Greek-Cypriot elementary teachers’ epistemological beliefs about mathematics

Teaching and Teacher Education 70 (2018) 47e57

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Greek-Cypriot elementary teachers’ epistemological beliefs about mathematics* Constantinos Xenofontos University of Nicosia, 46 Makedonitissas Avenue, CY-2417 Nicosia, Cyprus

h i g h l i g h t s  Mathematics teachers' beliefs are culturally conditioned.  Epistemological beliefs about the nature of mathematics.  Elementary teachers in the Republic of Cyprus.  Qualitative collective case study.  Uncovers the cultural specificities of elementary teachers' epistemological beliefs about mathematics.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 May 2017 Received in revised form 29 October 2017 Accepted 7 November 2017 Available online 15 November 2017

Drawing on data from the Republic of Cyprus, this paper uncovers elementary teachers' epistemological beliefs about mathematics. Twenty-two experienced teachers were invited to individual semi-structured interviews. Thematic data-driven analyses identified three themes and eight sub-themes, which I discuss, taking their socio-cultural context into consideration. This study suggests that applying predetermined frameworks directly taken from the literature when examining teachers’ epistemological beliefs in mathematics can be problematic, as they might hinder other culturally specific beliefs from emerging. In closing, this paper presents some implications for the results on teacher education and professional development, as well as ideas for future research. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Epistemological beliefs Elementary teachers Mathematics Republic of Cyprus

For more than thirty years, mathematics teachers' beliefs and their relationship to instructional practices have been put under the microscope, based on the assumption that the latter constitute manifestations of what one believes about the nature, the teaching, and the learning of the subject (see, for example, Chapman, 2002; Ernest, 1989; Raymond, 1997; Skott, 2013; Thompson, 1984). In acknowledgement of the definitional inconsistencies (Pajares, €rner, 2002) regarding the notion of beliefs (a term often 1992; To used interchangeably with similar ones, like attitudes, conceptions, views, images, perceptions and so on), here I follow da Ponte (1994), who argues that “[b]eliefs are the incontrovertible personal 'truths' held by everyone, deriving from experience or from

fantasy, having a strong affective and evaluative component” (p. 199). As such, comment Aguirre and Speer (1999), mathematics teachers' beliefs shape practice and orient knowledge. Beliefs are typically nested in clusters called belief systems (Op't Eynde, De €rner, 2002) or as organising strucCorte, & Verschaffel, 2002; To tures, according to Thompson (1992). These systems are not necessarily logically structured, making it possible that an individual may hold beliefs that are incompatible or inconsistent (Andrews & Hatch, 2000; Richardson, 2003). I shall not engage here in a detailed account of the nature of mathematics-related beliefs in general, as this would go beyond the scope of this paper.1 Exploratory in nature, the purpose of this paper is to examine Greek-Cypriot elementary teachers' epistemological beliefs about mathematics. Epistemological beliefs, a

* This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. E-mail address: [email protected].

1 For more on this, see the work of authors like Ernest (1989), Mason (2004) and Pajares (1992).

1. Introduction

https://doi.org/10.1016/j.tate.2017.11.007 0742-051X/© 2017 Elsevier Ltd. All rights reserved.

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particular type of beliefs, are concerned with the nature and source of knowledge (Perry, 1970) and vary significantly across disciplines (Schommer-Aikins, Duell, & Barker, 2003). In mathematics education research, while the relationship between teachers' general beliefs and their instructional practices is seen as complex and cannot be described in terms of cause-and-effect (Beswick, 2005; Raymond, 1997; Skott, 2013; Thompson, 1984), mathematics teachers' epistemological beliefs have been found to have significant connections with teaching efficacy beliefs (Chrysostomou & Philippou, 2010; Ertekin, Dilmaç, Yazıcı, & Peker, 2010) and are seen as strong predictors of one's instructional choices (Beswick, 2012; Garegae, 2016; Leung, 2006). Comparable observations have been made from pupils' perspective, whose epistemological beliefs appear to have a great impact on their attainment in mathematics (Garofalo, 1994; Rastegar, Jahromi, Haghighi, & Akbari, 2010). It is, therefore, not surprising that many recent studies in the field of mathematics teacher education have been designed to help participants enhance their epistemological beliefs (i.e. Charalambous, Panaoura, & Philippou, 2009; SchommerAikins, Unruh, & Morphew, 2015; Viholainen, Asikainen, & Hirvonen, 2014; Zakaria & Musiran, 2010). The focus of this paper is on experienced in-service elementary teachers in the Republic of Cyprus and on their epistemological beliefs about mathematics. More specifically, the initial aim of the analysis process was to provide answers to the following research question: What epistemological beliefs about mathematics are held by experienced in-service elementary teachers in the Republic of Cyprus? After the completion of data analysis, this initial question was refined and broken down into three more specific questions: (1) According to teachers' beliefs, what is mathematics? (2) What are teachers’ beliefs about how mathematical knowledge can be verified and how we come to know it? (3) What other epistemological beliefs are held by the teachers that do not fall under any of the first two research questions? In the second question, “how we come to know it” refers to the process(es) of internalisation of mathematical knowledge in individuals and/or groups of people. An alternative phrasing would be “how do we know what we know”. A retrospective revision of questions is very common in qualitative studies, in which the ongoing process of posing, revisiting, and revising the initial questions has a central role in understanding and uncovering the perspectives of others (Agee, 2009; van der Vaart, van der Zouwen, & Dijkstra, 1995). Before I discuss the findings of this study, I will turn to the wider international literature and consider both the philosophical question about the nature of mathematics and the research on mathematics teachers’ epistemological beliefs, with an emphasis on how various models categorise such beliefs. These theoretical considerations are important for positioning my work in the international research dialogue, through the presentation of a Greek-Cypriot perspective. Subsequently, the findings of this study are discussed, and some implications and suggestions for future research are provided. 2. Theoretical considerations Epistemological questions about mathematics are expressed and examined by both professional mathematicians and philosophers, with the former typically engaged in foundational research, while the latter contribute to the philosophy of mathematics (Horsten, 2016). Such enquiries are not new. In fact, the first discussions in this area date back to the fourth century BC and are attributed to Plato and Aristotle (Dossey, 1992). For Plato, mathematical objects had an existence of their own, where clear lines distinguished the ideas of the mind and their representations, as perceived by the senses. On the contrary, Aristotle, a student of

Plato, perceived mathematical ideas as idealisations performed by the mathematician, as a result of experience with objects. These two schools of thought later became associated with absolutist and fallibilist standpoints, respectively (Ernest, 1991). On the one hand, absolutism perceives mathematical knowledge as a body of knowledge that contains absolute, certain, and unchallengeable truths; and, fallibilism, on the other, sees mathematical knowledge and truth as corrigible. The “absolutism vs fallibilism” debate was intensified with the work of contemporary philosophers of mathematics, like Imre Lakatos (1976) and Thomas Tymoczko (1986), both members of the fallibilist school. As a consequence, more elaborate paradigms emerged (i.e. namely logisism, symbolism, intuitionism, quasi-empiricism, and so on) to secure the foundations of mathematics (Handal, 2003). Subsequently, alternative paradigms have been proposed, as, for example Ernest’s (1991) ~ ez's (2000) theory of social constructivism and Lakoff's and Nún embodied mathematics. Further discussion on various paradigms, however, would fall outside the remit of this paper. It is important to note from the above that each paradigm has a different position on what mathematics is and where it comes from, and that even in the circles of professional mathematicians, various epistemological opinions are held (Ernest, 1999). As a result, this diverse mixture of conceptions about the nature of mathematics has influenced the ways researchers in mathematics education, school teachers, and the general public see the teaching and learning of mathematics (Dossey, 1992). Yet, many scholars highlight how there are notable differences between professional mathematicians' and school teachers' (both primary and secondary) ways of working with € rfler, 2003; mathematics (i.e. Beswick, 2012; Boaler, 2008; Do Ernest, 1991). As Boaler (2008) points out, “[t]he erroneous thinking behind many school approaches is that students should spend years being drilled in a set of methods that they can use later” (p. 31), while for professional mathematicians, “[m]athematics is all about illuminating relationships such as those found in shapes and in nature” and constitutes “a powerful way of expressing relationships and ideas in numerical, graphical, symbolic, verbal and pictorial forms” (p. 19). Such disparities could be attributed to the different ways epistemological beliefs and practices developed by these two communities of practitioners (professional mathematicians and school teachers), as well as to the curricular emphases on procedural or conceptual knowledge, as set by policy makers (Garegae, 2016). The literature on mathematics teachers' epistemological beliefs is quite extensive and, even though there is no standard taxonomy (Mura, 1993), a variety of empirically grounded frameworks are proposed by colleagues. The first and most utilised frameworks are those of Ernest (1989, 1991) and Lerman (1990). Ernest describes three types of epistemological beliefs, namely the Platonist view, the instrumentalist view, and the problem solving view. From the Platonist view, mathematics is seen as a static but unified body of knowledge, whose structure and interconnections between various topics are fundamentally important. The instrumentalist view holds that mathematics is a useful and essentially unrelated collection of facts, rules and skills. Finally, the problem solving view sees mathematics as a dynamic and creative human invention and a problem-driven field of enquiry. Similarly, Lerman (1990) describes a bipolar framework, with apparent similarities to that of Ernest. On the one hand, there is the fallibilist paradigm, whereby teachers offer learners problem solving opportunities, through which the processes of mathematics might be developed. In this paradigm, which is related to Ernest's problem solving view, the social construction of mathematical knowledge is acknowledged. On the other hand, Lerman talks about the absolutist paradigm. Similar to Ernest's Platonist view, this paradigm sees mathematics as “an immutable body of knowledge, where creativity occurs only at the

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cutting edge of research, and teachers transmit their knowledge in the well-defined and replicable ways that their teachers used before them” (Andrews & Hatch, 1999, p. 206). Drawing on Ernest and Lerman, other researchers propose refined frameworks that emerge from praxis, as, for example, Andrews and Hatch (1999, 2000) and their study on English and Hungarian secondary teachers, and Goos (2006), who reflects on her own experiences as a mathematics teacher educator, and on the epistemological beliefs of her Australian student-teachers. These two sets of categories are described briefly in Table 1. In turn, Beswick (2012) uses Ernest's three views of mathematics and examines relationships between secondary teachers' beliefs about the nature of school mathematics and mathematicians' mathematics. As she concludes, “beliefs about mathematics can usefully be considered in terms of a matrix that accommodates the possibility of differing views of school mathematics and the discipline” (p. 127), as it is possible for a teacher to hold, for instance, Platonist beliefs about mathematics as a scientific discipline, while, have beliefs about the nature of school mathematics which are problem-solving oriented. The results of many comparative studies have established that, generally, mathematics teachers' beliefs, both of pre-service (i.e. Applebaum, Freiman, & Leikin, 2011; Xenofontos, 2014; Xenofontos & Andrews, 2012, 2014) and of experienced in-service (i.e. Andrews, 2007; Correa, Perry, Sims, Miller, & Fang, 2008; Kardanova, Ponomaryova, Safuanov, & Osin, 2014) are culturally conditioned. This means that various contextual factors, such as educational policies, the structure of the educational system, societal values and aspirations about education in general, and mathematics education in particular, historical and political issues, and so on (Wong, Taha, & Veloo, 2001) have a strong impact on teachers' beliefs to such an extent that more differences can be observed across educational systems, countries, and cultures than within the country addressed each time. Simply put, it is more likely that a teacher holds similar beliefs about mathematics, its teaching and learning with colleagues in the same country than with teachers in other countries. Research findings suggest that teachers' epistemological beliefs about mathematics are also

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culturally located. Andrews and Hatch (2000), who compared English and Hungarian secondary teachers' beliefs and identified both similarities and differences, argue that “the differences, rather than the similarities, are more likely to account for differences in pupil attainment” (p. 58e59) in international studies like TIMSS. More recently, drawing on data from IEA's Teacher Education and Development Study in Mathematics (TEDS-M), Felbrich, Kaiser, and Schmotz (2012) observe significant differences among future elementary teachers' epistemological beliefs across the 15 participating countries, which they associate with the ideas of Hofstede (1986) regarding collectivistically and individualistically oriented nations. For Hofstede, in individualistic societies, individuals are expected to take care of only themselves and their immediate families, while in collectivist societies, individuals expect their relatives or members of a particular in-group to look after them in exchange for unquestioning loyalty. As concluded by Felbrich et al. (2012), prospective teachers from individualistically oriented nations tend to have more dynamic beliefs about the nature of mathematics, whereas their peers in collectivistic countries appear to hold more static beliefs (see discussion above about Ernest, 1989, 1991; Lerman, 1990). 3. This study 3.1. Some information about the context of this study Since this study adopts a position regarding the impact of sociocultural factors on the formation of mathematics teachers' (epistemological) beliefs, providing some general information about the context of the study is necessary. The educational system of the Republic of Cyprus is highly centralised (Charalambous, Delaney, Hsu, & Mesa, 2010), in which schools are considered segments of the domain of the government, and not of the community. For this reason, the national curriculum and mandated textbooks of mathematics (Campbell & Kyriakides, 2000; Xenofontos & Papadopoulos, 2015) are prepared by the Ministry of Education and Culture, and are followed by all state schools. In

Table 1 Various epistemological beliefs about mathematics held by teachers, as described by Andrews and Hatch (1999, 2000) and Goos (2006). Andrews and Hatch (1999, 2000) Mathematics as a personal economic tool: Mathematics might facilitate a person's managing his or her household accounts. Mathematics as a diverse and pleasurable activity: People engage with mathematics and, more importantly, its processes in various ways and, in so doing, gain pleasure. Mathematics as an essential life-tool: An understanding of mathematics allows people to comprehend the world and make informed decisions. Mathematics as a service provider: Mathematics is essential for other areas of human activity such as science, commerce, industry and technology and, as such, was seen as a collection of results to be used rather than understood. Mathematics as a curricular determination: Mathematics is the prerogative of teachers rather than governments. Goos (2006) Mathematics as number: Mathematics is about understanding and working with numbers and symbols, or rules that apply to numbers. Mathematics as a tool: Mathematics is important in our everyday and working lives, a skill needed for many jobs: the base for science, accounting, engineering, and provides the fundamentals for everyday living. Mathematics as logical thinking Mathematics involves learning how to think logically and is a way to think and solve real life problems. Mathematics as a language: Mathematics is a creative language or a language of numbers and symbols that provides a means of communication. Mathematics as patterns: Mathematics is the science of patterns and relationships or is about numbers and patterns. Mathematics as a way of interpreting the world: Mathematics is a daily life experience (you see it everywhere) or a way of describing everything in life. Mathematics as beauty: Mathematics is inescapable, fun, exciting, awe-inspiring.

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general, the educational system of the Republic retains within its traditions a colonial legacy of the island's past as a former British colony (Xenofontos & Andrews, 2014); although the national curriculum adopted until 2010 was politicised (Koutselini-Ioannidou, 1997) with strong national(ist) elements, due to the historical and cultural connections of the Greek-Cypriot community with Greece (Xenofontos & Papadopoulos, 2015). Since 2010, the educational system of the Republic has been undergoing major curricular reform of all school subjects and levels (from kindergarten to upper secondary education). Acknowledging the national(ist) dimensions of the previous curriculum, the new curriculum sets the goal of creating democratic and humane schools (MoEC, 2010), an objective intended to be promoted through all school subjects, and whose results remain to be seen. Also, due to immediate employment, relatively high salaries, and the social status attached to the profession (Papanastasiou & Papanastasiou, 1997), in previous years, teacher education programmes and subsequently the elementary teaching profession, attracted high-school graduates with the highest GPAs (this situation has changed in recent years). To enrol in a state-funded initial teacher education programme at the Cyprus Pedagogical Academy (later evolved into the Department of Education at the University of Cyprus) or at the pedagogical departments of any university in Greece, high-school students have to succeed in highly competitive university entry exams organised by the Cyprus Ministry of Education and Culture. Taking an exam in mathematics, however, was not compulsory at the time the participants of this study had begun their undergraduate studies (see Charalambous et al., 2009). While an exam in the Modern Greek language and literature was compulsory, for the rest of the examination subjects one could choose to be examined in the science subject area (i.e. mathematics, physics, chemistry), the humanities subject area (i.e. history, ancient Greek, Latin), or the economics area (i.e. mathematics, political economy, accounting). These three areas correspond to the main study branches of upper-secondary education in the Republic of Cyprus (that is, the science branch, the humanities branch, and the economics branch). In other words, high-school graduates from the science or economics area had taken advanced mathematics before entering their undergraduate teacher education programmes. This system was in effect until the academic year 1998e1999, later replaced by a system of selecting individual school subjects (i.e. mathematics, history, physical education) rather than branches (clusters of subjects). 3.2. Methodological considerations Typically, for the examination of mathematics teachers’ epistemological beliefs, factor analytic techniques are employed, aiming at exploring or confirming factors based on predetermined items (see, for example, Amirali & Halai, 2010; Andrews & Hatch, 1999, 2000; Charalambous et al., 2009; Chrysostomou & Philippou, 2010; Schommer-Aikins, Duell, & Hutter, 2005). While important in making generalisations to the wider population, such approaches (like all approaches, of course) carry limitations. An important limitation is what Andrews and Hatch (1999) themselves indicate in their work: namely, the results are largely determined by the design of the questionnaire. In other words, the selection of items always has important consequences on the obtained results (Fabrigar, Wegener, MacCallum, & Strahan, 1999), as other contextual factors might be lost or remain hidden. Also, unless the results from a dataset of a particular cultural origin (i.e. teachers from a single country) are compared and contrasted with datasets from other contexts, as in the case of Andrews and Hatch (2000) and Felbrich et al. (2012), it is likely that researchers will fall into the trap of assuming universality, applicability, and generalisability

of the results to other contexts, especially since mathematics as a scientific discipline is largely, yet erroneously, considered acultural or pancultural (Barton, 1996). A few papers in this area follow individual case study methodologies and examine the epistemological beliefs of a small number of teachers. Thompson’s (1984) seminal work on three in-service teachers, for instance, is one of the most cited papers in the published literature. A similar comparison is a more recent example of Beswick (2012), a study investigating the epistemological beliefs of two secondary mathematics teachers, one experienced and one relatively new to the teaching profession. While important in unveiling deep understandings of (epistemological) beliefs and their enactment in classrooms, studies of this type, however, “lack generalisability, unless viewed as elements of a critical mass of similar studies” (Xenofontos & Andrews, 2012, p. 74). The approach followed here falls between the two camps described above. This project employs a collective case study methodology (Goddard & Foster, 2002; Yin, 2009), an approach that “involves more than one case, which may or may not be physically collocated with other cases” (Goddard, 2010, p. 164). In collective case studies, “cases are chosen because it is believed that understanding them will lead to better understanding, perhaps better theorizing about a still larger collection of cases” (Stake, 2005, p. 446). Furthermore, to indicate the importance of taking the individual's voice into account, participants were encouraged to share personal stories as elementary mathematics teachers within the spirit of narrative research. Privileging the individual's voice, narrative research strives to understand the ways in which informants construct stories to make sense of their professional world (Swidler, 2000) and intends, among other things, to establish honesty and trust between the researcher and participants (Connelly & Clandinin, 1990; Litchman, 2013). 3.3. Participants, data collection and analysis The participants of this study were 22 in-service elementary teachers in the Republic of Cyprus. All together they comprise the collective case of this project (Goddard & Foster, 2002; Stake, 2005; Yin, 2009). The gender ratio (18 women to 4 men) of the sample is consistent with the gender ratio of the general elementary teacher population of the Republic.2 Participants were recruited by following a snowball sampling approach (Goodman, 1961; Noy, 2008), which means the first two participants, who were my direct contacts, recommended several other colleagues, who, in their turn, referred the rest. Furthermore, they can all be labelled as typical teachers. The question of how many interviews are required in qualitative research in order to achieve data saturation has no standard answer (Baker & Edwards, 2012). Fusch and Ness (2015), however, suggest that data saturation is reached when there is enough information to replicate the study, when the ability to obtain additional new information has been attained, and when further coding is no longer feasible. In this study, up until the eleventh interview, I could identify new issues in the participants’ responses. Yet, from the twelfth until the fifteenth interview, no new issues emerged. It was, therefore, my professional decision to conduct double the number of interviews from the point at which new issues stopped occurring. The number 22 was thought to be satisfactory, especially when compared to studies like those of Guest, Bunce, and Johnson

2 According to the United Nations' Encyclopedia of the Nations, data from 2008 show that 82.39% of the elementary teacher population of the country corresponds to female teachers. For more information, see http://www.nationsencyclopedia. com/WorldStats/Gender-primary-education-teachers.html.

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Table 2 The participants. Pseudonym

Gender

Years of teaching experience

Advanced mathematics in high-school

Anna Antonis Athina Despina Electra Elena Evangelia Flora Georgia Julia Katerina Lamprini Loukia Maria Marilena Nikolas Pavlos Savina Stella Tasoula Vasia Yiannis

Female Male Female Female Female Female Female Female Female Female Female Female Female Female Female Male Male Female Female Female Female Male

10 24 8 14 12 8 16 9 23 14 23 14 11 16 18 12 10 15 27 16 22 15

No Yes Yes No Yes Yes Yes No Yes No Yes Yes No Yes No Yes Yes No No No No Yes

(2006), who observed saturation within the first twelve interviews. Below Table 2 presents biographical information about the participants, their given pseudonyms, gender, years of teaching experience, and whether they had taken advanced mathematics in highschool, which also means they entered their initial teacher education programme having taken (or not) an exam in mathematics (Charalambous et al., 2009). The only criteria of selecting the teachers were (a) their willingness to participate and (b) that each of them had at least eight years of teaching experience. At the time the data collection was conducted, the choice of 8 years as a minimum required experience, verified that participants belonged to the generation of Greek-Cypriot teachers considered the “best” high-school achievers and who received their initial teacher education at either the highly regarded Cyprus Pedagogical Academy, the Department of Education at the University of Cyprus, or at the pedagogical department of any Greek state university.

This paper draws on data from a larger project based on detailed semi-structured interviews concerning participants' mathematicsrelated beliefs. In particular, an adapted version of Ernest’s (1989) triadic model was utilised (beliefs about the nature of mathematics, beliefs about mathematics teaching, beliefs about mathematics learning), with an explicit addition of a self-efficacy dimension. As such, the interview protocol included questions around three areas/issues: (1) the nature of mathematics (epistemological beliefs), (2) school mathematics (beliefs about curriculum, teaching, and learning), and (3) self-efficacy beliefs (the self as a learner and the self as a teacher). Below, Table 3 illustrates examples from the interview protocol in regards to these three areas/ issues. Readers must keep in mind that, as the interviews were semi-structured, not all questions were necessarily posed to all participants. Rather, the aim was to cover the three main areas/issues with each participant, and to do this, the protocol provided the

Table 3 Sample questions from the semi-structured interview protocol. Area/issue

Sub-issue

Epistemological beliefs

Nature of mathematics   How we know what we  know   The curriculum  

Beliefs about school mathematics

Teaching

Learning Self-efficacy beliefs

The self as learner

Examples of questions

        

The self as teacher

   

What is mathematics? How do you understand this concept? What is a source of mathematical knowledge? Where does mathematical knowledge come from? How do we know what we know in mathematics? How does mathematical knowledge come to us? How can we be sure about what we know in mathematics? Why is mathematics a school subject? If you could change anything in the mathematics curriculum (add, remove, redesign), what would you change and why? Is there a personal story you would like to share in relation to the mathematics curriculum? What is the teacher's role in the mathematics classroom? What strategies do you use to help your pupils learn mathematics? Are there any factors that prohibit teachers from helping children in mathematics? If yes, what are these? Is there a personal story you would like to share in relation to mathematics teaching? In which ways do children best learn mathematics? What is the role of the pupil in the mathematics classroom? How do you see yourself as a mathematics learner? Can you give some examples from your experience (as a pupil, a university student, etc.)? How do you feel when you have to solve non-routine mathematical problems? How competent would you say you are? How do you manage difficulties you may encounter during non-routine problem solving? How comfortable do you feel with teaching mathematics? How competent do you feel in helping children learn mathematics? What are your strong and weak points as a mathematics teacher?

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researcher with the security of having some backup questions in case the discussion got stuck or lost its flow. Although the analyses and findings of this paper are largely based on participants' responses to questions from the first area/issue (that is, epistemological beliefs), the whole protocol is presented here because, as explained below, teachers' responses to questions from other areas/ issues alluded to their own epistemological beliefs. Therefore, I'd like readers to have a more complete picture of sample questions related to all areas/issues covered in the interviews. The interviews were conducted individually and in Greek, lasted on average for 40e45 min, and were held at nonworking time and at the places of each participant's choice. A thematic, data-driven, analysis was employed (Boyatzis, 1998), as no predetermined and specific coding scheme was utilised. Of course, I do acknowledge that it’s highly unlikely that researchers distance themselves from theory when performing data-driven analyses. Even at an unconscious level, ideas from the reviewing of the literature are always present when data are left to, in a way, speak for themselves. Following the ideas of coding and categorisation (Kvale & Brinkmann, 2009; Miles & Huberman, 1994), and exploiting the constant comparison process outlined by Strauss and Corbin (1998), several categories (sub-themes) were identified, which were later clustered under more general themes. The findings presented in this paper are concerned with participants' epistemological beliefs. However, they did not only emerge from questions explicitly inviting participants to talk about their epistemological beliefs. Teachers' epistemological beliefs were inferred from participants' responses to questions about school mathematics and/or themselves. A relative example is presented below in an extract taken from an interview with Evangelia. The researcher asked the teacher a question with an explicit focus on her mathematical self-efficacy, as a learner. Evangelia's response alluded to her epistemological beliefs, as well.

teachers, mathematics is seen as a humanly invented tool that emerges from a need of humankind to understand the world, an idea captured by Ernest's (1991) social constructivism. Julia, for example, argues that mathematics “emerge from our needs to adapt to society, to survive, to go out there and have perceptions of the world”. In Yiannis' view, “the invention of mathematics is the job of experts, that is, professional mathematicians” who “try to describe their observations of the world, and in their attempt to do so, they create codes, symbols, they get involved in trials and errors, and when they prove something, that knowledge becomes acceptable, unless someone is there to question it and prove it wrong”. Beliefs about the fallibilist character of mathematics are very strong in the responses of these informants. Evangelia's quote below summarises the beliefs of the eight teachers: Mathematical knowledge emerges through studies and attempts of professional mathematicians or people before becoming professional mathematicians, who all want to solve problems of this world (…) Mathematical knowledge is still being invented. It's not a static thing. Who knows, tomorrow one could find ways of refuting an established mathematical formula, just because a new one that works more accurately is invented. Also, Evangelia makes reference to her responsibilities as a teacher and states that “mathematics constantly changes, and as teachers, we need to become updated all the time. I've been teaching for 16 years now; yet I cannot claim I know it all. I need to update my mathematical knowledge all the time”. While all eight teachers here share views about mathematics similar to what Elena calls “the tool for solving most everyday problems”, none of them makes reference that adhere to a view of

Researcher: So, if I got this right [note: from the discussion before], you feel quite competent as a mathematics learner, right? Evangelia: In order for someone to claim that he or she has an excellent relationship with mathematics, 10e20 years of working with maths are not enough. You have to spend lifetime with mathematics. If I had to assess my own mathematical competence, on a scale of 1e10, hm, I'd say around 6, because I really believe that I still have lots to learn. And this is because mathematics is not static. It constantly develops and changes all the time. You have to update yourself all the time. And I can't really do this from the position of an elementary school teacher.

4. Findings and discussion Data analyses identified eight sub-themes regarding participants' epistemological beliefs. It should be noted that these subthemes, with an exception of the first two, are not mutually exclusive, since a teacher may hold beliefs pertaining to more than one sub-theme. These sub-themes were later clustered under three more generic themes, which are concerned with participants’ beliefs about (a) what mathematics is, (b) how mathematical knowledge is verified and how we come to know it, and (3) other epistemological beliefs. Below, the three themes and their respective sub-themes are presented with the support of representative quotes by the teachers, and discussed in connection to the literature.

4.1. Teachers’ beliefs about what mathematics is 4.1.1. Mathematics as a humanly invented tool for understanding the world Eight of the participants share views pertaining to a dynamic, constantly evolving nature of mathematics, along the lines of the fallibilistic paradigm (Ernest, 1989; Lerman, 1990). For these

mathematics as a personal economic tool for managing household accounts (Andrews & Hatch, 1999, 2000). On the contrary, mathematics is seen as a way of interpreting the world (Goos, 2006) and as an essential life-tool (Andrews & Hatch, 1999, 2000), the understanding of which allows people to comprehend the world and make informed decisions. 4.1.2. Mathematics as a static body of knowledge that exists on its own Absolutist beliefs about the nature of mathematics are expressed by seven teachers, who claim that mathematics is a static body of knowledge that exists on its own, a belief consistent with Platonist perspectives (Dossey, 1992; Ernest, 1989; Lerman, 1990). A typical quote is that of Antonis, who argues that: [a] professional mathematician studies mathematics and what other mathematicians have discovered in the past, and if he's3

3 The choice of the masculine gender is Antonis'. The original quote is in Greek, which is a gendered language. It is common for Greek speakers to use the masculine and refer to both male and female. For example, “ο maqhtή2” (pronounced “o mathitis”), literally means “the male student”, and is often used to refer to both male and female by implication.

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charismatic, he will discover something new. But mathematics exists on its own. We just haven't discovered everything yet. It exists around us; the big question is how a mathematical idea that already exists will be perceived, and how a new theory will be developed. This is how maths is. Let me give you an example. Say, I want to travel at the speed of light or through time holes, mathematics allows these to happen; it's that we, humans, haven't discovered it yet. Interestingly, all these seven teachers hold a belief that mathematics is discovered because of “our everyday needs, like handling money” (Marilena) or “cooking, shopping, paying bills and so on” (Georgia). In other words, they perceive discovered mathematical knowledge as a personal economic tool (Andrews & Hatch, 1999, 2000), the practicality and applicability of which is apparent. 4.1.3. Mathematics as a way of thinking Nine of the teachers point out that mathematics is a way of thinking, and, as such, it helps in the general intellectual development of people. Of these nine teachers, three shared fallibilistic beliefs (see first sub-theme above), only one expressed Platonic beliefs (see second sub-theme above), while the five remaining participants did not take any position on whether mathematics is socially constructed or a static body of knowledge. Characteristically, Electra comments that “[m]athematics constitutes, I believe, the basis on which a person can develop other skills for coping with the needs of society, in general. It's a way of thinking. It's what we call intellect. One needs mathematics to be complete as a human being”. In a similar vein, Flora argues that school mathematics should not be just about counting or performing simple sums. Mathematics is a way of rational thinking. By developing this way of thinking, you're able to find solutions to problems in general, not only mathematical problems. Through mathematics, you learn how to put your thoughts in order. It helps in the development of the mind and in how to think rationally. Goos (2006) identifies a similar belief among Australian preservice teachers, who see mathematics as logical thinking, which consequently helps in the solving of real-life problems. This view of mathematics is expressed by scholars, Mason, Burton, and Stacey (1982), Tall (1991), and Harel and Sowder (2005), who write about what it means to think mathematically and how this can be developed. 4.1.4. Mathematics as a scientific discipline Seven participants explicitly refer to mathematics as “science” or “a scientific discipline”. A somehow similar belief is expressed by Australian pre-service teachers who see mathematics as the science of patterns and relationships (Goos, 2006). However, the responses of these six teachers here could be clustered in two further categories, corresponding to mathematics as either a pure or an applied €rd, 2005). discipline (Blum & Niss, 1991; Lingefja On the one hand, four of them talked about mathematics as a pure science with rigor, stringency, and structure, in a similar manner expressed by supporters of formalism (Dossey, 1992; Ernest, 1991; Handal, 2003). From a formalist perspective, mathematics constitutes a formal system in which the truths are represented by formal theorems. Along these lines, Loukia claims that “mathematics is a scientific discipline, a pure science that contains series of theorems and formulas”. Antonis agrees and states that “mathematics is definitely the superior science of all”. The other three teachers, however, perceive mathematics as an

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applied science that examines the applicability of mathematical ideas in general, and numbers in particular in solving everyday problems. This view resonates with what Goos (2006) calls mathematics as number. In this spirit, Marilena claims that “mathematics is a science that deals with numbers and calculations, which we need in our daily lives”, while Katerina talks about “a science that brings numbers into life so that we can find solutions to everyday situations”. 4.2. Beliefs on how mathematical knowledge is validated and how we come to know 4.2.1. The unquestionable validation of mathematical knowledge by experts Regardless of their perspectives on whether mathematics is discovered or invented, eleven teachers comment that the validation of mathematical knowledge is the job of experts, that is, professional mathematicians. Once knowledge is validated through proofs, they all claim, the general public accepts it as it is and does not question it. In a characteristic response, Yiannis states that: [p]rofessional mathematicians validate their work with proofs, research, and reflection. When they are confident about the validity of an idea, they present it to other mathematicians. Once the community of the mathematicians approves it, then the idea passes on to the general public. We accept these ideas as they are. It's not our job to check their validity. We are teachers, they are mathematicians. The idea of validating mathematical knowledge through proofs and logical steps appears to be compatible with the principles of logicism (Dossey, 1992; Ernest, 1991; Handal, 2003), the school of thought that regards pure mathematics as part of logic, and that all mathematical truths can be proved from the axioms and rules of inference of logic alone. Furthermore, the teachers' comments here €rfler (2003), who argues that mathematics is the are in line with Do product of the activity of mathematicians, whereas the mathematics education community does not perform mathematical studies but meta-studies of mathematics, in the way that musicology is the study of music as a given human phenomenon. All eleven participants argue that it is not their job as teachers of mathematics to question the mathematical knowledge since it is validated by professionals. On the contrary, they declare that they teach mathematics without challenging concepts, procedures, proofs and their validity. Maria, for instance, explains that in the teaching of mathematics “we have some rules and formulas, which we take for granted, since others have proven them”. Katerina agrees and asks, “what is there to be questioned in the mathematics we teach in elementary school? Nothing! So I don't bother questioning the mathematical knowledge. Ok, a secondary teacher can show some proofs to the pupils, but I don't think it's necessary”. In Hofstede’s (1986) terminology, this could imply a relatively high degree of power distance, in the sense that the “less powerful” (teachers) accept that “power” (validation of mathematical knowledge) is held by the “more powerful” (mathematicians). Indeed, mathematics itself seems to possess an enormous authority that is passed on to practitioners like teachers, who, in turn, pass it on to learners (Amit & Fried, 2005). Also, it is not surprising that the teachers here believe that proving is not necessary for younger learners. This is a common belief that, consequently, endorses the marginal position that reasoning and proving have in elementary classrooms (Stylianides, Stylianides, & Shilling-Traina, 2013), even though research evidence supports that elementary school children demonstrate skills related to mathematical proof (Flegas & Lemonidis, 2013).

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4.2.2. Mathematics arrives to us through formal schooling A large majority of the teachers' claims that learners acquire mathematical knowledge through formal schooling. In particular, of the 22 participants, 17 express comments regarding the role of formal education and schooling in the acquisition and development of one's mathematical competence and skills. Their responses in relation to this issue are highly homogeneous. For instance, Georgia reports how “[m]athematical knowledge comes to us through our formal education. Our system provides an organised mathematics education”. Flora draws on her personal experiences and argues that she learned mathematics “through school. There were things I didn't understand when I was studying pure maths, but that I understood in other school subjects like statistics or physics”. In Elena's words, “we learn mathematics at school. That's why we need good teachers. Because it's at school where we receive the first knowledge of mathematics. Later, we use mathematics outside school”. Interestingly, while some of the teachers talk about the applications of mathematics to everyday practical situations or mathematics as a tool for understanding the world, none of them mention anything about learning mathematics outside school. In general, participants appear to disregard the well-documented view that mathematics learning takes place, not only in the formal settings of schools but at home (Andrerson, 1998; Bachman & Dierking, 2011; Leder, 1992) and in other informal contexts (Greiffenhage & Sharrock, 2008; Pais, 2013). From a different point of view, teachers’ comments about schooling could be a reiteration of the value formal education has in Cypriot society. In fact, the Republic of Cyprus has one of the highest percentages of tertiary degree holders among the EU countries (European Commission, 2016). Teachers, here, appear to see themselves as part of a chain in a formal schooling system; hence, they perceive themselves as “responsible for helping children learn mathematics at school because this is the first place one learns mathematics” (Anna). 4.3. Other epistemological beliefs 4.3.1. The legacy of the ancient Greek mathematicians Seven of the teachers make strong claims about the origins of mathematics, which they associate with Ancient Greece and ancient Greek mathematicians. Athina states that “mathematical knowledge comes from antiquity, from ancient Greek mathematicians, like Thales, Euclid, and Pythagoras. If we study the history of sciences then we can trace the origins of mathematical knowledge to Ancient Greece”. Ancient Greek mathematicians, claims Stella, invented mathematical knowledge. They were mathematicians and philosophers at once. They studied the subject in a lot of depth. And that knowledge arrived to us through the centuries. We are just teachers. Our job is procedural, just to transmit this great knowledge to children. For thousands of years nobody has ever questioned the work of Ancient Greek mathematicians. They were great! Interestingly, three of the teachers who talk about the legacy of the ancient Greek mathematicians (Julia, Nikolas, and Stella) appear to hold beliefs about mathematics as a humanly invented tool for understanding the world (see section 4.1.1). It can, therefore, be assumed that these three teachers believe that mathematics was invented by ancient Greeks and can still be used nowadays for understanding the world. When these responses are seen alongside those of the other four teachers (Athina, Electra, Loukia, Marilena), all seven teachers’ beliefs appear to pertain to Planonist (Ernest, 1989, 1991) or absolutist (Lerman, 1990) perspectives about mathematical knowledge. Whether or not it is believed that

mathematics is invented or discovered by ancient Greek mathematicians, it is believed that knowledge arrived to contemporary times intact and unchanged. This is somehow worrying considering that approximately one in three teachers in this sample of 22 believes that mathematics comes from ancient Greece, that mathematical knowledge has not changed from antiquity until today, and that the contribution of other civilisations (i.e. the Babylonians, Indians, Chinese, and so on) to the discipline of mathematics is practically insignificant. The educational system of the Republic of Cyprus is, to a large extent, Hellenocentric (Koutselini-Ioannidou, 1997; Papadakis, 2008), and this can be seen, among other places, in the strong connections and similarities of the Greek and GreekCypriot secondary mathematics textbooks, and particularly, in the ways that the history of mathematics is presented (Xenofontos & Papadopoulos, 2015). 4.3.2. Applications of mathematics to other disciplines As argued by seven teachers, mathematics is strongly connected to other scientific disciplines; in fact, it constitutes the basis on which other fields are built. Similar observations are made by Andrews and Hatch (1999, 2000), who call such a belief “mathematics as a service provider”, and Goos (2006) who places views of this kind under a more general belief, that is “mathematics as a tool”. According to Nikolas, “if mathematics didn't exist, then none of the other scientific disciplines would exist. Physics, chemistry, engineering, all these sciences need mathematics”. Similarly, Athina comments that: [t]he relationship between mathematics and physics is, I think, obvious, because physics uses formulas, numbers, and measurements. All natural sciences need mathematics because they analyse natural phenomena on a theoretical level with theories that are, in essence, mathematical. But mathematics has applications to other disciplines, like informatics. Mathematics is a basic science with many applications to other, more practical scientific fields. Lambrini is the only teacher who talks about applications of mathematics to disciplines other than natural and applied sciences, like the humanities and social sciences. In her view, “accounting, economics, philosophy, psychology, sociology, music, all these disciplines use mathematics. The only fields that don't have any connections to mathematics are language and literature. Or maybe they have, but I don't see them”. Lambrini does not appear to be aware of the applications of mathematics in fields like syntax and computational linguistics (see, for example, Steedman, 2000; Wall, 1974), or the extensive work associating mathematics with fiction and literature for children, teenagers, and adults (see, for example, Koellner, Wallace, & Swackhamer, 2009; Padula, 2005; and, Schiro, 1997). 5. Concluding thoughts Having taken a sociocultural perspective and acknowledging that (mathematics) education in general and (mathematics) teachers' beliefs in particular are cultural products, this paper set to uncover the epistemological beliefs about mathematics held by Greek-Cypriot elementary teachers. In so doing, no predetermined model on mathematics teachers' epistemological beliefs was utilised. The work of scholars like Ernest (1989, 1991), Lerman (1990), Andrews and Hatch (1999, 2000), and Goos (2006), has been extremely valuable and helpful in analysing and interpreting the findings of this study. However, it was my deliberate choice not to follow a theory-driven analysis approach, which would simply allow me to examine whether the data from this study pertained to

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the dimensions of predetermined coding scheme. On the contrary, I wanted to investigate the cultural specificities of the participants' epistemological beliefs. The adoption of a data-driven approach was successful in the sense that it has let culturally specific beliefs come to the surface, as, for example, beliefs about the legacy of ancient Greek mathematicians. This belief is unique to the contextual particularities of this study and would not have emerged otherwise. This paper, therefore, concludes that the application of predetermined frameworks taken directly from the literature when examining teachers' (epistemological) beliefs in mathematics can be problematic, as they might hinder other culturally specific beliefs from emerging. The issue of the cultural conditioning of teachers' epistemological beliefs about mathematics is underresearched, as only few studies adopt an explicitly socio-cultural perspective that takes the geographical and/or cultural setting into consideration (see, for example, Andrews & Hatch, 2000;; Felbrich et al., 2012). To this end, this paper provides evidence supporting the idea that, since mathematics as a scientific discipline is neither acultural nor pancultural (Barton, 1996), teachers’ epistemological beliefs about mathematics are not and cannot be devoid of cultural matter. On a first level, none of the teachers in this study appear to hold both fallibilistic and absolutistic beliefs about the nature of mathematics. They either express beliefs that mathematics is a humanly invented tool for understanding the world (fallibilism, social constructivism, problem-solving orientation) or that it is a body of knowledge that has an existence of its own (absolutism, Platonism). This is, seemingly, contradictory to the findings of studies like those of Amirali and Halai (2010) in Pakistan, and Beswick (2012) in Australia,4 who have found that it is possible for teachers to hold absolutistic and fallibilistic beliefs at once. These authors' conclusions, of course, reflect the view that belief systems are not necessarily logically structured; therefore it is possible for a teacher to hold incompatible or inconsistent beliefs (Andrews & Hatch, 2000; Op't Eynde et al., 2002; Richardson, 2003; Thompson, €rner, 2002). However, a closer examination of partici1992; To pants' responses in this study reveals that even teachers with explicitly fallibilistic beliefs can hold Platonist beliefs as well. This can be observed, for instance, in the responses of teachers like Julia, Nikolas, and Stella (see sections 4.1.1 and 4.3.1) who, despite arguing that mathematics is a human invention for understanding the world (a somehow fallibilistic belief), they also believe that mathematical knowledge arrived unchanged to contemporary times from who they see as the inventors, the ancient Greek mathematicians (Platonist, absolutistic belief). In other words, this study provides further evidence, from a Cypriot perspective, reinforcing the idea that inconsistent beliefs can be nested in the same belief system of a person. The methodological approach of a small-scale collective case study taken here is, in a sense, phenomenographic, meaning that it is concerned with the variation of the ways a phenomenon is perceived by a group of people (Marton, 1981). While effective in mapping the topic under scrutiny, this approach carries certain limitations, like, for example, the extent to which the findings are prone to generalisations, as well as their transferability to other contexts (Yin, 2009). Of course, claiming that the findings reported here are transferable to other cultural/educational contexts could

4 In Beswick’s (2012) paper, the country in which her study took place (Australia) is not explicitly addressed. However, she writes that “[t]he two cases reported here are drawn from eight that were part of a larger study, other aspects of which have been reported elsewhere (e.g. Beswick, 2005, 2007)” (p.132). In her 2005 paper, Australia (more specifically, the Australian state of Tasmania) is explicitly mentioned.

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contradict my position regarding the cultural location of mathematics teachers' beliefs and practices. I hope that the Cypriot perspective of this paper will enable dialogue among colleagues with similar concerns in other countries. There is a need for more studies that address the cultural context in which epistemological beliefs are formulated, developed, and enacted. Studies of this kind could also shed more light on the complex relationship between mathematics teachers' beliefs and instructional practices. Also, within the context of Cyprus, further quantitative research could be carried out in order for generalisations to be made. The participants in this study hold a bachelor degree in Elementary Education. In the Republic of Cyprus, it is compulsory for those who teach at the secondary level (ages 12e18) to hold a bachelor degree in Mathematics. It would, therefore, be interesting to examine the extent to which teachers' epistemological beliefs about mathematics are similar or different across school levels, within the same cultural context and educational system. Furthermore, due to the relatively small sample and qualitative nature of this study, any attempts to indicate connections between the themes and sub-themes would be methodologically inadequate. Besides, generalisations of this kind would go beyond the main aims of this paper. Nevertheless, future research could utilise the findings of this study in order to develop scales that would quantify teachers’ epistemological beliefs and, with the employment of appropriate statistical techniques, identify correlations and causations. Since 2010, the educational system of the Republic of Cyprus has been undergoing major curricular reform for all school subjects and levels (from kindergarten to upper secondary education), which has been gradually implemented. Without shifting teachers' beliefs, however, changes can “be cosmetic, that is, a teacher can be using new resources, or modify teaching practices, without accepting internally the beliefs and principles underlying the reform” (Handal & Herrington, 2003, p. 62). As stated in the introduction of this paper, many studies have established strong relationships between mathematics teachers' epistemological beliefs, their self-efficacy beliefs, and their instructional practices. Even though a number of studies report attempts to change prospective mathematics teachers' epistemological beliefs, it is important for pre-service teacher education and in-service professional development programmes to take teachers’ epistemological beliefs more explicitly into consideration and to make more targeted efforts to shift them towards fallibilistic (Lerman, 1990) and problem-solving-oriented (Ernest, 1989, 1991) perspectives.

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