importance of qualitative research in mathematics

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Approaches to qualitative research in mathematics education.

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Angelika Bikner-Ahsbahs, Christine Knipping, and Norma Presmeg, editors

• Table of Contents

Over the last twenty years, qualitative research has fought hard to earn legitimacy as opposed to the critical eyes of the positivist approach. Slowly these have emerged as complementary paradigms of scientific inquiry. It is still not uncommon, however, for colleagues and students to ridicule qualitative research methods as being loose, unscientific, or illegitimate. Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods is a clever gift for the skeptics who believe that pursuing truth is only possible through traditional empirical research.

In the book are 19 total chapters that cover the theories and methodologies of qualitative inquiry for research in mathematics education. Each chapter includes an abstract, keywords, sections and references. Together they cohesively showcase the variety of theories and methods within the broad framework of qualitative research and focus on connecting theories and research methods along with rich research examples. The theories and methods covered in the book include grounded theory, ideal type construction, theory of argumentation, the Vygotskian semiotic approach, networking of theories, mixed methods, multilevel analysis, qualitative content analysis, triangulation, and design-based research.

The book is too thick to be read in several days, but this is not a bad thing. Still, I doubt that the book can serve as an effective text for undergraduate students. Given the academic and technical nature of this book, it will better serve doctoral students in mathematics education, particularly students interested in examples of theoretical framework, design, and methods for qualitative study; as well as mathematics education researchers interested in gaining a current snapshot of advanced qualitative methodologies in the field.

It should be noted that each chapter stands alone: the chapters are neither interconnected nor presented in sequence. So readers may search for theories or methods in the subject index and read their chapters of interest. The author index is useful as well. For example, I was interested in how students develop abstract knowledge in the mathematics classroom. I searched the subject index and found an entry for abstractions, which led me to Chapter 8, titled “The Nested Epistemic Actions Model for Abstraction in Context.” The abstract for the chapter stated, “abstraction in context is a theoretical framework for studying students; processes of constructing abstract mathematical knowledge as it occurs in a context that includes specific mathematical, curricular and social components as well as a particular learning environment (p.185).” I was hooked and kept reading. The chapter provided an outline of the theoretical framework of Abstraction in Context, background information on the methodology, and a detailed account of how the theory and methodology supported one another in research design along with findings and analysis.

The work of the contributors inspires researchers in the field of mathematics education to replicate the studies and, more importantly, creates opportunities to further reflect on the ways theories inform qualitative research designs and methods. Whether the editors meant to achieve this or not, one thing is clear from reading the collective scholarly work: the book offers endless possibilities for our field to pursue truth beyond statistical significance in the phenomena of teaching and learning mathematics.

Woong Lim ( [email protected] ) is an Assistant Professor of Mathematics Education at University of New Mexico. His research interests include interrelations between language and mathematics, content knowledge for teaching, and social justice issues in mathematics education.

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Education Journal

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Qualitative Approaches in Mathematics Education Research: Challenges and Possible Solutions

Sashi Sharma

Department of Mathematics, Science and Technology Education, Faculty of Education,The university of Waikato, Hamilton, New Zealand

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Despite being relatively new in mathematics education research, qualitative researchapproaches need special attention as attempts are being made to enhance the credibility and trustworthiness of this approach. It is important that researchers are aware of the limitations associated with these methods so that measures are put in place to try and minimize the effects of these limitations Philosophical roots and key features of this paradigm are outlined. Qualitative methods such as the interview approach in research literature as a data gathering tool are considered next. Challenges faced by qualitative researchers in terms of reliability, validity and generability are considered. Examples are provided to illustrate methodological problems and solutions related to qualitative methods.

Research Methods, Qualitative Research, Data Collection, Quality Criteria, Limitations, Possible Solutions

Sashi Sharma. (2013). Qualitative Approaches in Mathematics Education Research: Challenges and Possible Solutions. Education Journal , 2 (2), 50-57. https://doi.org/10.11648/j.edu.20130202.14

Sashi Sharma. Qualitative Approaches in Mathematics Education Research: Challenges and Possible Solutions. Educ. J. 2013 , 2 (2), 50-57. doi: 10.11648/j.edu.20130202.14

Sashi Sharma. Qualitative Approaches in Mathematics Education Research: Challenges and Possible Solutions. Educ J . 2013;2(2):50-57. doi: 10.11648/j.edu.20130202.14

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Approaches to qualitative research in mathematics education : examples of methodology and methods

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• Part 1: Grounded theory methodology.
• Chapter 1: Anne R. Teppo. Grounded Theory Methods.
• Chapter 2: Maike Vollstedt. To see the wood for the trees: The development of theory from empirical interview data using grounded theory.-
• Part 2: Approaches to reconstructing argumentation.
• Chapter 3: Gotz Krummheuer. Methods for reconstructing processes of argumentation and Chaptericipation in primary mathematics classroom interaction.
• Chapter 4: Christine Knipping and David Reid. Reconstructing argumentation structures: A perspective on proving processes in secondary mathematics classroom interactions.-
• Part 3: Ideal type construction.
• Chapter 5: Angelika Bikner-Ahsbahs. Empirically grounded building of ideal types. A methodical principle of constructing theory in the interpretive research in mathematics education.
• Chapter 6: Angelika Bikner-Ahsbahs. How ideal type construction can be achieved: An example.-
• Part 4: Semiotic research.
• Chapter 7: Luis Radford and Cristina Sabena. The question of method in a Vygotskian semiotic approach.-
• Part 5: A theory on abstraction and its methodology.
• Chapter 8: Tommy Dreyfus, Rina Hershkowitz and Baruch Schwarz. The nested epistemic actions model for Abstraction in Context: Theory as methodological tool and methodological tool as theory.-
• Part 6: Networking of theories.
• Chapter 9: Ivy Kidron and Angelika Bikner-Ahsbahs. Advancing research by means of the networking of theories.
• Chapter 10: Angelika Bikner-Ahsbahs and Ivy Kidron. A cross-methodology for the networking of theories: The general epistemic need (GEN) as a new concept at the boundary of two theories.-
• Part 7: Multi-level-analysis.
• Chapter 11: Geoffrey B. Saxe, Kenton de Kirby, Marie Le, Yasmin Sitabkhan, Bona Kang. Understanding learning across lessons in classroom communities: A multi-leveled analytic approach.-
• Part 8: Mixed Methods.
• Chapter 12: Udo Kelle and Nils Buchholtz. The combination of qualitative and quantitative research methods in mathematics education-A "Mixed Methods" study on the development of the professional knowledge of teachers.-
• Part 9: Qualitative Content Analysis.
• Chapter 13: Philipp Mayring. Qualitative Content Analysis: Theoretical background and procedures.
• Chapter 14: Bjorn Schwarz. A study on professional competence of future teacher students as an example of a study using Qualitative Content Analysis.-
• Part 10: Triangulation and cultural studies.
• Chapter 15: Ida Ah Chee Mok and David J. Clarke. The contemporary importance of triangulation in a post-positivist world: Examples from the Learner's Perspective Study.-
• Part 11: Design research as a research methodology.
• Chapter 16: Arthur Bakker and Dolly van Eerde. An introduction to design-based research with an example from statistics education.
• Chapter 17: Michele Artigue. Perspectives on design research: The case of didactical engineering.
• Chapter 18: Erin Henrick, Paul Cobb and Kara Jackson. Educational design research to support system-wide instructional improvement.
• Part 12: Looking back.
• Chapter 19: Angelika Bikner-Ahsbahs, Christine Knipping and Norma Presmeg. Appendix.- References.- Index of keywords.
• (source: Nielsen Book Data)

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Approaches to Qualitative Research in Mathematics Education

2015, Advances in Mathematics Education

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This book by-passes both psychology and sociology to present an original social theory centered on seeing mathematical learning by everyone as an intrinsic dimension of how mathematics develops as a field in support of human activity. Here, mathematics is defined by how we collectively talk about it. Drawing on psychoanalytic theory, the student is seen as participating in the renewal of mathematics through their contributions to our collective gaze on mathematics as the field responds to ever new demands. As such learning takes a critical stance on the standard initiations into current practices often promoted by formal education. In the field of mathematics education, researchers have moved from psychology where individual students were seen as following natural paths of development through existing mathematical knowledge, to socio-cultural models predicated on students being initiated into the human world and understood through the reflective gazes this world has of itself, such as those found in comparisons of student learning in different countries. This book addresses the domain, purpose and functioning of contemporary research in mathematics education and is an original contribution to this theme. The book is aimed at a mathematics education research audience. It continues a dialogue with existing publications, seen widely as a cutting edge and will also be of interest to students and practitioners in the fields of qualitative research, social theory and psychology. New book- Tony Brown A contemporary theory of mathematics education research https://www.springer.com/gp/book/9783030550998#aboutBook A preface providing an action-packed overview is freely downloadable as a pdf: https://link.springer.com/book/10.1007%2F978-3-030-55100-1

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Using Focus Groups to Guide Action Research in Mathematics Education

Mr Marc Jacobs, Reading University

Abstract The aim of this doctoral research is to determine how students learn mathematics successfully and what strategies work best in secondary classrooms. Mathematics classrooms and teachers’ practice were investigated through several research methods. One method was student focus group interviews to reveal students’ views of teacher practice. It is widely appreciated (Dowker, 2009; Wilson & Räsänen, 2008) that there is limited research available on effective strategies for supporting students with their learning in mathematics compared to that available for literacy. Wilson and Räsänen (2008) suggest that there are several reasons for this limited research including the cost in terms of monetary considerations and implementation, particularly with large numbers. Therefore, implementing any form of successful mathematical investigation and intervention is challenging when attempting to use a strategy that works across a modern secondary school with a diverse population of teachers and students. This paper reports on the use of focus group interviews for the purpose of obtaining data from students about their learning experiences in mathematics lessons and their views of their teachers’ practice. Teachers’ practice is fundamental to successful learning experiences and Ball (1988) acknowledges that their practice is influenced by their beliefs about teaching and learning, about their students, and about their context thereby shaping how they teach.

Keywords: mathematics, focus groups, research methods

Introduction In the past few years, focus group interviews have been used increasingly in fields other than market research, where the technique was first developed (Berg, 1995). Berg (1995:65) also notes, focus group interviews have traditionally been dismissed as part of the “vulgar world of marketing research”. However, it is a method that is increasingly being appreciated for the advantages it offers to researchers in other data collection situations (Morgan, 1993; Gibbs, 1997; Barbour & Kitzinger, 1998). During the 1990s, one begins to see what a reversal in the elitist attitude that may be that focus group interviewing belongs to the somehow vulgar realm of marketing research. Instead, social scientists have begun regarding the approach with greater respect. Sussman, Burton, Dent, Stacy and Flay (1991: 773) state that “focus group methodology is one of the most widely used qualitative research tools in the applied social sciences.” Similar arguments have been offered by Basch (1987, 1989) and by Stewart and Shamdasani (1990). Clearly, there are some advantages to the use of this data-collecting orientation in certain situations. This research aims to explore how students believe they learn mathematics successfully and what classroom strategies work best by initially drawing upon reviews of this method and then subsequently using it with a group of secondary student participants. This paper is part of a wider doctoral project entitled Intervention in Mathematics: Creating successful strategies to ensure success in Secondary Schools where focus groups were used as one of the methods of data collection.

Definition of focus groups There are many definitions of a focus group interview and Kitzinger (2005) suggests the focus group method is an ‘ideal’ approach for examining the stories, experiences, points of view, beliefs, needs and concerns of individuals. The method is especially valuable for permitting the participants to develop their own questions and frameworks as well as to seek their own needs and concerns in their own words and on their own terms. Powell, Single and Lloyd (1996) define a focus group interview as a group of individuals selected and assembled by researchers to discuss and comment on, from personal experience, the topic that is the subject of the research. Morgan (1997) notes that a focus group (also called a focus group interview or a focus group discussion) is a form of group interviewing but it is important to distinguish between the two. Group interviewing involves interviewing a number of people at the same time and the emphasis is on questions and responses between the researcher and participants. Focus groups, however, rely on interaction within the group based on topics that are supplied by the researcher. The key characteristic, which distinguishes group interviewing from focus groups, is the insight and data produced by the interaction between participants (Morgan, 1997). A focus group is a form of qualitative research in which a group of people is asked about their perceptions, opinions, beliefs, and attitudes towards a phenomenon. Furthermore, focus groups are a form of group interview that capitalises on communication between research participants in order to generate data. Focus groups explicitly use group interaction as part of the method (Kitzinger, 1994). This means that instead of the researcher asking each person to respond to a question in turn, people are encouraged to talk to one another by asking questions, exchanging anecdotes and commenting on each other’s experiences and points of view (Kitzinger, 1994). During the 1980s focus groups reappeared in social sciences after being absent for some time and are now commonly used in cross-cultural research in a variety of fields, such as academic, policy-related or marketing research. Malhotra (1996:171-172) remarked that “focus groups are the most important qualitative research procedure. They are so popular that many marketing research practitioners consider this technique synonymous with qualitative research.” Even though this statement refers to the mid-1980s in the USA, it still has relevance today, although the arrival of new techniques, among them online focus groups, has certainly redesigned the overall picture.

The origins and history of focus group research The first use of group interviews was in the 1920s by social scientists Emory Bogardus and Walter Thurstone who used it to develop survey instruments. The methodology is mainly attributed to Bogardus who in 1926 described focus groups in his social psychological research to develop the social distance scale (see Wilkinson 1998). During World War II Robert Merton and Paul Lazarsfeld used group interviews to assist the allied forces in the development of propaganda materials, training manuals and to understand social issues. In the 1950s, focus groups became commonplace among marketers to understand customers while social scientists continued to prefer formal survey research. Sociologist Robert Merton worked with colleagues on the effectiveness of focus group in the years following World War II and the group (Merton, Fiske & Kendall) later wrote a seminal text entitled The Focused Interview: A manual of problems and procedures which was published in 1956. Of particular interest in the post-World War II era was the study of mass-mediated ‘propaganda’. The term focus group replaced group interview as the name of this technique. Over the last twenty years, there has been a steadily increasing interest in establishing qualitative research’s place in the academy which has resulted in the growing use of focus groups, especially in social sciences. In the 1980s focus groups re-emerged as a distinct research method in the social sciences (Conradson, 2005). Kamberelis and Dimitriadis (2005) state that focus groups have been popular and used extensively in several disciplines. Many social scientists and other professionals have found this qualitative approach very useful. Political scientists, for example, employed focus groups to examine the public perceptions of political candidates and their opinions on particular political issues (Madriz, 2003; Gaiser, 2008). During President Ronald Reagan’s administration in the 1980s, focus groups were adopted to learn about the perceptions of relations between the United States and the Soviet Union and their citizens (Stewart, Shamdasani & Rook, 2007). Focus groups were also used by the New Labour government from the early 1990s to early 2000s in the UK to examine British opinions about health spending, education policy and military action. The aim was to explore ‘a better understanding of the multiple and sometimes conflicting perspectives held by the public on particular issues’ (Conradson, 2005:130). The use of focus groups has been established over a period of time so their value to researchers for uncovering significant information is of interest.

Advantages of focus groups Focus groups are valuable when you want to consider not only people’s personal accounts of reality but also the way they negotiate these accounts with others, therefore showing divergence or convergence between their views. Cambridge and McCarthy (2001) describe a focus group interview as a group dynamic that can help build confidence, safe environments that are not threatening or intimidating and peer support and validation, enabling all people, regardless of perceived competence, to contribute to research discussions. Focus groups appear to be of value for all members of society and this success was evidenced by Fraser and Fraser (2001) whose research engaged participants with communication difficulties. Focus groups require both individual contributions and group dynamics and they found that with participants with communication difficulties groups smaller than the six-ten usually recommended were better and that the addition of an interpreter familiar with the participants’ communication was also important. Moreover, Fraser and Fraser (2001) found that participants’ ability to interact with others in a group was more important to success than their various types of communication challenges such as the ability to produce more than a few words, reliance on Makaton sign language, or even repetitive language. They concluded that ‘focus groups are a very good method for some people with learning disabilities in some situations but not in others; it is important to be able to distinguish this before setting up the group’ (Fraser & Fraser, 2001:225). So, there are clearly some limitations that need to be considered before the method is developed as a data collection instrument.

Limitations of focus groups Like any other research method focus groups do not suit all research aims and there have been times when they were found to be inappropriate or problematic. For example, compared to individual interviews, focus groups may not be as efficient in providing maximum depth on a particular issue. A particular disadvantage of a focus group is the possibility that the members may not express their honest and personal opinions about the topic at hand (Smithson, 2008). They may be hesitant to express their thoughts, especially when their thoughts oppose the views of another participant. Smithson (2008), a researcher who uses focus groups extensively, contends that some research topics are unsuitable for focus group environments. For example, topics which are seen as too personal (such as living with HIV/AIDS, sexuality, infertility, financial status, divorce, domestic violence and abortion) may be better carried out by other methods such as individual interviews. In institutional contexts (such as the workplace or schools), people may be reluctant to express their opinions or discuss their personal experiences in front of colleagues. Hopkins (2007) and Krueger and Casey (2009) found that often focus groups are criticised for only offering a shallower understanding of an issue than those obtained from individual interviews. In a focus group discussion, personal information and experiences may not be discussed thereby reducing the natural narrative that emerges from rich discourse. An example of this is Hopkins’ (2007) qualitative research project about the life and times of young Muslim men living in Scotland which showed that they revealed personal experiences of racism during individual interviews far more than they did in focus group discussions. The fact that focus groups are driven by the researcher’s interests can also be a source of weakness. What may be of intense interest to the researcher may be a non-issue to the participants. However, the fact that the researcher creates and directs the groups makes them distinctly less naturalistic than participant observation so there is always some residual uncertainty (Morgan 1996) about the accuracy of what the participants say. In particular, there is a very real concern that the researcher, in the name of maintaining the interview’s focus, will influence the group’s interactions. This problem is hardly unique to focus groups because the researcher influences all but the most unobtrusive social science methods. In reality, there is no hard evidence that the focus group researcher’s impact on the data is any greater than the researcher’s impact in participant observation or individual interviewing. Indeed, the dyadic nature of individual interviewing would seem to create at least as many opportunities for researcher influence. The concerns for focus groups include both a tendency toward conformity, in which some participants withhold things that they might say in private, and a tendency toward ‘polarization’ in which some participants express more extreme views in a group than in private (Sussman, Burton, Dent, et al., 1991). It is clear, however, that for some types of participants discussing some types of topics the presence of a group will affect what they say and how they say it. This is an inevitable aspect of focus groups that should be considered as a potential source of weakness for any given research project. Morgan (1988) states that the researcher, or moderator as they are often termed, has less control over the data produced than in either quantitative studies or one-to-one interviewing. The researcher has to allow participants to talk to each other, ask questions and express doubts and opinions while having very little control over the interaction other than generally keeping participants focused on the topic. By its nature focus group research is open-ended and cannot be entirely predetermined. On a practical note, focus groups can be also difficult to assemble. It may not be easy to get a representative sample and focus groups may discourage certain people from participating, for example, those who are not very articulate or confident, and those who have communication problems or special needs. The method of focus group discussion may also discourage some people from trusting others with sensitive or personal information. In such cases, personal interviews or the use of workbooks alongside focus groups may be a more suitable approach. Finally, focus groups are not fully confidential or anonymous, because the material is shared with the others in the group (Morgan 1997) and this has ethical implications that need to be considered when developing a methodological approach.

Types of focus groups There are different formations of focus groups and this section explores those types. Traditional Focus Groups are more straightforward, question-oriented groups. Usually, there is a ‘warm-up’ then the concept, idea, situation or product is presented to the group for their reaction. A neutral moderator who probes for issues of interest and follows up on interesting or relevant comments made by the participants’ guides this process. The key factors to successful traditional groups include clearly defined research issues, an experienced moderator who understands the issues at hand and decisions to be made; and diligent recruiting (Morgan, 1984). Projective Focus Groups bear a resemblance to traditional focus group discussions in that they are an informal, subtly structured conversation on a specific subject lead by a neutral moderator. They differ in the methods used to explore thoughts and feelings about the subject, and in the emotional depth that can be reached using these methods. Projective Groups rely more on indirect questioning and strongly emphasize the interpretation of group input. Some of the techniques that may be used in Projective Focus Groups include collage-building, brand personification, guided journey and pictorial symbols (Morgan, 1984). Projective Focus Groups are used extensively in exploring brand image and the development of creative concepts for products, services and advertising. A few of the questions addressed in Projective Focus Groups have included: Is this the right name for the product? What feelings are evoked by our brand? What mood should our advertising and collateral material invoke? I have used the traditional focus group interview as I wanted to hear what students think – literally. There is nothing more powerful than hearing first-hand what students have to say about how they learn mathematics.

Uses of focus groups Morgan and Kreuger (1993) state that the main purpose of focus group research is to draw upon respondents’ attitudes, feelings, beliefs, experiences and reactions in a way in which would not be feasible using other methods such as observation, one-to-one interviewing, or questionnaire surveys. These attitudes, feelings and beliefs may be partially independent of a group or its social setting but are more likely to be revealed via the social gathering and the interaction which being in a focus group entails. Compared to individual interviews, which aim to obtain individual attitudes, beliefs and feelings, focus groups elicit a multiplicity of views and emotional processes within a group context. The individual interview is easier for the researcher to control than a focus group in which participants may take the initiative. Compared to observation, a focus group enables the researcher to gain a larger amount of information in a shorter period of time (Morgan & Kreuger, 1993). Observational methods tend to depend on waiting for things to happen, whereas the researcher follows an interview guide in a focus group. In this sense, focus groups are not natural but organised events. Morgan and Kreuger (1993) suggest that the method is particularly useful when there are power differences between the participants and decision-makers or professionals, when the everyday use of language and culture of particular groups is of interest, and when one wants to explore the degree of consensus on a given topic.

Ethical issues According to Homan (1991), ethical considerations for focus groups are the same as for most other methods of social research. For example, when selecting and involving participants, researchers must ensure that full information about the purpose and uses of participants’ contributions is given. Being honest and keeping participants informed about the expectations of the group and topic, and not pressuring participants to speak is ethical practice. At the outset, moderators will need to clarify that each participant’s contributions will be shared with the others in the group as well as with the moderator. Participants need to be encouraged to keep confidential what they hear during the meeting and researchers have the responsibility to anonymise data from the group.

The Research Study

a) Selecting participants Miles and Huberman (1994) explain that most focus groups rely on purposive sampling with researchers selecting participants on the project and on the potential contributions of participants. Alternatively, participants can be randomly selected from a larger group that should be able to give insight into the topic. For example, if someone wanted to know more about a particular religious congregation purposive sampling, such as obtaining a church membership listing and randomly selecting parishioners to participate, would be an efficacious approach (Patton, 1990). This action research project has a wider population of 240 students in Year Seven (2013-2014) but the focus group interviews included 10 participants selected anonymously; one child from each of ten mathematics sets was selected to form two focus groups of five. The members of a focus group were invited because they are known to have experience from a particular context which in this case was secondary mathematics classrooms.

b) Structure Researchers such as Kitzinger and Barbour (1990), Lindlof (1995), Kreuger (1998), Green and Hart (1999) and Brown (1999) disagree about the practicable number of participants for a successful focus group. Many experienced moderators prefer a group ranging from eight to twelve suggesting further that the group should consist of four to twelve if the group is homogeneous and six to twelve if heterogeneous. A balance between the need to have sufficient participants for a lively discussion and the unwieldy milieu of a large group is the goal of the researcher.

c) The role of the moderator The moderator’s management of the focus group can determine the success or otherwise of the method regardless of the context. Morgan (1998) describes the moderator as the person who has the task of leading the focus group. This leadership or management involves: a) setting the scene; b) explaining the purpose of the focus group; c) introducing participants to the topics for discussion; d) keeping the group on time; e) focused on the topics; f) encouraging participation from all the group members; and, g) ensuring that all the key issues are addressed (Morgan, 1998). It is useful to have a note-taker recording all discussions so the moderator can give all their attention to the group. The notion of conducting a focus group interview effectively includes an assumption that the interview will be facilitated. The moderator had assumed most of the practical roles concerned with the planning of the physical environment of the interview room and the organisation of equipment and refreshments. The moderator also took responsibility for the welcoming of participants on the day and therefore began the process of setting participants at their ease and opening up channels of communication. According to Kitzinger (1995:299) the moderator ‘leads’ the focus group, their role is only to keep the discussion on track and should not influence the opinions of the group, this has been referred to as “structured eavesdropping”. During the start of proceedings of the focus group, the moderator’s first question is critical in breaking the ice. After each participant has said something it becomes easier to make further contributions and feel that their opinion is valued. With the use of focus groups in this research, it was particularly important to avoid domination by any particular participant, making sure that everybody had their say and enabling some level of consistent data collection between focus groups. This study worked with young teenagers and it became apparent that the moderator’s role was to ensure that all children felt their ideas were valued and that no one child dominated the discussion. d) Running the Focus Groups Two focus groups were held as part of the initial data collection period of this study. I undertook the role of the moderator and my supervisor took the role of the assistant who was responsible for the audio recording of the event and note-taking during the discussions to capture non-verbal signals and nuances. There were five participants in each of the two focus groups which were conducted during the students’ regular mathematics lesson time in a quiet library space on their campus.

e) Conducting and Observing the Focus Group The moderator and assistant sought to provide a friendly introductory environment which was established as the students arrived at the library. The moderator introduced himself and his assistant. Thanks, were extended to the students for attending and the purpose of the meeting was explained. The conventions of the group discussions were outlined together with reassurances about guarantees of confidentiality. Any initial anxieties or questions about the proceedings were invited. Each participant was asked to introduce themselves before the questions were addressed. The focus groups went smoothly and generated a great deal of data within the allocated timeframes. Moderator intervention was mainly restricted to prompts, probes and moving the discussion on when a particular issue had been exhausted. An example of moving them on to the next topic was when one of the participants could not think of a time he had a ‘good learning experience’; I told him that I would give him time to think and that I would come back to him. He then told us about a ‘good learning experience’. There was no domination in the group. The assistant moderator contributed or intervened in the discussion and sat at the main table to support the clarity of the Student contributions and to witness the discussion. Once the formal proceedings were brought to a close, participants were once again thanked for their contributions to the focus group.

Ethics Focus group research raises a number of ethical issues. We were particularly concerned to ensure confidentiality in and after the discussions. To this end no questions probed for any personal or sensitive information. Anonymity remains paramount and pseudonym use ensures that participants cannot be identified in any publications. Data from tapes and transcripts of the interviews are retained by the researcher and all data stored on university computers.

Analysis of the data There are a variety of methods of analysing data for focus groups (see Johnson & Christensen, 2004). The audio recordings were transcribed and the data together with the notes were discussed with the researcher’s supervisor. The discussions on a number of topics revealed the high level of detail which focus groups can engender as a result of the group interaction. For example, in the first focus group discussion on the topic of ‘how maths should be taught at secondary school,’ this led to a lively debate in which the students were very open about their own views and experiences. An example of this was, all the participants in the focus group wanted to express their views that maths should be interactive, fun and hands-on. Any reservations that we had that students would be reluctant to open up in front of their peers on such a sensitive issue were not borne out. Although it might be expected that participants would be guarded concerning their knowledge around maths teaching, they revealed that they felt trapped and teachers are unable to relinquish textbook teaching. The participants were therefore particularly interested in hearing the experiences of their peers as they were all taught by different teachers. The students reflected upon the use of textbooks as the primary resource, in most lessons and they reflected on how they felt in using textbooks on a daily basis, four times a week. It is commonly assumed that textbooks (with accompanying teacher guides) are one of the main sources for the content covered and the pedagogical styles used in classrooms. It is not surprising, then, that considerable attention has focussed on textbooks, including the economic and political circumstances of their production (Apple, 1986 and 1992), their linguistic features (Castell et al, 1989) and their sociological features (Dowling, 1996). Students in this study were put in sets for mathematics during their first year in secondary school according to their results in national curriculum tests. Once in those sets, they followed the same national curriculum but from different starting points and with different endpoints in mind. Textbooks reflected this way of organising students so that in any year group, a particular textbook scheme might have different textbooks aimed at different sets of students. Teachers used textbooks regularly, and almost all that use in lesson times was for students to practice exercises selected by the teacher following from teacher explanation of a particular skill or technique. Listening to the students and their concerns regarding the use of textbooks and their need for more ‘hands-on’ activities, gave me sufficient information to enact change in the mathematics classrooms as part of the Action Research for this project.

Discussion The main purpose of focus group research is to draw upon respondents’ attitudes, feelings, beliefs, experiences and reactions in a way in which would not be feasible using other methods, for example, observation, one-to-one interviewing, or questionnaire surveys. Focus groups rely on interaction within the group based on topics that are supplied by the researcher (Morgan 1997: 12). Hence the key characteristic which distinguishes focus groups is the insight and data produced by the interaction between participants. This is to ensure that participants have a specific experience of or opinion about the topic under investigation; that an explicit interview guide is used; and that the subjective experiences of participants are explored in relation to predetermined research questions. An example is when the participants were asked ‘Which class or which teacher helped you?’ and one of the participants responded: Well, there’s, like, most of them, they’re really nice and supportive, but some lessons are, like, a bit, like, they help a few people that are, like, really struggling, but they never really help the rest of the people. Like, they focus on a few people and that’s about it. But that’s only a few. That’s, like, three classes or two. And another participant replied: Yes, most of my teachers are quite supportive in that way, but like A, some of them can be a bit focusing on some people and, like, thinking other people can do well, so it doesn’t mean they’re… It means they’re not struggling. So they, sort of, put you aside and, like, …, you’re fine, you can do it yourself, even when you’re struggling on the topic.’ Therefore, focus groups are particularly useful when there are power differences between the participants when the everyday use of language and culture of particular groups is of interest, and when one wants to explore the degree of consensus on a given topic (Morgan & Kreuger, 1993). An advantage of focus groups to clients, users, participants or consumers is that they can become a forum for change, both during the focus group meeting itself and afterwards. For example, in this research the participants were asked: ‘… to design how maths should be taught at Secondary School, how students can really learn well…’ and they replied: Like, maybe a bit more interactive lessons because, like, it’s really, like, when they’re, like, with their friends and they can learn with their friends, but then still be with someone that they hang around with and, like, and then, but still have, like… Sort of, sometimes it’s really like they could be really interactive because people… Like, textbooks, example, are a little bit boring and, like, put you off, like, you just, sort of, read that. Like, say the teachers maybe help a bit more because, like, they say I can’t really explain this to you. It’s a bit, like, it’s, kind of, annoying when they can’t really do that. Not in my test, but, like, just in general classwork. So yes. I think they should make lessons more interactive because we usually always do textbook work and when, like, you’re stuck in your seat and you’re stuck in a textbook it gets, like, really boring, so I’d like to every once in a while, like, have an interactive lesson. A focus group is a small-group discussion guided by a trained leader. It is used to learn more about opinions on a designated topic, and then to guide future action that can bring change to an organisation (Morgan & Kreuger 1993). A main advantage of this method derives from group relations evident in the sessions. Students were encouraged to explain, challenge and share their honest views on questions asked. Listening to the ideas, opinions and experiences of others demands that we interrogate our own beliefs, and this was evident in these interviews. The reason for a significant level of candidness may derive from a move in the influence of the relationship between the researcher and participants. In focus groups, the researcher is in a marginal position and participants are amongst their peer group. This seems to make participants more willing to discuss topics openly in their own language than they would in one-to-one research environments. Here the manner was particularly useful as a means of allowing students to express their views and experiences without hindrance by the constraints inherent in one-to-one discussions with a researcher. Thus, focus groups are an effective way of ascertaining detailed views and experiences on relevant mathematical questions asked. Although the groups were guided by an agenda, they were able to ‘snowball’ their views on issues, presenting a wider context for their own position. For example, views on motivations and learning often generated a wider discussion. The advantage of this method over face-to-face interviews is that each speaker provides a platform for another to contribute, rather than responding only to a predetermined list of questions. Here participants were prepared to add to or qualify what had been said previously, providing a much more complete picture of their mathematical world. Participants provided long, detailed narratives about their experiences in mathematics, which often revealed their views and motivations. However, the analysis of the data emerging from group interaction can also provide a rich understanding of how students learn mathematically and, on balance, I would argue that focus groups should be used more widely. From the experience so far, the method generated data of range, depth, specificity and personal context which can stand alone or complement other research methods. Ultimately, using focus groups with students in education can help close the ‘culture gap’ between researchers and the subject they seek to understand.

Conclusion This paper has examined the use of focus groups as a method of understanding teachers’ practice in secondary mathematics through the interrogation of students’ ideas, beliefs and experiences. The results illustrate that a series of ongoing focus groups should provide a valuable longitudinal viewpoint. Although focus groups seem to have been used seldom in educational research, the results of the groups reported here illustrate the contribution that this method can make as a method to provide a picture of the views and experiences of students in a secondary school.

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The future of mathematics education since COVID-19: humans-with-media or humans-with-non-living-things

Marcelo c. borba.

Mathematics Education Area, Department of Mathematics, São Paulo State University (UNESP), 24A Avenue, 1515, Bela Vista, Rio Claro, São Paulo, 13506-900 Brazil

The COVID-19 pandemic has changed the agenda of mathematics education. This change will be analyzed by looking at three trends in mathematics education: the use of digital technology, philosophy of mathematics education, and critical mathematics education. Digital technology became a trend in mathematics education in response to the arrival of a different kind of artifact to the mathematics classroom. It was thrust into the spotlight as the pandemic suddenly moved classrooms online around the world. Challenges specific to mathematics education in this context must be addressed. The link between the COVID-19 pandemic and digital technology in education also raises epistemological issues highlighted by philosophy of mathematics education and critical mathematics education. Using the notion that the basic unit of knowledge production throughout history is humans-with-media, I discuss how humans are connected to the virus, how it has laid bare social inequality, and how it will change the agendas of these three trends in mathematics education. I highlight the urgent need to study how mathematics education happens online for children when the home environment and inequalities in access to digital technologies assume such significant roles as classes move on-line. We need to understand the political role of agency of artifacts such as home in collectives of humans-with-media-things, and finally we need to learn how to implement curricula that address social inequalities. This discussion is intertwined with examples.

Introduction

It is not possible to predict the state of the COVID-19 crisis at the time this article reaches the reader. The effects of the pandemic, and the response to it, have been shocking—with lockdowns, masks, and respirators, etc.—and have left most people at a loss. Some “world leaders” say that the virus is “just a cold,” while others say we may take months or years to have things “back to normal.” There are even those who say that COVID-19 is just a test for a much more serious health crisis that may be still to come. What is certain is that throughout the world, things have changed dramatically and suddenly. The virus has hit all classes of society, though of course it has hit the poor harder. But what are the effects of the pandemic in mathematics education? One effect that was almost universal was a tendency to “go online”: shop online, meet friends online, and learn online.

We have moved online because COVID-19 is caused by an invisible virus; it has no cure; and, without a clear pattern, it can cause the death of one person in a few days and cause almost no symptoms in another. Moreover, one may be infected and transmitting but asymptomatic for several days and then become very ill all of a sudden. Though not all “leaders” have taken their advice, most experts and the World Health Organization (WHO) recommend social isolation as the main tool to control, slow down, and hopefully stop the pandemic. All of a sudden, teachers, professors, and educational managers at all levels were put under pressure to develop (mathematics) education online, as the virus can be transmitted through physical contact—both between humans and between humans and non-living things.

Since the beginning of the official history of the International Commission on Mathematical Instruction (ICMI) in 1908, only war has interrupted the International Meetings of Mathematics Education (ICME), according to Menghini et al. ( 2008 ). This year, the ICMI decided to suspend ICME-14 1 for a different reason: due to the risk of spreading the coronavirus, traveling and gathering in groups would be unsafe. Some would say that ICME-14 was suspended due to a different kind of war: instead of generals in the background, and soldiers in the field, ready to kill or die, we have the whole of humanity trying to fight this non-living being, a virus. It is debatable whether the war metaphor is appropriate or not for this health crisis, but terminology aside, the crisis can lead us to some reflection on mathematics education. This essay will raise some questions to the mathematics education community that were caused by this non-living-thing: the virus SARS-CoV-2, which causes COVID-19.

Engelbrecht et al. ( 2020 ) reported that they had to change the conclusion of their survey paper on digital technology in March to April of this year, as it occurred to the authors that the paper could become dated even sooner than other digital technology survey papers. In normal times, such papers become old because digital technology changes so fast, and we rarely even have the time to implement a given technology in the classroom before a new one comes up. However, at this point, everything may become outdated, because we cannot predict the evolution of the COVID-19 crisis, nor whether a new crisis will follow it. The authors decided to include discussion about COVID-19 in the introduction and conclusion of the paper. At the end of the paper, they write:

The question is, what has this [COVID-19] to do with mathematics education and digital technology? Besides the impact on conferences and on the transforming mathematics classroom we may have to ask broader questions: Digital technology intensified traveling and our way of living, so it is also partly responsible for the present crisis. Is it possible that the use of digital technology can generate a similar crisis in mathematics education? Conversely, if the crisis lasts for a long period, would digital technologies be able to provide alternative ways to implement mathematics education? There is not much research on online mathematics education for young children, but if the crisis lasts for a long time, are we going to implement it without sufficient research? If the current crisis is over soon, are we going to develop research on mathematics education for a possible “COVID-2X” crisis? In this paper, among others, we have anthropomorphized media, talking about agency. The notion of humans-with-media as the collective that produces knowledge, may synthesize it, as we discussed in this paper. The COVID-19 virus (SARS-CoV-2) is a non-living being: can we talk about the impact (agency) of COVID-19 on mathematics education and on the world? Engelbrecht et al. ( 2020 , p.838)

This paper will deal with the questions from this excerpt in the following sense: I will discuss how new trends of mathematics education may arise or change with the ongoing crisis, and I will draft responses to some of these questions. Trends in mathematics education can be understood as a response, an answer, to some problem, as suggested by D'Ambrosio and Borba ( 2010 ). A working group, or a conference on a given trend within mathematics education, emerges as a response to new demands. I will use the theoretical construct of humans-with-media to connect the COVID-19 crisis to three different trends: the use of digital technology, philosophy of mathematics education, and critical mathematics education. In the context of the trend of digital technology, I will discuss the possibilities and drawbacks of having more and more online education, as well as the new demand for this trend. In doing so, I will revisit the notion of humans-with-media and its perspective of collective knowledge production involving humans and non-human actors such as computers and SARS-CoV-2. This will put new issues on the agenda for philosophy of mathematics education, focusing on the agency of “things” and humans’ relation to this virus thing. Finally, I will give a brief history of the trend of critical mathematics education, and I will raise an agenda provoked by COVID-19 for these three trends in mathematics education. I believe that these discussions may be important for us to understand the moment we are living in, beyond mathematics education itself. They can also help to set an agenda of research and action in the classroom for those interested in these trends and their connection to the pandemic.

Digital technology and mathematics education

Taking into consideration the notion of trends, presented above, the trend that studies the link of mathematics education and “new technologies”—informatics, communication and information digital technology, and alike—has been present in conferences for more than 30 years. At ERME 2 and SBEM 3 (Borba, 2018 ), at ICMEs (Menghini et al., 2008 ), and at PME 4 , there are always working groups, discussion groups, and panels on the subject, because, as authors such as Jim Kaput ( 1991 ,  1992 ,  1998 ) have pointed out, we need to understand how to use computers in mathematics education. Borba et al. ( 2016 ) prepared a survey that was presented at ICME-13 and put forward four phases for the use of digital technology in mathematics education. The four phases themselves show the strength and the length of this movement, which has involved many researchers, teachers, and students.

The first two phases, symbolized, respectively, by Logo and by curriculum-topic software (e.g., Cabri-Géomèetre), are not so important for the discussion in this paper, as the Internet became the big star during the pandemic. The third phase of the use of digital technology was characterized by the emergence of the Internet and online courses. This phenomenon became important around the turn of the century, depending on the country. Some so-called developed countries saw the Internet become popular in the mid-1990s and in some other countries, like Brazil, very early this century. Brazil was one of the first countries to start online courses at the graduate level, at a time when other countries were very protective of their face-to-face education.

The current fourth phase is characterized by the arrival of fast Internet, which reshaped the possibilities of online education. As this phase has developed, Engelbrecht et al. ( 2020 ) have pointed out that different forms of blended learning are important, in particular for teacher education. The term “hybrid” has become more important to express the combination of face-to-face mathematics education and online education:

A wide array of media and technology is available to create new hybrid forms of teaching. The integration of technology enables educators to create learning experiences that actively and meaningfully pull students into course content. “This technology may form thinking collectives (Lévy, 1993 ) with teachers that can break the walls of the regular “cubic” classroom that is associated with lecturing.” (Engelbrecht et al., 2020 , p.838)

If we consider a trend as an effort to find answers to a given issue, COVID-19 has pushed forward the agenda of the digital technology trend in mathematics education. With the need for social isolation, it became necessary to offer education to children and undergraduates at home. In most of the world, the first semester of education in 2020 was suspended or went online. Many are now discussing different kinds of hybrid education as health conditions allow students and teachers to go back to school and universities. But although we have plenty of research on implementing education online on undergraduate education (Engelbrecht & Harding, 2002 , 2004 , 2005 ), this is not the case for education for children. In the survey articles mentioned above, and in conference working groups, hardly any research has been presented on online education for children. As this theme develops, (mathematics) education will have to deal with structural issues, such as the participation of parents or responsible others in education.

In Brazil, newspapers say that teachers are “going crazy” with demands from students coming from WhatsApp and other social networks, as students and parents in their home cannot deal with school tasks. Grading is another problem: can we grade students so young online? Is help from parents allowed? This type of question has not yet been researched. In Brazil, some research groups such as GPIMEM 5 are trying to document what is happening in some state systems as a first step for research and understanding of online education for children. In the state of São Paulo, a new app, CMSP 6 , was created in less than 30 days for 200 thousand teachers and 3.5 million students to somehow have access to education. The app operates in conjunction with two preexisting TV channels, one operated by the state and another by a consortium of universities (Paz, 2020 ).

Teachers and administrators were able to supervise students through the app to some degree, and students were having three classes a day instead of five, as the state is trying to implement education through other platforms as well (Secretaria de Educação do Estado de São Paulo (São Paulo State Department of Education)—SEED, 2020 ). But this was a very complex moment: teachers had to go online without enough time to be prepared, and at the same time, they had to deal with their regular problems: São Paulo is the richest state in Brazil but pays its teachers a terribly low salary compared to other professionals, as pointed out to me in an online interview with a teacher who preferred to stay anonymous. Underpaid teachers now have to deal with students 24 h a day, 7 days a week, which includes dealing with students’ “personal” problems—including problems associated with the chronic social inequality in Brazil. Teachers with low salaries are not likely to have the best mobile phones, laptops, or Internet plans. Teachers who may teach fifty 50-min classes a week may deal with hundreds of students. It is likely that such problems are occurring in other countries as well, as differences between the “haves” and “have-nots” exist throughout the world, and are amplified by COVID-19, as described by the historian Walter Scheidel (Canzian, 2020 ).

Crisis is also a chance for change: teachers who teach 50 classes per week will not have time to learn to use digital technology for teaching. With many states and city educational systems forced to go online because of the pandemic crisis, the argument to use technology is very strong. It is likely that we will have a lot of research associated with this new reality. For the purposes of this article, I was not able to collect data systematically, but informal reports from teachers suggest that the reality of teaching young teenagers and children online will have to be investigated. As mentioned before, there is hardly any research on online education associated with levels below high school, which can be verified in many survey papers related to the theme (Engelbrecht et al., 2020 ). But the focus cannot only be on teachers. How do children experience this version of home schooling? There are also many jokes on social networks about parents losing control as they become home-teachers at the same time as they had to implement the home-office, so the role of parents in online mathematics education may be another area for research. Involvement of parents in mathematics education has been a theme of some research, including involvement associated with the use of digital technology (Ford, 2015 ; Wilson, 2013 ). However, this was in informal or blended settings, such as festivals (Domingues, 2020 ). Now we have new challenges, including to report and discuss how online assessment was developed (or not developed). Inviting students to produce mathematical videos was a research project developed before the pandemic. Having students expressing mathematical knowledge with videos, or doing research with videos, was not a solid trend in the literature. However, video production may be an alternative for education during and after the pandemic. Instead of focusing on test results, we can have students producing videos online to express what they have learned in conditions such as the pandemic. Videos can be produced collectively, with help of parents, friends, and different media. Differences in resources, including degree of parental aid received, can be considered by teachers and school systems in a “non-ranking” type of assessment.

Production of digital mathematical videos by students and teachers is growing in Brazil (see Fig. ​ Fig.1 1 for an example), and with the onset of the pandemic, an online “library” with more than 600 videos ( https://www.festivalvideomat.com/ ) has been used as a resource for teachers and students in their classes and as inspiration for the kind of task students and teacher may produce. Moreover, issues that have been the subject of previous research may gain new life: in a recent review paper (Engelbrecht et al., 2020 ), it became clear that different technologies used in a class, from the blackboard to the most modern mobile phone, are not necessarily only mediators but also actors. This is an epistemological issue, and it is part of a trend that has been discussed within the psychology of mathematics education and the philosophy of mathematics education.

Mud Sea: Modelling and Mathematics Education. Source: https://www.youtube.com/watch?v=YpCteGqjxd0&list=PLiBUAR5Cdi63gZoTSrJ9qXeiZQEH2wFBL

Philosophy of mathematics education and agency in the notion of humans-with-media

“Why do we have education? What are the relations between education and society? How do we know?” These are the basic questions of philosophy of education. For more than 20 years, there have been working groups on the philosophy of mathematics education (Bicudo & Garnica, 2001 ). “How do we learn?” is connected to “How do we know?,” and thus questions regarding epistemology, the theory of knowing, have also been debated by psychology of mathematics education discussion groups. Both domains of research may be seen as trends, as they seek foundations for mathematics education, and they discuss how mathematics education is articulated in the classroom, the research that is developed about it, and its “return” to practical settings: settings, like the classroom, which for many months have been on hold by the coronavirus pandemic. Several authors have discussed classrooms and schools and the artifacts produced there. For example, Villarreal and Borba ( 2010 ) have shown how mathematics is produced by collectives of humans-with-artifacts throughout the history of mathematics.

D'Ambrosio and Borba ( 2010 ), besides conceptualizing a “trend” as a response to a given problem, have argued that trends are intertwined, using the metaphor of a tapestry. It is unsurprising, then, that the discussion about who is the agent of knowledge is discussed in more than one trend: in digital technology working groups and in philosophy of mathematics education and psychology of mathematics education discussion groups or conferences. Different mathematics education authors (e.g., Faggiano et al., 2017 ) have claimed that computers, for instance, have agency. Inspired by the work of Lévy ( 1993 ) and on the phenomenological approach that humans are “being-with-others,” the notion of humans-with-media has been developed over the course of many years. The notion of reciprocal modeling was the first step (Borba, 1993 ). My early work on this showed not only that different media shape humans (an idea shared with many) but also gave some empirical evidence of how humans shape technology, specifically a piece of software about functions. Being part of the design software team and a mathematics educator developing research, I could see this “collaboration” between, on the one hand, a piece of software—full of the ideas of a multidisciplinary team, presented at meetings of developers, mathematics educators, teachers, and so on—and, on the other hand, how high school students would interact with the software (and with me, a teacher-researcher). A high school student, for instance, was influenced by what I said and by the design of the piece of software Function Probe (Confrey, 1991 ), and he also shaped the piece of software in ways that were not predicted by the multidisciplinary team that had developed the software. This student did not use the commands the design team had created but used the size of the computer screen and other measuring artifacts to coordinate algebra and graphs. Borba and Villarreal ( 2005 ) synthesized how the notion of humans-with-media could be understood based on the work of Lévy ( 1993 ), Lave ( 1988 ), and Tikhomirov ( 1981 ). This led to the notion that knowing was not social solely in the sense that it involves more than one person, but that it also involves things.

The notion of humans-with-media was proposed to emphasize that production of knowledge is a result of a collective of humans and things. From Tikhomirov and Lave came the idea that knowing was goal oriented and that values were involved. Later, in Borba ( 2012 ), discussions about the values, emotions, and media involved in knowing mathematics with GeoGebra (or whatever software was available) were extended to the idea that media and technology themselves change notions of what humans are. Media are therefore constitutive not only of what we know but also of what we are. Kaptelinin and Nardi ( 2006 ) also analyzed the idea of extending agency to non-humans. These authors compared the capacities to produce effects, act, and fulfill intentions of different agents: things (natural), things (cultural), non-human living beings (natural), non-human living beings (cultural), and human beings as social entities.

Agency, therefore, should not be seen as binary, as either present or absent, but having different levels. I see this notion of agency as a “fuzzy” one, as in fuzzy mathematics, in which we may have degrees of agency. In such a mathematics, for instance, my jeans are not just blue or not (zero or one), but they are, for instance, 0.6 blue. Kaptelinin and Nardi ( 2006 ) suggest three dimensions of agency: based on necessity (action is taken based on biological and cultural reasons), delegated (things or living beings act as the perceived intentions that are delegated by other humans and things), and conditional (actions of things or people which result in unintended effects).

The notion of humans-with-media, which is consistent with a more complex view of agency, has been challenged, in many instances, by arguments that want to preserve the power of a human as the center of any action. In these views, intentionality and action come from somewhere that is not social. Much of mathematics education, cognitivist or not, is based on such a “one-knower” view. From such a perspective, the agent of knowing is a single person, or collective of humans, even though most researchers would recognize the influence of artifacts, environment, and social cultural factors.

The notion that both humans and non-humans have agency is part of an effort to model artifacts—in particular, pieces of software, hardware, and the Internet of Things (i.e., things that are connected to the Internet)—as the historical, social, and cultural factors in the collective that produces knowledge. It stresses a view that knowledge is produced (both from a philosophical and a psychological perspective) by humans-with-artifacts. With a perspective in which things have agency, artifacts are labeled media as they are thought to communicate. This argument was more easily applied for technologies of intelligence (Lévy, 1993 ): humans-with-graphing-calculators were easier to accept as having agency than humans-with-libraries or humans-with-classrooms.

Regardless of whether readers value online mathematics education or not, they may at some point use their memory of a regular classroom to claim that face-to-face interaction is fundamental to any learning that occurs in mathematics education. Alternatively, one may use the notion of a “distributed classroom”: an office for one student, the bedroom for another, and some kind of computer center for others. But everyone would recognize that classrooms are changing. We have described this as a classroom in movement (Borba et al., 2014 ).

What constitutes the unit of knowing is an endless, philosophical discussion: is it a single person? Is it social because it involves more than one person? Is it social because it has a goal and it involves humans and non-human actors? It is an endless discussion, like most philosophical discussions. However, it seems that the emergence of SARS-CoV-2 gives strength to one perspective on knowing because, according to authors such as Racaniello ( 2004 , p.1), “Viruses are not living things. Viruses are complicated assemblies of molecules, including proteins, nucleic acids, lipids, and carbohydrates, but on their own they can do nothing until they enter a living cell. Without cells, viruses would not be able to multiply. Therefore, viruses are not living things.” Yet despite being non-living, the virus has dramatically changed the way humans live. Viruses are closely connected to us: they cannot exist for long apart from living things, like humans, who have cells; the symptoms of COVID-19 arise under certain conditions when the virus is inside human cells. We can say that the virus has agency in the sense that it has changed the way we have to do things. This analogy helps us to understand how certain things are much more likely to happen if certain actors are present. To use the metaphor of the virus, software also needs humans to “survive.” Software, and later on the Internet, has changed the environment of educational settings, in a similar way to how SARS-CoV-2 has suddenly turned children’s bedrooms into classrooms.

Latour ( 2020a , b ), another inspiration for the notion of humans-with-media, presents his concern with the virus crisis in a way that relates to the discussion in this paper:

But there is another reason why the figure of the “war against the virus” is so unjustified: in the health crisis, it may be true that humans as a whole are “fighting” against viruses — even if they have no interest in us and go their way from throat to throat killing us without meaning to. The situation is tragically reversed in ecological change: this time, the pathogen whose terrible virulence has changed the living conditions of all the inhabitants of the planet is not the virus at all, it is humanity! But this does not apply to all humans, just those who make war on us without declaring war on us. For this war, the national state is as ill-prepared, as badly calibrated, as badly designed as possible because the battle fronts are multiple and cross each one of us. It is in this sense that the “general mobilization” against the virus does not prove in any way that we will be ready for the next one. It is not only the military that is always one war behind. (Latour, 2020a , b , para.8)

Latour, without saying so explicitly, foregrounds the agency of this virus: SARS-CoV-2 spreads through humans to survive and reproduce, and this action provokes reaction—agency—from humans. Of course, every comparison or metaphor has its limits. But the coronavirus has transformed our lives—we still do not know for how long—in a dramatic way. Computers—now represented by mobile phones, which are much more potent computers than the ones used at the end of the last century by the minority of students who had access to them—have changed the way we can experience mathematics, in particular the way we can “experiment” with mathematics. The Internet has become a community, an agent, and an artifact. Videos that are produced and shared by students with digital technology soon themselves become a part of new collectives of humans and media that are involved in producing knowledge. Souto and Borba (Souto & Borba, 2016 , 2018 ) have discussed how the notion of humans-with-media, which had its origins in activity theory (Tikhomirov, 1981 ), is now about to change the third generation of activity theory, breaking the rigidity of the triangles espoused by Engeström ( 2002 ) and Sannino and Engeström ( 2018 ) (Fig. ​ (Fig.2 2 ).

The structure of an activity system. Source: Sannino & Engeström, 2018

This version of the humans-with-media construction has been called system-of-humans-with-media (Souto & Borba, 2018 ) to emphasize even more the notion that the collective of humans and non-humans is goal oriented and embedded in a community that has rules (Fig. ​ (Fig.2). 2 ). Considering media as agent has made it possible to think of the rigid triangles of the third generation of activity theory as dancing triangles, or as a GIF, in which the Internet, for instance, could be jumping from the instrument corner to the subject corner and/or to the community corner. Such an animation can be found on the GPIMEM website, in order to overcome the limits of the printed text ( https://igce.rc.unesp.br/#!/pesquisa/gpimem---pesq-em-informatica-outras-midias-e-educacao-matematica/animacoes/triangulo-sannino--engestrom/ ).

It is hard to know, as mentioned before, where the developments of the current health crisis will take us, but it seems that thinking about agency of non-living things as discussed in this section will be part of it. Questioning what the definition of “living things” is may be another consequence, which, of course, goes beyond what has been called the psychology of mathematics education or philosophy of mathematics education. But it will be relevant to some questions that perhaps were put aside or never asked before, questions such as: What are the specific roles of spaces/artifacts such as the classroom, face-to-face environments made for the intense use of Internet in education, and the “online classroom?” If the pandemic lasts even longer, what do we really mean by “face-to-face?” What does it mean to discuss affection in mathematics education without physical contact (e.g., hand shaking, hugging, kissing the cheek), so important in many parts of the world? The whole discussion about humans-with-media may gain a new dimension, as suggested in this section, related to some of the basic questions of philosophy of (mathematics) education. The pandemic foregrounds the role of home and the role of different parents and different social conditions in collectives that construct knowledge, in activity systems that produce knowledge. The idea of seeing fuzzy agency in non-humans should be developed further to include not only good access to internet, but to housing, which is a site of brutal inequality in Brazil and elsewhere. This famous photo (Fig. ​ (Fig.3) 3 ) illustrates the extent of inequality in Brazil, which, from the educational point of view, suggests that different housing may have different agency in constructions of knowledge, in particular in situations such as the one we lived during the pandemic. Housing matters in knowledge construction. Trying to solve a mathematics problem in a crowded house in a slum is very different than doing so in a spacious, luxurious apartment with a veranda.

Social inequality. Source: Of “Com 1% do país concentrando 28% da renda, Brasil não tem como dar certo...” L. Sakamoto, 2020. Recovered from https://noticias.uol.com.br/colunas/leonardo-sakamoto/2020/12/15/com-1-do-pais-concentrando-28-da-renda-brasil-nao-tem-como-dar-certo.htm?fbclid=IwAR3cAed7k9bb4qhHWhi7uAtZVhgCLFz9J-yx1dPuoW5rAS1xqVfgey6YrOc

In this sense, SARS-CoV-2 has pushed homes into the center of a collective that produces knowledge. Once again, we ask all the basic questions of the philosophy of mathematics education and psychology of mathematics education. What is the role of mathematics education? What is the role of the different education of parents in mathematics education? What is the role of non-living things, such as viruses, pieces of software, and homes, in the way we know and learn mathematics? A question that may be more critical is: What is the role of mathematics education for resisting inequality in the world?

Critical mathematics education and coronavirus

The trend of critical mathematics education (CME) responds to the main problem of social inequality in (mathematics) education and struggles against the view that mathematics is a branch of science that is separate from social, cultural, and political issues. CME’s role in the community of mathematics education is to remind us all about social inequality and other types of inequalities. CME may be said to have been officially born in 1990, in a meeting at the Cornell University in the USA (Powell, 2012 ; Torisu, 2017 ). There, the Critical Mathematics Educators Group was founded, with several members 7 , focusing on the key phrase “social justice.” Powell ( 2012 ) reports on how at ICME 6, in Budapest, Hungary, there was a meeting of researchers and how after the Cornell meeting, the group began to meet regularly, starting at ICME 7, in Quebec, Canada.

Present at the Quebec meeting was Skovsmose ( 1994 ), who also wrote about the development of critical mathematics education in Europe. Skovsmose shows the connection of this branch of CME in Europe to the Frankfurt School of Critical Education, one of the main representatives of which was Adorno, whose main issue was seeking an education that would prevent Nazism from occurring again. Today, critical mathematics education is more than important, in a moment in which countries such as the USA, Brazil, and Italy have far-right or fascist leaders, who have praised some of the fascist leaders of the twentieth century.

In the Cornell meeting, issues of social inequality, the role of mathematics in society, the ideology of certainty, and research methodologies appropriate to CME were presented (Borba, 1991 ; Borba & Skovsmose, 1996 ; Skovsmose & Borba, 2004 ). Since the 1990s, in Africa, authors such as Paulus Gerdes, from Mozambique, developed curricula and research about African traditions in mathematics and how to incorporate them into mathematics education (Gerdes, 2010 ; Torisu, 2017 ).

Development of curricula and pedagogical perspectives that highlight social inequality, gender and racial inequity, and the ideology of certainty was the initial focus of CME. More recently, environmental issues, and issues that were treated in other trends (e.g., mathematics education to the deaf or the blind), were brought into the agenda of CME. In sum, CME is a trend that shows that education is not neutral: it can promote equality or inequality. There are indicators already from Forbes that social inequality is growing during this pandemic: the billionaires are becoming even richer (Gavioli, 2020 ). The owners of Facebook and Amazon are among them! There is no need to be a mathematician to understand that this concentration of wealth upward means that the rest of the people have less. The owners of tech companies stand to gain as people move more and more online: their companies run online social networks, run online shopping services, and store digital data in online systems worldwide.

As I have already illustrated, social inequality is also growing in schools. As most schools and universities suspend face-to-face classes and go online one way or another, the issue of access has been a barrier to some and a trampoline to even more social inequality. Some universities in Brazil even opted not to resume education online because of inequitable access; but of course, as the university is not the only source of knowledge, online education also may have caused more social inequality. Here is an example from (mathematics) education in Brazil of a Catholic school located on the outskirts of a midtown city in the state of Sao Paulo: the school does not charge tuition for students, as parents do not earn enough income to feed their families; violence is also part of the daily experiences of these children. Teachers are paid above average (considering Brazilian standards), and from interviews with them, it is easy to see their engagement in fighting social inequality. Classes were first suspended in mid-March 2020 and resumed online afterwards, at different moments of April, depending on the school. Two teachers, Luiz Felipe Trovão (mathematics educator) and Karla Cristina Stropa Goulart (science educator), who were asked to answer an open question about their experience with teaching during the pandemic, reported how hard it was to communicate with students. Most students did not have access to the Internet. When they had access, they did not have the money to buy credits to operate the Internet 8 . The school tried to overcome this problem by providing chips with credits or sending printed didactical material to the children. But with less interaction with teachers, and without an environment to study in poor homes, through no fault of the teachers or the school, very little mathematics education or science education occurred. Trovão said that it is almost impossible to teach geometry online without proper interaction: homes, Internet access, etc.

The billionaires are becoming even richer; the poor are having even more difficulty accessing mathematics education: this may foreground the need that children will have, after the pandemic, to understand what happened. Mathematics educators may have to explore some tough topics: exponential functions to explain the spread of the coronavirus and how the richest grew even richer. Mathematics will not be enough, but a new agenda will be generated. Freire’s ( 1968 ) work about the pedagogy of the oppressed will be even more important. Putting together the agenda for the three trends, one should consider, for example, the role that home, as a physical and emotional “thing,” has in the pandemic school. We have collectives of home-parents-internet-student-teacher as the minimal unit of the collective agent who produces knowledge. Home and parents, things and humans, have added more to social inequality and to discussions about how to use digital technology in mathematics education.

Humans-with-media, seen as an activity system, provides a dynamic epistemological view that we can use to understand the different social aspects (in the micro- and macrolevels) of the research of digital technology. Simultaneously, in acknowledging agency in a wide variety of things, not only computers, it will be possible to structurally show social inequality: homes equipped differently cannot be assessed the same way. Children will suffer even more injustice than they suffer in school, if differences in Internet access, the comfort of home, etc., are not considered in assessment and teaching. Research under this frame, in digital technology, critical mathematics education, assessment, ethnomathematics, and other trends, may help to bring light to more epistemological discussion that is not value-free.

The three trends interacting

During the pandemic, “Lives” have become a craze in Brazil: presentations streamed over the Internet by artists, educators, and others. First, artists began holding Live presentations to incentivize people to stay at home. Soon after, other types of workers, such as mathematics educators, started holding our own Live presentations. During this pandemic, I have given many Lives produced by collectives that included Geogebra, the Internet, my home, and various broadcast software. The discussions of the mathematics of the pandemic and the sigmoid curve and its derivative were used in possibly thirty Lives. Figure ​ Figure4 4 is a screenshot from a short video that shows this curve dynamically: https://igce.rc.unesp.br/#!/pesquisa/gpimem---pesq-em-informatica-outras-midias-e-educacao-matematica/animacoes/curva-epidemica-no-geogebra/ .

COVID-19 flattening curve. Source: https://www.youtube.com/watch?v=XYNMuaPm654&list=PLiBUAR5Cdi60qXjrzAVdhOgufWVuWTdl2&index=16

The derivative of the sigmoid was used to explain why it was both possible and important to “flatten the curve.” Different curves, with faster or slower growth, were associated to the roles of prevention, social status, and different kinds of homes. Examples of this type of “virtual classroom,” outside of the school/university context, illustrate how much the three trends analyzed in this paper can be powerfully intertwined. This calls for research to understand what kind of mathematics education is being experienced by those who synchronously or asynchronously viewed the Lives.

Discussion and conclusion

Most of mathematics education is supported by empirical papers. In the 1970s, most research was quantitative, and data was used to “prove” that a given method of teaching was better than another. Empirical data had the same role it plays to this day in a good part of what is considered science: there were control groups and experimental groups, and the methodology was based on (or reduced to) statistical treatment and conclusions. Later last century, and earlier this century, qualitative research has swung the pendulum in another direction. Qualitative research sees data as a voice, as a complement that should be added to other evidence in order to make (“prove”) a point (Borba et al., 2018 ). Truth was assumed to be explicitly contingent and subject to change long before the COVID-19 pandemic brought so many instabilities to our beliefs. As arguments grew apart from data, a wide set of reactions, including some from powerful funding agencies, emerged. For example, there were funding agencies that require quantitative data in a project. Now the notion of mixed methods is prevalent, even though it is not clear what the role of the data or the view of “truth” is in much of the research published.

Essays such as this paper serve the purpose of discussing ideas and presenting bases for research papers, so that we can know (in the different directions briefly presented above) about mathematics education, in the different epistemological positions that characterize our community. In this sense, this paper is a result of a reflection on how three trends could have their agendas transformed by SARS-CoV-2. Of course, other trends, such as ethnomathematics or early-grades mathematics education, will also be affected. The issues raised throughout this paper should be transformed by readers and should themselves become the objects of research. In this paper, I choose to deal with digital technology, philosophy of mathematics education, and critical mathematics education because the pandemic seems to have played a significant role in the changes of the agendas of these three trends. It seems important to raise new issues in these trends.

Digital technology is now a theme of concern (or research) for everyone (Engelbrecht et al., 2020a , b ). The amplification of the starkness of inequality under the pandemic cannot be ignored (except for those who believe that the Earth is flat and that hydroxychloroquine is a miracle cure for COVID-19), and the rise of the home office, associated with home schooling, confinement, and lockdown, may help many to think about philosophical issues regarding the role of “place” in knowing/learning and notions such as humans-with-media.

In the paragraphs above, I have pointed at my choices in identifying important trends. Why did I say “I” instead of “we,” which would refer to a collective of humans-with-media? It is a good question, and a tentative answer, in another domain of discussion (qualitative research and its influence in the classroom), was given in Borba et al. ( 2018 ). The authorship of a paper or a book may be individual, but it is a result of a collective endeavor of “endless” humans-with-media. This paper 9 has one author, but it involved the active participation of one doctoral student (Juliana Çar Stal), three teachers who lent me their speech (Karla Cristina Stropa Gourlart, Luiz Felipe Trovão, and one who wanted to remain anonymous), the reviewers, the editors of this special issue, members of the research group I belong to, the more than 100 members of the graduate program in mathematics education at UNESP 10 , Rio Claro, friends, the computer, the word processor, the home, the office, and, of course, the pandemic, COVID-19. We hope we can discuss this at the next ICME and that it does take place in 2021!

1 https://www.icme14.org/static/en/index.html

2 European Society for Research in Mathematics Education

3 Brazilian Society of Mathematics Education

4 Psychology of Mathematics Education Annual Meeting

5 Grupo de Pesquisa em Informática, outras Mídias e Educação Matemática [Group for research in informatics, other Media and Mathematic Education] - GPIMEM website: ttps://igce.rc.unesp.br/#!/gpimem

6 CMSP–Centro de Mídias da Educação de São Paulo. Recovered from https://centrodemidiasp.educacao.sp.gov.br/

7 Alan Bishop, Arthur Powell, Claudia Zaslavsky, David Henderson, Dorothy Buerk, Europe Sign, George Gheverghese Joseph, Kelly Gaddis, Marcelo Borba, Marilyn Frankenstein, Marty Hoffman, Munir Fasheh, Paul Ernest, and Sam Anderson

8 In Brazil, most people will not have unlimited access to Internet in their cell phone. Especially if you are poor, you typically buy credits for Internet and pay as you go.

9 The content of this article is partially financed by the research Grants by CNPq, 400590-2016-6 and 303326-2015.

10 São Paulo State University

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The Combination of Qualitative and Quantitative Research Methods in Mathematics Education: A “Mixed Methods” Study on the Development of the Professional Knowledge of Teachers

• First Online: 01 January 2014

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• Udo Kelle 6 &
• Nils Buchholtz 7  

Part of the book series: Advances in Mathematics Education ((AME))

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Research about education in mathematics is influenced by the ongoing dispute about qualitative and quantitative research methods. Especially in the domain of professional knowledge of teachers one can find a clear distinction between qualitative, interpretive studies on the one hand and large-scale quantitative assessment studies on the other hand. Thereby the question of how professional knowledge of teachers can be measured and whether the applied constructs are developed on a solid theoretical base is heavily debated. Most studies in this area limit themselves to the use of either qualitative or quantitative methods and data. In this chapter we discuss the limitations of such mono-method studies and we show how a combination of research methods within a “mixed methods design” can overcome these problems. Thereby we lay special emphasis on different possibilities a mixed methods approach offers for a mutual validation of both qualitative and quantitative findings. For this purpose, we draw on data and results coming from an empirical study about a teacher training program in mathematics, where quantitative data measuring the development of professional knowledge of student teachers were related to qualitative in-depth interviews about the training program.

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Beck and Maier distinguish slightly differently between the “normative” and the “interpretive paradigm” going back on Wilson ( 1970 ).

It should be clear from the preceding discussion that this is not so much a problem of quantitative research per se —it may occur if one strictly follows a hypothetico-deductive approach (which is for many reasons advisable if quantitative methods are applied) and if researchers lack empirically contentful hypotheses, workable theories and/or specific knowledge about the domain under study. The latter is often not so much the fault of uninformed researchers but a consequence of the fact that social action is often structured by culture-bound rules and “local knowledge”.

A methodological adjustment of the treatment groups by measures of treatment evaluation (e.g. propensity score matching) has been omitted so far as the use of elaborate statistical methods to determine treatment effects appeared disproportionate due to the small group sizes. Furthermore, the group differences in Abitur grades are not significant and the relationship of school-related pre-cognitions considering the attendance at Advanced or Basic course merely reflects the pre-cognitions of local convenience samples.

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Kelle, U., Buchholtz, N. (2015). The Combination of Qualitative and Quantitative Research Methods in Mathematics Education: A “Mixed Methods” Study on the Development of the Professional Knowledge of Teachers. In: Bikner-Ahsbahs, A., Knipping, C., Presmeg, N. (eds) Approaches to Qualitative Research in Mathematics Education. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9181-6_12

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