies: ask participants to record their experiences, thoughts, and activities in a diary over a specified period. This method provides a deeper understanding of user experiences, uncovers patterns, and identifies areas for improvement.

Five-second testing: participants are shown a design, such as a web page or interface, for just five seconds. They then answer questions about their initial impressions and recall, allowing you to evaluate the design’s effectiveness.

Surveys: get feedback from participant groups with structured surveys. You can use online forms, telephone interviews, or paper questionnaires to reveal trends, patterns, and correlations.

Tree testing: tree testing involves researching web assets through the lens of findability and navigability. Participants are given a textual representation of the site’s hierarchy (the “tree”) and asked to locate specific information or complete tasks by selecting paths.

Usability testing: ask participants to interact with a product, website, or application to evaluate its ease of use. This method enables you to uncover areas for improvement in digital key feature functionality by observing participants using the product.

Live website testing: research and collect analytics that outlines the design, usability, and performance efficiencies of a website in real time.

There are no limits to the number of research methods you could use within your project. Just make sure your research methods help you determine the following:

What do you plan to do with the research findings?

What decisions will this research inform? How can your stakeholders leverage the research data and results?

Recruit participants and allocate tasks

Next, identify the participants needed to complete the research and the resources required to complete the tasks. Different people will be proficient at different tasks, and having a task allocation plan will allow everything to run smoothly.

Prepare a thorough project summary

Every well-designed research plan will feature a project summary. This official summary will guide your research alongside its communications or messaging. You’ll use the summary while recruiting participants and during stakeholder meetings. It can also be useful when conducting field studies.

Ensure this summary includes all the elements of your research project. Separate the steps into an easily explainable piece of text that includes the following:

An introduction: the message you’ll deliver to participants about the interview, pre-planned questioning, and testing tasks.

Interview questions: prepare questions you intend to ask participants as part of your research study, guiding the sessions from start to finish.

An exit message: draft messaging your teams will use to conclude testing or survey sessions. These should include the next steps and express gratitude for the participant’s time.

Create a realistic timeline

While your project might already have a deadline or a results timeline in place, you’ll need to consider the time needed to execute it effectively.

Realistically outline the time needed to properly execute each supporting phase of research and implementation. And, as you evaluate the necessary schedules, be sure to include additional time for achieving each milestone in case any changes or unexpected delays arise.

For this part of your research plan, you might find it helpful to create visuals to ensure your research team and stakeholders fully understand the information.

Determine how to present your results

A research plan must also describe how you intend to present your results. Depending on the nature of your project and its goals, you might dedicate one team member (the PI) or assume responsibility for communicating the findings yourself.

In this part of the research plan, you’ll articulate how you’ll share the results. Detail any materials you’ll use, such as:

Presentations and slides

A project report booklet

A project findings pamphlet

Documents with key takeaways and statistics

Graphic visuals to support your findings

• Format your research plan

As you create your research plan, you can enjoy a little creative freedom. A plan can assume many forms, so format it how you see fit. Determine the best layout based on your specific project, intended communications, and the preferences of your teams and stakeholders.

Find format inspiration among the following layouts:

Written outlines

Narrative storytelling

Visual mapping

Graphic timelines

Remember, the research plan format you choose will be subject to change and adaptation as your research and findings unfold. However, your final format should ideally outline questions, problems, opportunities, and expectations.

• Research plan example

Imagine you’ve been tasked with finding out how to get more customers to order takeout from an online food delivery platform. The goal is to improve satisfaction and retain existing customers. You set out to discover why more people aren’t ordering and what it is they do want to order or experience. 

You identify the need for a research project that helps you understand what drives customer loyalty. But before you jump in and start calling past customers, you need to develop a research plan—the roadmap that provides focus, clarity, and realistic details to the project.

Here’s an example outline of a research plan you might put together:

Project title

Project members involved in the research plan

Purpose of the project (provide a summary of the research plan’s intent)

Objective 1 (provide a short description for each objective)

Objective 2

Objective 3

Proposed timeline

Audience (detail the group you want to research, such as customers or non-customers)

Budget (how much you think it might cost to do the research)

Risk factors/contingencies (any potential risk factors that may impact the project’s success)

Remember, your research plan doesn’t have to reinvent the wheel—it just needs to fit your project’s unique needs and aims.

Customizing a research plan template

Some companies offer research plan templates to help get you started. However, it may make more sense to develop your own customized plan template. Be sure to include the core elements of a great research plan with your template layout, including the following:

Introductions to participants and stakeholders

Background problems and needs statement

Significance, ethics, and purpose

Research methods, questions, and designs

Preliminary beliefs and expectations

Implications and intended outcomes

Realistic timelines for each phase

Conclusion and presentations

How many pages should a research plan be?

Generally, a research plan can vary in length between 500 to 1,500 words. This is roughly three pages of content. More substantial projects will be 2,000 to 3,500 words, taking up four to seven pages of planning documents.

What is the difference between a research plan and a research proposal?

A research plan is a roadmap to success for research teams. A research proposal, on the other hand, is a dissertation aimed at convincing or earning the support of others. Both are relevant in creating a guide to follow to complete a project goal.

What are the seven steps to developing a research plan?

While each research project is different, it’s best to follow these seven general steps to create your research plan:

Defining the problem

Identifying goals

Choosing research methods

Recruiting participants

Preparing the brief or summary

Establishing task timelines

Defining how you will present the findings

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How to apply

Once you have chosen your postgraduate research programme, you can follow our instructions to apply online.

Applying for postgraduate research studies is easy and can be done quickly online. Simply complete and submit an online application on the University application page.

We welcome applications for postgraduate research all year round, but please check your intended programme's start date to ensure that your application is received on time.

To find out what we need from you to process your application as well as advice on the application process, please visit the University website below.

Do I need to submit a research proposal?

A research proposal is optional for PhD applications in the Department of Mathematics. While you're welcome to submit one, we're perfectly happy for you to merely indicate the area(s) of mathematics (such as fluid mechanics, financial mathematics, number theory, numerical analysis, etc.) you're interested in, and/or list the names of any specific academics who you'd like to work with. If you omit the latter we will forward your application to all members of staff who have an interest in the research areas you've listed. Coming up with a research project that is not only worthwhile but also do-able (within the duration of a typical 3½ year PhD) is not a trivial exercise, and we generally encourage applicants to work on this jointly with the potential supervisor.

Do I need to submit a personal statement?

As with research proposals, providing a personal statement is optional, though we do encourage you to write one. Explain your motivation for wanting to study for a PhD, elaborate on your areas of interests, etc. If you do choose to include a personal statement, then this should be because you have interesting and relevant information to tell us.

Give us specifics rather than generic statements. So, rather than stating that you have been fascinated by mathematics from an early age, tell us about specific projects that you may have worked on in the course of your undergraduate degree, during summer research internships, or your MSc thesis, say. How did these give you insight into what research is like, and why did you enjoy them? 

It is important that any personal statement is written by you, and is not copied (or even be "inspired by") personal statements of others that are available on the internet and elsewhere.

What should the references say?

References form an extremely important part of the application. While an excellent undergraduate degree is obviously desirable, an applicant's performance in undergraduate exams is not always a good indicator of their suitability for research.

We therefore like to see references that comment on your suitability for research and explain the research activities that you've been involved in to date.

We recommend taking every available opportunity to obtain research experience (eg: in a final year project, a summer internship, or an MSc research project), and for many students the enjoyment of this experience forms a large part of their motivation to study for a PhD.

Choose referees who can comment on this; your project supervisor or any other academic who has seen you undertake research or project work are obvious choices. Other good referees tend to be academics who have noticed you in their lectures (maybe your interest in the material meant that you asked lots of good questions  – academics tend to remember this).

What happens after I've submitted my application?

If you have already been in contact with a supervisor, have agreed a research project and have been interviewed, you should get a decision very quickly. We will simply check that the anticipated supervisor is willing to take you on as a student and that the paperwork (transcript, referees reports, etc) confirms their positive impression of you. If so, you will receive an academic offer.

If you haven't had a formal interview, we will arrange one with the supervisor and a second academic, usually the co-supervisor. This can be done by video call (eg Skype), though if possible, you should try to visit the Department if you have not done so already.

If you have not been in contact with a supervisor prior to applying (and, as indicated above, this is not a bad thing), we will forward your application to all potential supervisors who work in the area(s) you expressed an interest in. If they are interested (in principle) to supervise you and have the capacity to take on more PhD students, they will contact you to discuss possible projects in more detail. If we can't find a suitable supervisor for you, we'll let you know. We're a big Department and have people working in many areas of mathematics, but there is a limit to the number of students individual academics can sensibly supervise.

Head to the University website to follow our advice along the application process before submitting your application.

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Writing the Research Plan for Your Academic Job Application

By Jason G. Gillmore, Ph.D., Associate Professor, Department of Chemistry, Hope College, Holland, MI

A research plan is more than a to-do list for this week in lab, or a manila folder full of ideas for maybe someday—at least if you are thinking of a tenure-track academic career in chemistry at virtually any bachelor’s or higher degree–granting institution in the country. A perusal of the academic job ads in C&EN every August–October will quickly reveal that most schools expect a cover letter (whether they say so or not), a CV, a teaching statement, and a research plan, along with reference letters and transcripts. So what is this document supposed to be, and why worry about it now when those job ads are still months away?

What Is a Research Plan?

A research plan is a thoughtful, compelling, well-written document that outlines your exciting, unique research ideas that you and your students will pursue over the next half decade or so to advance knowledge in your discipline and earn you grants, papers, speaking invitations, tenure, promotion, and a national reputation. It must be a document that people at the department you hope to join will (a) read, and (b) be suitably excited about to invite you for an interview.

That much I knew when I was asked to write this article. More specifics I only really knew for my own institution, Hope College (a research intensive undergraduate liberal arts college with no graduate program), and even there you might get a dozen nuanced opinions among my dozen colleagues. So I polled a broad cross-section of my network, spanning chemical subdisciplines at institutions ranging from small, teaching-centered liberal arts colleges to our nation’s elite research programs, such as Scripps and MIT. The responses certainly varied, but they did center on a few main themes, or illustrate a trend across institution types. In this article I’ll share those commonalities, while also encouraging you to be unafraid to contact a search committee chair with a few specific questions, especially for the institutions you are particularly excited about and feel might be the best fit for you.

How Many Projects Should You Have?

While more senior advisors and members of search committees may have gotten their jobs with a single research project, conventional wisdom these days is that you need two to three distinct but related projects. How closely related to one another they should be is a matter of debate, but almost everyone I asked felt that there should be some unifying technique, problem or theme to them. However, the projects should be sufficiently disparate that a failure of one key idea, strategy, or technique will not hamstring your other projects.

For this reason, many applicants wisely choose to identify:

• One project that is a safe bet—doable, fundable, publishable, good but not earthshaking science.
• A second project that is pie-in-the-sky with high risks and rewards.
• A third project that fits somewhere in the middle.

Having more than three projects is probably unrealistic. But even the safest project must be worth doing, and even the riskiest must appear to have a reasonable chance of working.

How Closely Connected Should Your Research Be with Your Past?

Your proposed research must do more than extend what you have already done. In most subdisciplines, you must be sufficiently removed from your postdoctoral or graduate work that you will not be lambasted for clinging to an advisor’s apron strings. After all, if it is such a good idea in their immediate area of interest, why aren’t they pursuing it?!?

But you also must be able to make the case for why your training makes this a good problem for you to study—how you bring a unique skill set as well as unique ideas to this research. The five years you will have to do, fund, and publish the research before crafting your tenure package will go by too fast for you to break into something entirely outside your realm of expertise.

Biochemistry is a partial exception to this advice—in this subdiscipline it is quite common to bring a project with you from a postdoc (or more rarely your Ph.D.) to start your independent career. However, you should still articulate your original contribution to, and unique angle on the work. It is also wise to be sure your advisor tells that same story in his or her letter and articulates support of your pursuing this research in your career as a genuinely independent scientist (and not merely someone who could be perceived as his or her latest "flunky" of a collaborator.)

Should You Discuss Potential Collaborators?

Regarding collaboration, tread lightly as a young scientist seeking or starting an independent career. Being someone with whom others can collaborate in the future is great. Relying on collaborators for the success of your projects is unwise. Be cautious about proposing to continue collaborations you already have (especially with past advisors) and about starting new ones where you might not be perceived as the lead PI. Also beware of presuming you can help advance the research of someone already in a department. Are they still there? Are they still doing that research? Do they actually want that help—or will they feel like you are criticizing or condescending to them, trying to scoop them, or seeking to ride their coattails? Some places will view collaboration very favorably, but the safest route is to cautiously float such ideas during interviews while presenting research plans that are exciting and achievable on your own.

How Do You Show Your Fit?

Some faculty advise tailoring every application packet document to every institution to which you apply, while others suggest tweaking only the cover letter. Certainly the cover letter is the document most suited to introducing yourself and making the case for how you are the perfect fit for the advertised position at that institution. So save your greatest degree of tailoring for your cover letter. It is nice if you can tweak a few sentences of other documents to highlight your fit to a specific school, so long as it is not contrived.

Now, if you are applying to widely different types of institutions, a few different sets of documents will certainly be necessary. The research plan that you target in the middle to get you a job at both Harvard University and Hope College will not get you an interview at either! There are different realities of resources, scope, scale, and timeline. Not that my colleagues and I at Hope cannot tackle research that is just as exciting as Harvard’s. However, we need to have enough of a niche or a unique angle both to endure the longer timeframe necessitated by smaller groups of undergraduate researchers and to ensure that we still stand out. Furthermore, we generally need to be able to do it with more limited resources. If you do not demonstrate that understanding, you will be dismissed out of hand. But at many large Ph.D. programs, any consideration of "niche" can be inferred as a lack of confidence or ambition.

Also, be aware that department Web pages (especially those several pages deep in the site, or maintained by individual faculty) can be woefully out-of-date. If something you are planning to say is contingent on something you read on their Web site, find a way to confirm it!

While the research plan is not the place to articulate start-up needs, you should consider instrumentation and other resources that will be necessary to get started, and where you will go for funding or resources down the road. This will come up in interviews, and hopefully you will eventually need these details to negotiate a start-up package.

Who Is Your Audience?

Your research plan should show the big picture clearly and excite a broad audience of chemists across your sub-discipline. At many educational institutions, everyone in the department will read the proposal critically, at least if you make the short list to interview. Even at departments that leave it all to a committee of the subdiscipline, subdisciplines can be broad and might even still have an outside member on the committee. And the committee needs to justify their actions to the department at large, as well as to deans, provosts, and others. So having at least the introduction and executive summaries of your projects comprehensible and compelling to those outside your discipline is highly advantageous.

Good science, written well, makes a good research plan. As you craft and refine your research plan, keep the following strategies, as well as your audience in mind:

• Begin the document with an abstract or executive summary that engages a broad audience and shows synergies among your projects. This should be one page or less, and you should probably write it last. This page is something you could manageably consider tailoring to each institution.
• Provide sufficient details and references to convince the experts you know your stuff and actually have a plan for what your group will be doing in the lab. Give details of first and key experiments, and backup plans or fallback positions for their riskiest aspects.
• Hook your readers with your own ideas fairly early in the document, then strike a balance between your own new ideas and the necessary well referenced background, precedents, and justification throughout. Propose a reasonable tentative timeline, if you can do so in no more than a paragraph or two, which shows how you envision spacing out the experiments within and among your projects. This may fit well into your executive summary
• Show how you will involve students (whether undergraduates, graduate students, an eventual postdoc or two, possibly even high schoolers if the school has that sort of outreach, depending on the institutions to which you are applying) and divide the projects among students.
• Highlight how your work will contribute to the education of these students. While this is especially important at schools with greater teaching missions, it can help set you apart even at research intensive institutions. After all, we all have to demonstrate “broader impacts” to our funding agencies!
• Include where you will pursue funding, as well as publication, if you can smoothly work it in. This is especially true if there is doubt about how you plan to target or "market" your research. Otherwise, it is appropriate to hold off until the interview to discuss this strategy.

So, How Long Should Your Research Plan Be?

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Here is where the answers diverged the most and without a unifying trend across institutions. Bottom line, you need space to make your case, but even more, you need people to read what you write.

A single page abstract or executive summary of all your projects together provides you an opportunity to make the case for unifying themes yet distinct projects. It may also provide space to articulate a timeline. Indeed, many readers will only read this single page in each application, at least until winnowing down to a more manageable list of potential candidates. At the most elite institutions, there may be literally hundreds of applicants, scores of them entirely well-suited to the job.

While three to five pages per proposal was a common response (single spaced, in 11-point Arial or 12-point Times with one inch margins), including references (which should be accurate, appropriate, and current!), some of my busiest colleagues have said they will not read more than about three pages total. Only a few actually indicated they would read up to 12-15 pages for three projects. In my opinion, ten pages total for your research plans should be a fairly firm upper limit unless you are specifically told otherwise by a search committee, and then only if you have two to three distinct proposals.

Why Start Now?

Hopefully, this question has answered itself already! Your research plan needs to be a well thought out document that is an integrated part of applications tailored to each institution to which you apply. It must represent mature ideas that you have had time to refine through multiple revisions and a great deal of critical review from everyone you can get to read them. Moreover, you may need a few different sets of these, especially if you will be applying to a broad range of institutions. So add “write research plans” to this week’s to do list (and every week’s for the next few months) and start writing up the ideas in that manila folder into some genuine research plans. See which ones survive the process and rise to the top and you should be well prepared when the job ads begin to appear in C&EN in August!

Jason G. Gillmore , Ph.D., is an Associate Professor of Chemistry at Hope College in Holland, MI. A native of New Jersey, he earned his B.S. (’96) and M.S. (’98) degrees in chemistry from Virginia Tech, and his Ph.D. (’03) in organic chemistry from the University of Rochester. After a short postdoctoral traineeship at Vanderbilt University, he joined the faculty at Hope in 2004. He has received the Dreyfus Start-up Award, Research Corporation Cottrell College Science Award, and NSF CAREER Award, and is currently on sabbatical as a Visiting Research Professor at Arizona State University. Professor Gillmore is the organizer of the Biennial Midwest Postdoc to PUI Professor (P3) Workshop co-sponsored by ACS, and a frequent panelist at the annual ACS Postdoc to Faculty (P2F) Workshops.

Other tips to help engage (or at least not turn off) your readers include:

• Avoid two-column formats.
• Avoid too-small fonts that hinder readability, especially as many will view the documents online rather than in print!
• Use good figures that are readable and broadly understandable!
• Use color as necessary but not gratuitously.

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Mathematics Research Proposals Samples For Students

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Regents approve plans for new chemistry and applied mathematics facility

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Rendering of the newly approved chemistry and applied mathematics facility. This is a draft rendering, subject to change.

The University of Colorado Board of Regents today approved plans to construct a new academic and research facility aimed at advancing research and educational opportunities in the fields of chemistry and applied mathematics. The proposed facility, spanning approximately 79,200 square feet, will be situated on the southeast side of Business Field on Main Campus.

The project, with an estimated cost of $175.43 million, will be funded primarily through a combination of campus cash reserves and debt.

Key features

• Shelled space for future specialized quantum research laboratories and offices.
• A nuclear magnetic resonance (NMR) spectroscopic core facility
• Modern classroom, office, meeting and research lab space
• A centrally scheduled 200-seat auditorium
• Student study areas
• Upper-level roof terrace

Academic goals

• The facility will provide modern research laboratories to promote the Department of Chemistry ’s research mission, improve recruitment and strengthen retention. Foundational and interdisciplinary research fields include: analytical spectroscopy, environmental chemistry, experimental and theoretical physical/ biophysical chemistry, inorganic and materials chemistry, synthetic and physical organic/bio-organic chemistry.
• The facility will support the Department of Applied Mathematics ’ efforts to continue to provide a major research presence in computational and physical/biological mathematics and the statistical sciences, enhancing the department’s effort to be a leading program of applied mathematics in the United States.
• The facility will provide shelled spaces to be completed in the future for the growing field of quantum-chemistry research.
• It will serve as a central hub for cutting-edge research, student learning outcomes and sustainable innovation, and it will foster interdisciplinary collaboration among students and researchers.
• It will provide training opportunities for students and researchers, fostering interdisciplinary collaboration and innovation.

Sustainability considerations

• The chemistry and applied math building seeks to be one of the most energy-efficient research buildings in the history of CU Boulder by achieving an Energy Use Intensity (EUI) of approximately 100 kilo British thermal units (kBtu) per gross square feet of the building (kBtu/ft2), according to current modeling.
• Construction will follow Buy Clean Colorado Act guidelines, using eligible materials that do not exceed the maximum allowable global warming potential limit in each construction category. 
• The building’s mechanical system is being designed to use low-temperature hot water in preparation for the eventual connection to a future district energy loop heated by electricity, in support of the university’s overarching decarbonization plans.
• It will be the first building on campus to implement the use of cross-laminated timber (CLT) construction. The project will use CLT instead of concrete for the structure in non-research areas of the building. This will reduce the carbon impact of the structure.

What they’re saying

“The approval of the new chemistry and applied mathematics facility marks a significant milestone for our campus,” CU Boulder Chancellor Phil DiStefano said.

“This project underscores our commitment to advancing research and fostering interdisciplinary collaboration. It will not only provide state-of-the-art resources for our scholars but also pave the way for groundbreaking discoveries that will benefit both our campus community and society at large."

What’s next

Construction is slated to begin in October, with occupancy planned for late 2026/early 2027.

Vacated spaces within Cristol Chemistry & Biochemistry are anticipated to be assigned to the College of Media, Communications and Information (CMCI), while spaces vacated by Applied Mathematics within the Engineering Center will benefit the College of Engineering and Applied Science.

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