text:
Enrique Markku S Hannula
email: markku.hannula@helsinki@fi

Garcia Moreno-Esteva

Enrique Garcia Moreno-Esteva

Identifying subgroups of CERME affect research papers

Keywords: Affect, educational theories, graphs, literature reviews

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Introduction

The affect group in CERME has spent a lot of time and energy discussing the conceptual framework and terminology, leading to more extensive theorization of the area. Three theoretical frameworks have been especially influential in CERME for structuring the area of affect. The first is McLeod's 1992 framework that identified three main topics of research in mathematics related affect: emotions, attitudes, and beliefs. Moreover, the framework suggested that emotions are the most intensive, the least stable, and the least cognitive of the three, while beliefs are at the other end of the continuum and attitudes are in the middle.

An important synthesis of discussions in the group was Op 't Eynde's graphic representation of the conceptual field at CERME 5 (Hannula, Op 't Eynde, Schlöglmann & Wedege, 2007, Figure 1). This model identifies some new constructs that had been then discussed in CERME affect group: most importantly, the model identifies motivation as a dimension separate from affect, and meta-affective constructs. Moreover, the model highlights the local (classroom) and socio-historical contexts.

The model was discussed and developed by Hannula in his CERME plenary (2011) and further elaborated in the CERME special issue of RME (2012). This cube model (Figure 2) identified three dimensions that are relevant when discussing affective constructs. The first dimension addresses cognitive, affective, and motivational types of constructs. The second dimension separates the rapidly changing state-type constructs and more stable trait-type constructs. The third dimension identifies three levels of theorizing affect: psychological, social and embodied theories.

These distinctions identify separate research areas of mathematical affect. But how separate are they? Do the studies on attitudes, emotions, and beliefs form three separate research traditions or even 18 separate research areas, as Hannula's cube model suggests? Perhaps the studies on state and trait type affects are separate? Or are the different topic areas partially overlapping with diffuse borders, as Op 't Eynde's figure depicts it? One way to analyze the question empirically is to analyze the references different research papers share. We are aware that there is a lot of cross-referencing between authors of CERME affect papers and that there are some fundamental articles that keep being cited often. However, there has not yet been any systematic analysis on the cohesion of the research papers. We are going to make a network analysis of CERME papers in this research area. With this analysis we can identify whether there are some clusters of different research traditions within the CERME affect group. Moreover, we expect to identify important foundation works for each research tradition.

Our research questions are: (1) What possible subgroups can we identify among CERME research papers on affect when analyzing the cited authors of each paper? (2) What are the defining characteristics of each subgroup of papers?

Methods

Corpus and measures

Our corpus for analysis were the 100 research reports published in the affect groups in CERME conferences four to nine. The first two CERME conferences did not have an affect group and CERME3 one was left out because of some difficulties in the formatting that we did not have time to solve. We excluded from our analysis affect papers published in other groups, for example in groups of teacher beliefs or comparative studies.

Knowing the CERME publications in affect group, we knew that some authors had published papers that might fall into different research traditions. For example, many researchers have published papers both on teacher affect and student affect, and it could be possible that these fields would be empirically separate clusters. Therefore, we decided to search for clusters of papers published in CERME rather than identifying clusters of researchers.

When analyzing the lists of references, we had two basic options to identify links between papers. We could identify whether the same reference appeared in the reference lists of two papers, or we could link them whenever they had papers by the same author in their lists of references. Because our data corpus was modest in size, there was high probability that papers using the same theoretical framework might not share the same exact references even if they use papers by the same authors. Therefore, we decided to use the authors in the reference lists rather than the exact references as the method of connecting papers.

As a conclusion, our data consisted of research reports published in the proceedings and the authors appearing in their reference lists. For each of the papers, we created a connecting link from the paper to each of the authors mentioned in the references. For analyzing the connections between papers, we identified CERME papers and cited authors as vertices of a graph. Edges of the graph are the links that connected each CERME paper with the authors mentioned in the list of references, allowing multiple edges when a paper had several references by the same author. Hence, we produced a graph connecting all papers to their cited authors and this was then subjected to a mathematical analysis.

Analysis

The analysis included two stages: (1) To identify Graph Communities, and (2) to identify commonalities within the papers and authors of the selected Graph Communities.

We identified related papers using the FindGraphCommunities algorithm with modularity-based clustering to identify how papers and cited authors are related (Wolfram Alpha LCC, 2016). The modularity approach was originally developed by [START_REF] Newman | Finding and evaluating community structure in networks[END_REF] and the algorithm used in Wolfram Alpha is based on Fortunato's article (2010).

The input for the algorithm was the graph connecting research reports to authors cited in these papers.

The output was subgraphs called Graph Communities, each of which consisted of some of the research reports and authors. The algorithm chooses only one Graph Community for each of the graph's vertices. In other words, although the same author may originally appear in the reference lists of papers from many graph communities, in these subgraphs each cited author belongs exclusively to one Graph Community. This accentuates the differences between Graph Communities, especially with respect to those authors who are cited in papers in several communities. In addition, we are aware that the current method does give additional weight to authors cited several times in a paper.

The next stage of analysis was to identify which Graph Communities to include in further analysis. This stage was based on a visual inspection of Graph Communities to see how well they are connected. The five biggest Graph Communities are presented as graphs (Figures 3456), all affect papers published in CERME are represented as vertices with multiple edges. However, authors may have either one or multiple edges, depending on how many times they appeared in references. Another six Graph Communities are described but due to space limitation their graphs are no presented.

In the last stage of the analysis, we examined which papers and authors were represented in each of the Graph Communities.

Results

The algorithm identified 21 Graph Communities. We shall describe nine of them.

The first Graph Community 1 (29 papers; Figure 3) we call Foundation. It was the largest and the most cited authors in it included arguably the most influential researchers in the area of mathematical affect: McLeod (e.g. 1992), Schoenfeld (e.g. 1992) and Goldin (e.g. 2002). The most frequently cited authors in Foundation were active participants of CERME affect group: Hannula, Zan, Pehkonen, and Di Martino. Foundation is perhaps the most difficult to describe and may be best done by contrasting it with other Graph Communities. The Foundation's papers represent a large scope of research topics and theoretical frameworks, including papers that focus on beliefs, attitude, affect during problem solving, and emotions. This group also contains several papers that deal with dynamically changing affective states.

The second Graph Community (11 papers, Figure 4,) was given the label Self-efficacy. It had papers mostly from Cyprus and Turkey (e.g. [START_REF] Arslan | Turkish prospective middle grades mathematics teachers' teaching efficacy beliefs and sources of these beliefs[END_REF]. The shared theoretical framework of self-efficacy was indicated by numerous references to Pajares and Bandura.

The third Graph Community (11 papers, Figure 5) we named Motivation. Philippou and Pantziara were influential authors in this Graph Community. Seven of the papers included at least one of them as the author (e.g. [START_REF] Pantziara | Fear of failure in mathematics: What are the Sources[END_REF]. Also Waege appeared three times in this group as an author. This group shared motivation theory framework, and the most cited authors were well-known motivation theorists Midgley, Deci, Ryan, Pintrich, and Elliot. The Fourth Graph Community (8 papers, Figure 6) collected together papers on Teacher Development. Liljedahl was an important author in this group with four papers and the most cited authors include Liljedahl and Ball. The following six Graph Communities were smaller, each including 3 to 5 papers. Due to space limitations, these will be described only briefly. The four papers in Academic Emotions share Pekrun's (e.g. [START_REF] Pekrun | Academic emotions in students' self-regulated learning and achievement: A program of qualitative and quantitative research[END_REF] academic emotions framework and all these papers have been published in CERME8 or CERME9. All four papers in Metacognition were authored by Panaoura (citing e.g. [START_REF] Flavell | Speculations about the nature and development of metacognition[END_REF]. Teacher Beliefs had three papers from CERME8, citing, e.g. [START_REF] Fives | What do teachers believe? Developing a framework for examining beliefs about teachers' knowledge and ability[END_REF]. Resilience included 5 loosely connected papers without any frequently cited author. Meaning (citing e.g. [START_REF] Skovsmose | Meaning in mathematics education[END_REF] included four papers, and Identity (citing, e.g. [START_REF] Sfard | Telling identities: In search of an analytical tool for investigating learning as a cultural shaped activity[END_REF] four papers.

The remaining 14 Graph Communities included one or two papers each, altogether 16 papers. Nine of these papers were by authors who have ever published only once in the CERME affect group. Yet, these included also papers by frequent CERME participants (e.g. Hannula and Philippou).

Discussion

The analysis identified nine groups of CERME affect publications. Their defining features were a shared theoretical framework and often a research team. The largest group, Foundation, did not hold a theoretical framework clearly separating it from other groups. Rather, this group seemed to rely more on the seminal works in the field of mathematics related affect and cover a variety of research topics indicating that there is much cohesion in this research field.

How much are these identified groups of papers determined by having the same authors? Most researchers with several papers in the analysis had most of their papers in a single community and only Philippou appears in three different Graph Communities. Often, having publications in different Graph Communities seems to be explained by supervisors co-authoring their students' papers that may often have quite different theoretical frameworks than their own papers.

The current method did not allow overlapping of Graph Communities, which made it difficult to identify possible authors who have a cross-cutting importance across several graph communities.

However, looking at the total numbers of citations across all Graph Communities we found some such authors. The clearest examples were Ernest, who was cited 12 times in total, but not more than four times in any Graph Community, and Mason, who was cited 10 times but not more than twice in any Graph Community. Also, most authors described above as defining a Graph Community are cited in many papers of other communities. This suggests that results might identify groups more clearly, if we defined connections through specific cited research papers rather than cited authors. However, our current corpus might not be sufficiently large for that kind of analysis. Such analysis would be recommended when using a data corpus of at least a few thousand articles.

There are some methodological issues that we are aware of. We realized that summaries of the affect group from the previous CERME proceedings were cited often, inflating the number of citations by their authors. A more fundamental question is, that we have no measure for the reliability of The Graph Community analysis. With the current data corpus, our first analysis included an error that excluded 11 of the 100 papers. This was enough to produce a significantly different result: A subgroup of Foundation papers (Pehkonen and his students) was identified as a separate Graph Community and Teacher Development was not identified as a Graph Community. This suggests that the results of the analysis are quite sensitive to changes in data.

The first author of this paper has published several synthesizing articles on research in mathematics related affect. Using the graph analysis was an attempt to overcome possible personal biases in perceiving the structure of the research area. Our method of connecting CERME publications by authors appearing in their lists of references seems to have worked. It confirmed research on motivation research as a specific research domain. The analysis also identified specific research traditions on self-efficacy and academic emotions. While earlier reviews (e.g. [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF], identified beliefs and emotions as two areas within mathematics-related affect, the current analysis identified research on beliefs in three different groups: Foundation, Self-Efficacy, and Teacher Beliefs. Similarly, the current analysis identified Academic Emotions as separate group while most emotion papers were part of Foundation. These results suggest that in the [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF][START_REF] Hannula | Exploring new dimensions of mathematics related affect: Embodied and social theories[END_REF] model, the distinction between cognition (beliefs), motivation, and emotions is the most important one. On the other hand, one small Graph Community, Identity, can be considered to be characterized by its theoretical background being sociological. A possible new characterizing feature for research could be focus on the dynamics of change, exemplified by the research traditions Teacher Development and Resilience.

Figure 1 .Figure 2 .

 12 Figure 1. A graphic representation of the different dimensions of mathematics-related affect and their relationships, presented at CERME 5 (Hannula et al., 2007, p. 204)

Figure

  Figure 3. The Graph Community Foundation

Figure 5 .

 5 Figure 5. The Graph Community Motivation Figure 6. The Graph Community Teacher Development

Proceedings of CERME10


domain: math shs

text:
Eeva Haataja
email: eeva.haataja@live.fi

Anu Laine
email: anu.laine@helsinki.fi

Finnish educators' conceptions of the social-emotional needs of mathematically gifted high school students

Keywords: Mathematical giftedness, gifted students, social-emotional development

This article presents conceptions of social-emotional needs of mathematically gifted adolescents of certain Finnish educators. The article is based on a qualitative research study conducted with the methods of semi-structured interviews and participant observation in a Finnish high school that offers a special programme for mathematically oriented students. The study shows that the educators considered the most essential social-emotional needs of mathematically gifted students to be the need to be respected as unique personalities, to meet other gifted, to feed and guide the intrinsic motivation.

Introduction

The purpose of this study is to present and analyse the conceptions of the social-emotional needs of mathematically gifted students held by certain educators who work in a mathematically oriented high school in Finland. The research is conducted in a unique Finnish school community that is focused on educating students who exhibit particular interest in mathematics. The data of this research was collected by interviewing five educators of the school and by participant observation in the school. The research context provides valuable knowledge as it brings together the practices and theories of research on gifted education [START_REF] Ambrose | Unified, insular, firmly policed, or fractured, porous, contested, gifted education?[END_REF][START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF].

Giftedness is a very complex concept and varies both qualitatively and quantitatively within gifted persons [START_REF] Passow | The nature of giftedness and talent[END_REF], and multiple definitions of giftedness have been written over the decades [START_REF] Ambrose | Unified, insular, firmly policed, or fractured, porous, contested, gifted education?[END_REF]. This chapter presents the theoretical fundamentals on mathematically gifted adolescents and summarises former research on teachers' conceptions of giftedness and gifted education.

Being gifted means possessing promising potential in a certain domain of giftedness and being able to develop this potential into actual high performance. This domain, e.g. mathematics, is the field and the context in which the gifted activity occurs [START_REF] Cross | School-based conception of giftedness[END_REF]. Mathematical giftedness in particular means the ability to abstract numbers, variables and functions and the relations between them. For a mathematically gifted person, it also means courage, persistence and intrinsic motivation to go further and deeper in such mode of comprehension [START_REF] Gardner | Frames of mind: The theory of multiple intelligences[END_REF][START_REF] Reis | Underachievement in gifted students[END_REF][START_REF] Movshovitz-Hadar | Intellectual courage and mathematical creativity[END_REF][START_REF] Subotnik | The psychosocial dimensions of creativity in mathematics: Implications for gifted education policy[END_REF]. The development of such person is not only affected by personal characteristics, but also by the structure and properties of the individual's particular domain of talent [START_REF] Coleman | Social-emotional development and the personal experience of giftedness[END_REF].

Mathematically gifted students need mathematical activities, and with the support of their surroundings, they are able to take an active role in their own learning and develop into professional mathematicians [START_REF] Cross | School-based conception of giftedness[END_REF][START_REF] Usiskin | Development into the mathematically talented[END_REF]. They have also certain social-emotional needs… A gifted person is able to fulfil his whole potential only if his intrinsic abilities, the support of his social surroundings and the social-emotional dimensions are in balance [START_REF] Subotnik | The psychosocial dimensions of creativity in mathematics: Implications for gifted education policy[END_REF][START_REF] Usiskin | Development into the mathematically talented[END_REF]. Exceptionally gifted adolescents often experience dissimilarity and even unpopularity among their schoolmates [START_REF] Mönks | Dyssynchrony: Uneven development[END_REF][START_REF] Rimm | Peer pressures and social acceptance of gifted students[END_REF], especially when the giftedness takes place in the domain of mathematics [START_REF] Pettersson | Hur matematiska förmågor uttrycks och tas om hand i en pedagogisk praktik[END_REF].

Many professional educators still view giftedness as a fixed and innate characteristic of a person [START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF]. Some of teachers even consider the gifted as students who actually do not need training or instruction [START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF]. However, recent theoretical definitions of giftedness have shifted towards contextual and malleable conceptions (e.g. Ambrose, VanTassel-Baska, Coleman, & Cross, 2010;[START_REF] Cross | School-based conception of giftedness[END_REF]. Various researches illustrate this gap separating the theoretical knowledge of giftedness from the conceptions of professional educators [START_REF] Ambrose | Unified, insular, firmly policed, or fractured, porous, contested, gifted education?[END_REF]. Research further shows that there is also a gap between the teachers' conceptions of gifted education and the educational practices they conduct. Therefore there is need for in-depth research to further our understanding of teachers' conceptions of giftedness [START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF].

Teachers' conceptions have a significant role in supporting young gifted students in advancing their talent [START_REF] Mann | Creativity: The essence of mathematics[END_REF][START_REF] Mcnabb | Motivational issues: Potential to performance[END_REF][START_REF] Pettersson | Hur matematiska förmågor uttrycks och tas om hand i en pedagogisk praktik[END_REF]. Teachers tend to favor quite traditional conceptions of giftedness (Moon & Brighton, 2008). Generally, according to teachers, the most determining characteristic of a gifted student in the school context is a specific difference from others, which presents itself as the gifted student's capability to perform fast, intelligent and creative learning [START_REF] Kaya | Teachers' conceptions of giftedness and special needs of gifted students[END_REF][START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF][START_REF] Mattsson | Head teachers' conception of gifted students in mathematics in Swedish upper secondary school[END_REF]Moon & Brighton, 2008). It is interesting to note that teachers also associate mainly positive social-emotional characteristics with giftedness, such as enthusiasm, sensitivity and curiosity [START_REF] Kaya | Teachers' conceptions of giftedness and special needs of gifted students[END_REF][START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF][START_REF] Mattsson | Head teachers' conception of gifted students in mathematics in Swedish upper secondary school[END_REF].

There seems to be a gap between the theories and teachers' traditional conceptions of giftedness. Therefore, the purpose of this study was to examine the educators' conceptions of the socialemotional needs of the mathematically gifted adolescents in a school with successful practices. We were especially interested in the conceptions that were also enacted in their practices.

Methodology

In this qualitative research the data was collected by interviewing all the teachers who had regularly supervised the summer schools (two mathematics teachers, the biology teacher) as well as the principal and the healthcare officer. Additionally… by observing, filming and participating in the mathematics classes in the autumn of 2011 and an overnight school session in January of 2012. A qualitative semi-structured interview technique was used in order to give the interviewees the possibility to conceptualize and describe the topic in the way they prefer.

The researcher spent time with the student participants of the overnight school and participated in the social and mathematical activities of the event. Short informal discussions were conducted with the students while participating. As the data was collected in the authentic environment of the research subjects by means of participant observation, this study contains characteristics of ethnographic research [START_REF] Delamont | Ethnography and participant observation[END_REF]O'Reilly 2005).

The data was transcribed and analysed with inductive content analysis. Inductive content analysis means categorizing and combining units of the analysis into larger aggregates [START_REF] Elo | The qualitative content analysis process[END_REF]. A whole statement was chosen to be the unit of the analysis. The whole statements included one thought, conception or opinion varying in length from a couple of words to several sentences. These units were first categorized into codes. After that, as is done in inductive content analysis [START_REF] Elo | The qualitative content analysis process[END_REF], the codes were connected into categories and such categories into main categories, and finally the results were interpreted in the light of the theoretical background of the study.

Results

The purpose of the school is to gather together and educate adolescents interested in mathematics. The students of the school are offered a wide range of instruction in mathematics and diverse learning environments such as the Night of Mathematics and an annual summer school in Lapland. The educators interviewed in this study generally described their students as gifted. They defined mathematical giftedness as the ability to picture, learn and remember mathematical causations rapidly and with clarity. They described two types of giftedness appearing in the school: with multiple talents and those with a single exceptional talent. The students with multiple talents were interested in societal influencing and social activities. On the other hand, the exceptionally talented tended to impress their teachers with their commitment to studying and with their high level of mathematical reasoning skills.

Principal:

Roughly speaking there are those Renaissance talents who are widely talented and then those exceptionally gifted, who focus on the area of their deepest interest.

The uniqueness of the school's students was emphasized in the interviews. The interviewees were unwilling to stereotype the students and rather described their personalities, interests, social skills and profiles of giftedness as very individual.

Principal: I don't want to give any stereotyped answer here. I don't want to say that they are this kind or that kind.

According to the educators, many of the students have experiences and memories of feeling different and isolated during elementary school. Sometimes a change of school climate can be essential for a gifted adolescent.

Mathematics Teacher2: And we offer a community where you can discuss the Schrödinger equation during a break without being sneered at.

Biology Teacher: I just received a message where the parents were thankful because it has been so great [for him]. To be accepted in the group and let him be himself and encouraged and so on.

According to the interviews, the students with exceptional mathematical giftedness had more challenges in terms of social skills than those who were gifted in various fields. Moderately gifted students are usually relatively popular among their school mates and age peers, while the exceptionally gifted are more prone to being left alone [START_REF] Gross | Social and emotional issues for exceptionally intellectually gifted students[END_REF][START_REF] Rimm | Peer pressures and social acceptance of gifted students[END_REF]. Any school environment requires various social skills from students [START_REF] Payton | The positive impact of social and emotional learning for kindergarten to eighth-grade students: Findings from three scientific reviews[END_REF]. According to Mathematics Teacher 1, both "social sharks" as well as those who have "obvious problems in that respect" could be found among the students of the school.

Biology Teacher: Some of them have very poor social skills. …It is often related to this narrow field of giftedness.

Every student was welcome to participate in the social activities of the school to the extent of their own preferences. According to the interviews, one important social skill for the students is tolerating of all kinds of personalities. In the interviews, the diversity among the students was seen as an important part of the school's social climate.

Mathematics Teacher1: Of course one can choose to enjoy small groups or solitude.

Mathematics Teacher1: We have a vast variety of personalities and a tight community, which means that it becomes a tolerant community.

To study and associate with other gifted students was considered one reason behind the distinctive solidarity of the school community. These views are congruent with the literature [START_REF] Gross | Social and emotional issues for exceptionally intellectually gifted students[END_REF][START_REF] Rimm | Peer pressures and social acceptance of gifted students[END_REF]. Even though a variety of social skills existed among the students of the school, the common interests made social interaction easier.

Health-care officer: To find congenial people. And I know how the teachers describe, how they [the students] make experiments in the physics lesson or somewhere, the burning enthusiasm they show.

According to the interviewees, the students of the school were able to form close friendships with each other. The class-based structure and diverse range of informal activities formed the basis for the development of friendships at the school.

Biology Teacher: And then across the groups of each year's class, because on Mondays [when extra courses in mathematics are taught] and at overnight schools, they spend time together, there are no boundaries.

Associating and studying with other gifted students are emphasized both in these results and in the literature (e.g. [START_REF] Subotnik | The psychosocial dimensions of creativity in mathematics: Implications for gifted education policy[END_REF][START_REF] Rimm | Peer pressures and social acceptance of gifted students[END_REF]. The positive social climate of the school was constructed upon acceptance of the dissimilarities of students, diverse social interaction, shared experiences and interest in learning mathematics. These features were also seen as suitable for enhancing giftedness.

Intrinsic motivation is one of the most essential social-emotional characteristics for the development of mathematical giftedness (e.g. [START_REF] Mcnabb | Motivational issues: Potential to performance[END_REF][START_REF] Subotnik | The psychosocial dimensions of creativity in mathematics: Implications for gifted education policy[END_REF]. The importance of motivation was also emphasized by the mathematics teachers interviewed in this study.

Mathematics Teacher2: They are very motivated. And that is more determining than giftedness. Of course they need some kind of giftedness. But with some kind of basic giftedness you can go very far.

Even though motivation is often seen as a person's inner characteristic [START_REF] Subotnik | The psychosocial dimensions of creativity in mathematics: Implications for gifted education policy[END_REF], the interviewees highlighted the significance of peer support in connection with maintaining motivation.

The shared motivation and interest in mathematics was also apparent in the overnight school, where groups of students solved mathematical problems together while demonstrating amazing enthusiasm.

Health-care officer: It is the passion for [mathematics] that creates common good things in the class or the group or among the students.

Mathematics Teacher2: The social pressure can influence them one way or another… They support each other very much in studying.

The observations of the overnight school showed that the students were able to discuss their perfectionism, too. The conversation was humorous, and the participants were laughing at their perfectionistic characteristics.

Student1: I am not a workaholic at all! Student2: No you're not. You only scared all the freshmen with your stories last year.

Sometimes high motivation comes with negative phenomena such as unbeneficial perfectionism. The biology teacher and the healthcare officer had seen that achieving certain objectives or failing to do so may cause stress and exhaustion.

Principal: It [exhaustion] does not occur often, but someone every year.

Health-care officer: Often great giftedness and striving for perfection and achievements are a part of the personality. There is a risk of stress and fatigue and exhaustion.

The interviewees tended to see perfectionism as a practical problem of the educational system rather than a problem in the adolescents. The perceptions of the interviewees were summed up by the healthcare officer. According to her, negative perfectionism can be prevented and treated by guiding the students, being adaptive and offering constant care to the students.

Health-care officer: Flexibility and a flexible education system are what secure the path of the adolescents somehow. And also the caring, in particular, daily care.

Studying with the other gifted students in a supporting school climate was described to help the adolescents to form a realistic self-image and a strong self-confidence as people and mathematicians.

Mathematics Teacher1: It is easy to obtain perspective, [because] some really are incredibly good. --But nobody is the best of all.

Health-care officer: It is amazing to notice how they somehow gain self-confidence.

The gifted students were described as both ordinary and special at the same time, as they encounter the common social-emotional challenges of adolescence but also have special characteristics and needs due to their mathematically oriented and ambitious environment. The influence of the domain of mathematics on the development of mathematically gifted adolescents cannot be ignored.

Discussion

All qualitative research should be subject to realistic reflection on its general reliability [START_REF] Lincoln | Naturalistic inquiry[END_REF]. In this particular study, the use of triangulation of data collection increases the validity of the research. Nevertheless, a longer participatory observation could have offered more profound information on the social interactions of the students. Additional reliability was achieved by presenting the interviewees with the results of this study.

This research studied educators who possess particular experience in the context of Finnish education on teaching students who are recognised as gifted. Therefore its results differ from earlier studies on conceptions of giftedness of Finnish teachers [START_REF] Tirri | How Finland serves gifted and talented pupils[END_REF][START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF]. The interviewees of this study described giftedness as advanced performance and a modifiable characteristic of a student as well as of the whole school community. Finnish teachers tend to relate gifted students only with positive social-emotional characteristics, such as creativity and high level of motivation [START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF]. The interviewees of this study were prone to discuss also socialemotional challenges related to lives of the gifted adolescents. Still, the positive attitude to giftedness, enthusiasm for teaching and pride of their students were easily heard within the interviews. Research has also noted that the amount and especially the quality of cooperation with gifted students determines the teachers' conceptions of and approaches to giftedness [START_REF] Kaya | Teachers' conceptions of giftedness and special needs of gifted students[END_REF].

The concept of dissimilarity is widely included in definitions of giftedness as well as in the conceptions of giftedness commonly held by teachers. If someone is thought to be gifted, she is also seen as somehow, although often positively, different from others [START_REF] Shani-Zinovich | On being a gifted adolescent: Developmental, affective and social issues[END_REF]. In this research, the school was described as a meeting place for mathematically gifted students who are, in some way, different from many other adolescents. Therefore it is significant to understand the difference as a subjective experience of a gifted adolescent. Although the society, parents and teachers usually appreciate giftedness, exceptional talents often experience isolation within their age group [START_REF] Gross | Social and emotional issues for exceptionally intellectually gifted students[END_REF][START_REF] Rimm | Peer pressures and social acceptance of gifted students[END_REF].

The organization and the curriculum of national school systems should meet the needs of every student including the gifted ones [START_REF] Cross | School-based conception of giftedness[END_REF][START_REF] Kaya | Teachers' conceptions of giftedness and special needs of gifted students[END_REF]. The school investigated in this research does not represent a common high school in Finland. Neither does it reflect general Finnish attitudes toward special education of highly performing adolescents [START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF][START_REF] Tirri | How Finland serves gifted and talented pupils[END_REF]. A person's individual growth and particular social environment determine the social-emotional challenges of a mathematically gifted child instead of the mathematical giftedness per se [START_REF] Wilson | Social and emotional characteristics and early childhood mathematical and literacy giftedness: Observations from parents and childcare providers using the ECLS-B[END_REF]. Finnish teachers are highly qualified and skilled at differentiating learning contents both for fast and slow learners [START_REF] Laine | Finnish teachers' conceptions of giftedness[END_REF]. However, this research shows that even the most devoted and competent teacher cannot replace the need of meeting, studying and making friends with other congenial peers. As a conclusion, when planning education for gifted students the social-emotional aspects of gifted education and the gifted students' need for meeting congenial adolescents should be considered.

Proceedings of CERME10


text:
Annika Grothérus
email: annika.grotherus@liu.se

Emilia Fägerstam
email: emilia.fagerstam@liu.se

Impact of long-term regular outdoor learning in mathematics -The case of John

Keywords: Outdoor learning, mathematics anxiety, self-regulation, mathematical proficiency

This paper reports on a longitudinal study investigating the impact of long-term regular outdoor learning in mathematics in the school-grounds. An interview-based case study of John, a lower secondary school student, will be analysed. The case study will describe John's perceived experience of long-term regular outdoor learning in mathematics and its impact on affective and academic factors. The findings emphasise the positive outcomes of long-term regular outdoor learning in mathematics, indicating enhanced cooperative learning, reduced mathematics-related stress and anxiety, changed self-concept, and enhanced mathematical proficiency.

Introduction

The constant focus on textbooks and formal mathematical practice might invoke a view among students that mathematics is abstract, distanced and only useful in a classroom context, working only in the textbook [START_REF] Boaler | Open and closed mathematics: Student experiences and understandings[END_REF]. If students are not given the opportunity to engage with real-life problems in mathematics they will encounter problems applying their knowledge in an outside school context [START_REF] Desforges | Learning out of school[END_REF]. Mathematics taught in the classroom will have limited value if it is not transferable to students' everyday life and future academic and career endeavours. Current research on outdoor learning in mathematics demonstrates positive affective outcomes and possible academic benefits [START_REF] Daher | Characteristics of middle school students learning actions in outdoor mathematical activities with the cellular phone[END_REF][START_REF] Moffett | Outdoor mathematics trails: an evaluation of one training partnership[END_REF][START_REF] Noorani | Exposing the Fun Side of Mathematics via Mathematics Camp[END_REF]. In this paper, by analysing John's perceived experiences of long-term (3 year period) regular outdoor learning in mathematics, we explore any possible impact on both affective and academic factors. By affective factors we mean mathematics-related stress and anxiety, and motivation. With academic factors we refer to possible academic outcomes such as application and understanding of mathematical knowledge, enhancement of mathematical proficiency, learning strategies, self-regulation and self-concept.

Theoretical background

Outdoor learning

Outdoor education can be referred to as organised learning that takes place in the outdoors and is drawn up on the philosophy and theory and practises of environmental as well as experimental education. The embodied and multisensory experiences provided by well-organised outdoor learning are believed to enhance the individual's learning and understanding within a subject, in this case, mathematics. The variation of context between the indoor classroom activities and the outdoor activities enables rich opportunities for cooperative learning in real-life situations (e.g. [START_REF] Jordet | En undersøkelse om uteskolens didaktikk i et danningsteoretisk og erfaringspedagogisk perspektiv[END_REF].

Mathematical proficiency

In the present paper we use the framework of mathematical proficiency presented by [START_REF] Kilpatrick | Adding it up: Helping children learn mathematics[END_REF]. According to [START_REF] Kilpatrick | Adding it up: Helping children learn mathematics[END_REF], mathematical proficiency has five strands: conceptual understanding (comprehension of mathematical concepts, operations and Thematic Working Group 08 Proceedings of CERME10 relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently, and appropriately), strategic competence (ability to formulate, represent and solve mathematical problems), adaptive reasoning (capacity for logical thought, reflection, explanation, and justification) and finally, productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's efficacy) (Kilpatrick et al., 2001, p. 116).

Mathematics anxiety, motivation, self-regulation and self-concept

Mathematics is highly valued by society, and proficiency in mathematics has become increasingly important in order to become a fully functioning citizen, with resulting financial empowerment, making the ability to use mathematics in an out-of-school context even more important (OECD, 2004;[START_REF] Peterson | Globally challenged: Are US students ready to compete? The latest on each state's international standing in math and reading[END_REF]. A variety of studies have shown that many students have negative attitudes towards mathematics, which sometimes results in mathematics-related stress and anxiety [START_REF] Ashcraft | Math anxiety: Personal, educational, and cognitive consequences[END_REF][START_REF] Maloney | Math anxiety: Who has it, why it develops, and how to guard against it[END_REF]. Within a student, engaged in a mathematical activity, an on-going unconscious evaluation of the situation with respect to the student's self-concept and personal goals is taking place. Depending on the individual's evaluation of the outcome, the student may become emotionally affected being either motivated or not for further mathematical activities [START_REF] Hannula | Attitude towards mathematics: Emotions, expectations and values[END_REF][START_REF] Pintrich | A conceptual framework for assessing motivation and self-regulated learning in college students[END_REF]. According to PISA (OECD, 2013), [START_REF] Ryan | Promoting self-determined school engagement[END_REF] and [START_REF] Wigfield | Expectancy-value theory[END_REF] two forms of motivation to learn mathematics within self-regulative skills exist. The two forms are intrinsic motivation and extrinsic motivation. Intrinsic motivation is the motivation to perform mathematics merely for the joy gained from doing mathematics. While extrinsic motivation is the motivation to perform mathematics due to its importance in society, usefulness and the fact that mathematical knowledge will aid future career prospects and further academic studies. An individual with mathematics-related stress and anxiety is emotionally affected in a negative sense and tends to have a low self-concept (self-related cognitions of ability that can explain as well as predict achievement related behaviour, belief in one's abilities) as well as low levels of self-efficacy (a student's belief that he/she has the capability to perform a given mathematics task at a designated level) [START_REF] Bandura | Self-efficacy[END_REF]Bong & Skaalvik, 2003). Furthermore, these individuals also tend to lack the ability to self-regulate, resulting in avoidance behaviour and a decline in mathematics performance. As a consequence they are assigned lower grades and tend to have limited choices of possible future academic and non-academic career paths [START_REF] Wu | Math anxiety in second and third graders and its relation to mathematics achievement[END_REF].

One important factor to promote learning is students' capability to self-regulate [START_REF] Pintrich | A conceptual framework for assessing motivation and self-regulated learning in college students[END_REF]. If students' are provided with guidance on how to self-evaluate their own learning process and how to develop suitable strategies to promote their own learning during formal schooling, such guidance will enable acquired knowledge to remain updated after leaving school and to be used in an outside school context [START_REF] Pintrich | A conceptual framework for assessing motivation and self-regulated learning in college students[END_REF][START_REF] Zimmerman | Becoming a self-regulated learner: An overview[END_REF]. Self-regulation refers to the degree in which students are active participants in their own learning. It is an individual's ability to set mastery goals, mobilizing the efforts and resources the individual will need in order to achieve these goals. To reach these mastery goals, students use a wide range of self-regulatory processes and display a number of adaptive motivational factors such as self-efficacy and self-concept, where moderate levels of mathematics-anxiety may actually facilitate learning as well as performance [START_REF] Pintrich | A conceptual framework for assessing motivation and self-regulated learning in college students[END_REF].

Cooperative learning

Procedures in cooperative learning are designed to engage students actively in the learning process through inquiry and discussion with their peers in small groups, which need to be well organised with a clear structure to promote cooperative participation and learning (Davidson & Worsham, 1992, p. xii). Cooperative learning provides students with academic factors such as positive effects in mathematics performance and student achievement [START_REF] Whicker | Cooperative learning in the secondary mathematics classroom[END_REF]. Cooperative learning is a preferable method when helping individuals with mathematics-related anxiety to reduce their stress and anxiety. Furthermore, it is also an important feature of self-regulated learning because active participation is a crucial element of the self-regulation construct [START_REF] Clark | Formative assessment: Assessment is for self-regulated learning[END_REF][START_REF] Daneshamooz | Cooperative learning and academic hardiness on students' mathematical performance with different levels of mathematics anxiety[END_REF].

Focus of the study

The aim of the study was to investigate John's perceived experience of long term (three years) regular outdoor learning in mathematics in terms of possible impact of integrated outdoor mathematics activities on affective and academic factors.

Methodology

This paper is a part of a larger intervention research project aiming to explore the possible impact of outdoor teaching and learning in lower secondary school [START_REF] Fägerstam | Space and place: Perspectives on outdoor teaching and learning[END_REF]. Outdoor teaching and learning was implemented on a regular basis as a complement to ordinary classroom teaching during the entire lower-secondary school period of three years. The focus of this paper is to explore the possible impact of long-term regular outdoor learning in mathematics, and the case of John is presented as an example of its possible impact. The research is exploratory to its nature because there are few longitudinal studies on outdoor learning in mathematics. John is 15 years old, attending his third and final year of lower secondary school. John is a fictitious name that has been given to ensure the individual's anonymity. John's class had one of their four weekly mathematics lessons outdoors on a regular basis during the entirety of lower secondary school. The same mathematics teacher taught John during the three years of the intervention project. The school to which John and his class belong was situated in the suburbs of a medium-sized (approx. 85000 inhabitants) municipality in Sweden. The school, grade 7 to 9, was a normal sized school with approx. 450 students in six parallel classes. John was interviewed using semi-structured interview as a method. The interview was audio-recorded and transcribed using verbatim. Data was analysed thematically to identify recurrent patterns and commonalities using thematic coding [START_REF] Boyatzis | Transforming qualitative information: Thematic analysis and code development[END_REF]. Aspects of self-regulation skills were analysed based on concepts originally used in the PISA survey (OECD, 2004), namely intrinsic and extrinsic motivation, self-concept and mathematics anxiety. Representative illustrative quotes will describe the possible impact of regular outdoor learning in mathematics on self-regulation in mathematics and mathematical proficiency as well as John's perceived experience. The ethical guidelines and directives stipulated by The Swedish Research Council regarding good research have been followed [START_REF] Hermerén | Good research practice[END_REF].

The case of John

John is a fifteen-year-old boy who has had severe difficulties with learning and understanding mathematics since he started school. John was selected as a case due to his low levels of mathematics self-concept, expressed mathematics-related stress and anxiety and low levels of mathematical proficiency, which were reflected in his mathematical performance.

John's overall experience: Well-planned lessons, structure, intelligibility, and time

John emphasises the importance of well-planned lessons, structure and intelligibility. It is important, he says, that one knows what is expected and that everyone knows what to do.

It is of importance that everyone knows what he or she is supposed to do. It is important that the teacher gives a thorough briefing before the outdoor lesson. It is important that the teacher presents a well-organised picture of the task. You need to have a check before you start as well so you know what to do and that you do not just start directly and miss out on something that is of importance when solving the task that is presented to you and your group.

Through well-organised lessons and well-made tasks, it is easier to understand the mathematics and what is expected of you. John continues:

It is crucial that you understand what you are supposed to do, what kind of theory you need in order to solve the task that is presented to you. If the lessons are not well organised, the head, the brain, you get so, you disconnect, you start to think of other things.

Time is another aspect of John's experience of the long-term regular outdoor learning. He thinks that the teacher who teaches mathematics provides time to work with mathematics outdoors. However, he questions why other teachers in other subjects do not provide time and prioritise time to have some of the weekly lessons outdoors. I think that you should have outdoor lessons in other subjects too. Take biology for example. In biology there are so many "outdoors and environmental issues" so it would be a great possibility to work more outdoors. But, we are never outdoors during our lessons in biology, which I think is strange.

According to John, outdoor learning in mathematics provides more time and space to understand what and why you do things.

Indoors you seldom receive any help from the teacher. Often you just sit there for like ten minutes waiting for the teacher to have the time to help you. This results in you not raising your hand to ask for help, because it is quite meaningless. On the contrary, during the outdoor lessons in mathematics the group could either help each other or if the group needs assistance from the teacher, the teacher helps the whole group at the same time, which is really great.

Impact on affective factors

John experienced and expressed a change in self-concept. It is relaxing to work on a regular basis with mathematics outdoors. I get really stressed during the regular indoor lessons in mathematics and suffer from mathematics-related panic attacks. However, during the outdoor lesson in mathematics I really enjoy myself, I am more relaxed and do not suffer from mathematics-related anxiety attacks.

John experiences that he is enjoying himself and feeling more relaxed during the regular outdoor lesson in mathematics.

John also indicates that his extrinsic motivation has a tendency to hamper his achievement and performance in mathematics. He brings up the pressure to perform and the stress and anxiety the national tests in mathematics cause. However he also brings up a sense of changed perception of himself and emphasises the positive outcomes of variation of context for the learning of mathematics.

It makes you understand everything much better. You become more engaged and motivated. Regular outdoor lessons in mathematics provide you with more input and understanding of mathematical concepts. In addition, you feel better and enjoy the mathematics lessons more.

John appears to address the idea of losing one's self-confidence, which will lead to low self-concept.

John emphasises the importance of feeling engaged and motivated. To be extrinsically motivated tends to have a negative impact on understanding and learning mathematics. It rather makes you give up because you feel like a loser who cannot manage mathematics. However, by working to engage with real-life problems in mathematics with regular outdoor lessons in mathematics solving these real-life problems together with others provides, according to John, the possibility to become aware of one's true mathematical proficiency and that it might be enhanced. As demonstrated, John experienced that he began to enjoy mathematics more. He changed from being extrinsically motivated to become more intrinsically motivated and was more ready to face new, more challenging tasks because he started to believe in yohisur own abilities.

Impact on academic factors

John also stated that he had difficulties with negative numbers when using them and understanding the concept of them. During one of the outdoor lessons in mathematics they had worked with negative numbers. Before this outdoor lesson they had, during the indoor lesson, talked about negative numbers and worked with them in the textbook.

Well, we had one lesson when we worked with negative numbers, you know plus and minus and that kind of stuff. We did this exercise where you were supposed to run and put a piece of paper next to another on one of these big long things that looks like a row, and then there were also, you know, numbers in between and at the far end there was perhaps minus something and in the middle was zero. After a while you started to realise that it was kind of a huge thermometer. It was almost like the numbers became connected with each other. The visual picture and the practice of actually building the thermometer gave you a better understanding of the concept. It is strange but you actually need lots of self-confidence when it comes to learning and understanding mathematics.

As seen, working more visually and practically with negative numbers, strengthens John's conceptual understanding of negative numbers.

John emphasised the gain of cooperative learning. During the outdoor lessons in mathematics the students were supposed to collaborate when solving different problems they were given. During the outdoor lessons in mathematics, John was given a feeling of participation.

The class cooperated better when working with mathematics outdoors than indoors. Indoors the class seem to be more divided into certain groups. There is that group with the smart kids, who are good at mathematics, then the rest of us who are kind of left behind. During the outdoor lessons there tends to be more cooperative work, because all of us know that you must first begin to solve the given mathematics task on your own to begin with and then help the group by discussing the problem together. It is important that everyone can join in, and be given the possibility to explain how to solve the task at hand. Everyone should be given the opportunity to show his or her proficiency and share one's knowledge with others. It is a way to better understand how a given task can be solved. You become more engaged and motivated if you are allowed to participate and speak your mind regarding how you believe that the task at hand can be solved.

John emphasises that cooperative learning opens up opportunities for and development of adaptive reasoning. Through the outdoor lessons in mathematics, the students are given the possibilities to reason with each other. They have the opportunity to explain and try out their logical thoughts as well as justify their thoughts and chosen solution to the task at hand by reasoning with others.

Productive disposition

John expressed experiencing that he began, thanks to the regular outdoor lessons, to recognise and realise the importance of mathematics and that it is worthwhile to make the effort to understand the concepts of mathematics. Through cooperative learning he realised that he actually possesses a certain level of proficiency in mathematics. He realised that a task can be solved in several different ways.

You need self-confidence in mathematics and if you enjoy mathematics then you become more confident and more motivated. Indoors you just sit and understand nothing. But when you work with mathematics outdoors you understand how it all works.

Discussion

John's perceived experiences of regular outdoor learning in mathematics indicate that it is reasonable to assume that students can benefit from regular outdoor learning in mathematics. We may conclude that long-term regular outdoor learning in mathematics shows a tendency to have an impact on affective factors by altering the individual's self-concept, reducing mathematics-related stress and anxiety and resulting in a more engaged and motivated student, which is supported by the findings of previous studies (e.g. [START_REF] Moffett | Outdoor mathematics trails: an evaluation of one training partnership[END_REF]. In addition, long-term regular outdoor learning in mathematics shows a tendency to have an impact on academic factors and one major feature was the importance of variation of context. It is reasonable to assume that the possibility to work more visually and practically with real-life problems, which had first been presented theoretically during the indoor lessons, might be a key factor for enhanced conceptual understanding [START_REF] Kilpatrick | Adding it up: Helping children learn mathematics[END_REF]. We may conclude that engagement and motivation for learning mathematics is enhanced. Accordingly, students tend to become more engaged and motivated by regular outdoor learning in mathematics. Cooperative learning is a prominent part of outdoor learning in mathematics and is emphasised by John as a key feature for the enhancement of his adaptive reasoning [START_REF] Kilpatrick | Adding it up: Helping children learn mathematics[END_REF]. Previous findings have indicated that cooperative learning tends to help mathematics-anxious individuals reduce their stress and anxiety for mathematics [START_REF] Daneshamooz | Cooperative learning and academic hardiness on students' mathematical performance with different levels of mathematics anxiety[END_REF]. The main feature of cooperative learning is the opportunity to discuss and reason with others and justify one's mathematical thoughts on how to solve different mathematical problems. John stresses that cooperative outdoor learning in mathematics tends to make students aware of their true mathematical proficiency by being given the possibility to observe that a task hand can be solved in more than one way and that more than one "right" solution to the problem may exist. John experienced that when mathematics became less abstract and more transferable to his everyday life, learning mathematics becomes more joyful resulting in a more positive attitude towards mathematics making him engage instead of avoid learning mathematics (e.g. [START_REF] Maloney | Math anxiety: Who has it, why it develops, and how to guard against it[END_REF]. Long-term regular outdoor learning in mathematics gave John the possibility to develop the ability to selfregulate his learning [START_REF] Wu | Math anxiety in second and third graders and its relation to mathematics achievement[END_REF]. Finally, it is reasonable to assume that if a student is provided with a sense of control, that student may enhance the levels of self-concept and reduce mathematics related stress and anxiety.

  Outdoor lessons in mathematics make it easier for you to remember what you do and why. When you have your lesson in mathematics outdoors, the teacher explains clearly what to do. You are given a clear picture of what is expected of you and what the task at hand is about and what the aim of the lesson is. You will better understand what you do and why you do it. Moreover, it seems like the change of learning environment reduces the mathematics related stress and anxiety.

Proceedings of CERME10


text:
Wajeeh Daher
email: wajeehdaher@gmail.com

Osama Swidan

Amani Masarwa

Positioning and emotions in learning algebra: The case of middleachieving students

Keywords: Discursive framework, positioning, emotions, middle-achieving students

 to study middle-achieving students' emotions in learning, utilizing technology, the topic of 'the quadratic function as a product of two linear functions'. The students' learning was videoed and then transcribed to be analyzed according to the discursive framework. The research results indicate that members in the middle-achieving groups claimed the collaborator positioning in order to learn the mathematical topic using mainly behavioral, social and cognitive processes. Leaders claimed their positioning through carrying out processes related to the different aspects of learning, mainly the cognitive, metacognitive, meta-emotional, social and linguistic aspects. Dominant emotions in the groups' learning were frustration, enjoyment and content.

Introduction

The affective aspect is a growing area in educational research due to its relationships with other aspects of students' learning, especially the cognitive one. In the present paper, we study middleachieving students' emotions in relation to their positionings when they learned the quadratic function as a product of two linear functions using dynamic software. In a previous research, we examined students' positioning and emotions in one group learning geometry [START_REF] Daher | Discursive positionings and emotions in a small group's learning of geometric definitions[END_REF], where the group consisted of two high-achieving students and one middleachieving student. Following that study, we wondered how emotions and positioning are experienced by groups consisting of only middle achieving students. This paper intends to examine this issue, using the discursive framework [START_REF] Evans | Discursive positioning and emotion in school mathematics practices[END_REF]. [START_REF] Polya | how to solve it: A new aspect of mathematical method[END_REF] addressed the necessity to consider emotions as they influence the problem solving process. Later, especially in the 1980's, researchers considered affect as a significant component of students' mathematical problem solving (e.g., [START_REF] Mcleod | Affective issues in mathematical problem solving: Some theoretical considerations[END_REF]Schöenfeld, 1985). Emotion is one of the fundamental elements of the affective aspect [START_REF] Hannula | Affect in mathematics educationexploring theoretical frameworks[END_REF]. [START_REF] Hannula | Affect in mathematics educationexploring theoretical frameworks[END_REF] describes emotions as connected to personal goals. Furthermore, emotions, when managed appropriately, become a potential tool for effective thinking rather than a disturbance to this thinking [START_REF] Antognazza | The flow of emotions in primary school problem solving[END_REF][START_REF] Salovey | Emotional intelligence. Imagination[END_REF]). In the present research, we intend to study, by using the discursive emotions and positioning framework, middle-achieving students' emotions when they utilize technology to study Algebra.

Emotions in mathematics education

The discursive positioning framework for studying students' emotions

Positioning is defined as "the discursive process whereby people are located in conversations as observably and subjectively coherent participants in jointly produced storylines" (Davies & Harré, 1999, p. 37). Evans, et al. (2006) suggest this discursive positioning framework for studying students' emotions. This framework assumes that meaning making occurs in social practices, using semiotic resources. Social practices have an emotional dimension that helps maintain social identity. Moreover, empirical data in this framework is seen as text, the analysis of which demands attention to its context(s). This analysis entails a combination of structural and textual phases that each informs the other. The structural analysis considers the positions available to or claimed by the participants. Positions are associated with power in relation to others, as well as with differing values within the discourse, which creates spaces within which emotion may arise. Usually, there is more than one available position for a participant, either within a single discourse or several competing discourses. Evans, et al. (2006) describe the positionings taken care of in the structural analysis: Helper and seeker of help, collaborator and solitary worker, director of activity and follower of directions, evaluator and evaluated, insider and outsider.

The textual analysis considers the exchange of meanings. This phase has two functions [START_REF] Evans | Affect and emotion in mathematical thinking and learning[END_REF] showing how positionings in social interactions are actually taken up by the participants, and (b) providing indicators of emotional experience. The textual analysis has two stages. In the first stage, the focus is to identify the interpersonal aspects of the text that establish the positions of the participants. Indicators at this stage include reference to self and others, reference to valued statuses (e.g. claiming understanding or correctness), modality (indicating degrees of un/certainty), hidden agency (e.g., passive voice) or repetition. For example, leadership is indicated by demonstrating knowledge or meta-emotional behavior (e.g., trying to change the negative emotions of the group members). The collaborator position is indicated by the activity of a group member, as answering questions or doing actions in response to events occurring during the group learning.

The second stage of the textual analysis attends to (a) indicators of emotional experience that include: direct verbal expression (e.g., 'I feel anxious'), use of particular metaphors (e.g. claiming to be 'coasting' ), emphasis by words, gesture, intonation, or repetition, body language (e.g., facial expression); (b) indicators suggested by psychoanalytic theory, as indicators of defenses against strong emotions like anxiety, or conflicts between positionings (as 'Freudian slips'), surprising error in problem solving, behaving strangely (as laughing nervously), denial (e.g., of anxiety).

Research rationale and goals

In spite of mathematics education researchers' acknowledgement of the role of affective aspects in mathematical education in general and mathematical problem solving in particular, research related to this aspect is still not widespread [START_REF] Antognazza | The flow of emotions in primary school problem solving[END_REF]. We intend to study emotions in problem solving using the discursive framework developed by Evans, et al. (2006). In more detail, we intend to analyze the positionings taken by ninth grade students and their related emotions when learning in groups, with the help of GeoGebra, the quadratic function as a product of two linear functions. Doing so, we introduce to the discursive framework the different aspects of learning; as the meta-cognitive and meta-emotional aspects. This will shed more light on the factors that influence middle-achieving students' experiencing of positioning and emotions.

Research question

-How are positionings taken up by middle-achieving ninth grade students, working in a group to learn the quadratic function as a product of two linear functions, in the presence of technology?

-How are middle-achieving ninth grade students' emotions associated with the positionings that they claim, when learning with technology the quadratic function as a product of two linear functions?

Methodology

Research setting and participants

In a previous research [START_REF] Daher | Discursive positionings and emotions in a small group's learning of geometric definitions[END_REF]; we analyzed the affective aspect of one group's learning of mathematics, where the members were both high and middle achieving students. We wondered how the affective aspect would be affected in just middle achieving or high achieving groups of students. In the present research, we analyze this aspect in three groups of grade 9 middle-achieving students (ages between 14 and 15 years). One group consisted of three female students (Sana, Amal, Asil), and two consisted of two female students and one male student each (Fairouz, Noura, Salim) and (Alaa, Siham, Amin). All the participating students had not worked with GeoGebra before, and they were introduced to it in two hours' session before learning the quadratic function topic. Furthermore, the students had learned some issues in the topic of the quadratic function (the function's maximum or minimum, the vertex of the function and the domain of increasing/decreasing), but not the quadratic function as a product of two linear functions. The third author taught the three groups in a middle school in Israel.

Data collecting and analyzing tools

We collected our data using observations of the learning of the three groups. We also conducted interviews with their members. Every group's learning was videoed and at the end of each lesson, the three students in each group were interviewed individually regarding their positionings and emotions during learning. We analyzed the two types of collected data using the discursive analysis framework presented above. Moreover, we combined the analyses of the data collected by the two tools (observations and interviews). The findings section in this paper sheds light on this method.

Learning material

The three groups of ninth grade students worked with a sequence of activities; all related to the quadratic function as a product of two linear functions. Following is an example of these activities.

In the same coordinate system, we want to draw the three functions: y=x, y=x 2 + and y=x(x+2). -What are the algebraic characteristics of the linear function: y=x? -What are the graphical characteristics of the linear function: y=x? -What are the algebraic characteristics of the linear function: y=x+2? -What are the graphical characteristics of the linear function: y=x+2? -What are the algebraic characteristics of the linear function: y=x(x+2)? -What are the graphical characteristics of the linear function: y=x(x+2)? -What are the similarities and the differences between the characteristics of the two above linear functions and the characteristics of the quadratic function?

Note: Algebraic characteristics are related to the parameters of an equation, while the graphical characteristics are related to the intersection points with the axes, increasing or decreasing of a function, etc.

Findings

The present research aimed at characterizing middle-achievement students' positioning and emotions when learning algebra with technology. Doing so, we found that mainly the students had the leader and collaborator positionings during a lesson. We will describe how the students in the middle-achieving groups claimed each of the positionings and experienced their emotions and/or reported them in each positioning. Doing so, we will address the following aspects of learning that the positioning is related to: behavioral, cognitive, meta-cognitive, social and linguistic. The emotional aspect of learning will be considered in light of the taken positioning.

Collaborator's functioning

The middle-achieving groups utilized collaboration to learn the new mathematical ideas. This is exhibited in that generally the members of each of the participating groups claimed the collaborator positioning to pursue, with the help of the mathematical software, their learning of the quadratic function. This claiming resulted in making the group's members learn enthusiastically to understand the appropriate mathematical relations. This resulted in the group's members enjoying the activity and being content when arriving at its solution. Thus the collaborator's positioning helped make the students' emotions concerning their learning experiences positive ones.

To claim the collaborator positioning, the group members were involved with behavioral processes (working with GeoGebra), social processes (group discussions), as well as cognitive processes (mathematical reasoning). These three types of processes, not only helped the group members claim the collaborator positioning, but at the same time, supported their attempts, as described above, to perceive the new mathematical ideas. In the interview, the students associated their behavioral processes with positive and negative emotions: enjoyment of their work when the software helped them solve the mathematical problem, and frustration when finding difficulty to operate the software. Excerpt 1 shows this claiming the collaborator's positioning.

A1 Sana

We need to find the intersection with x for the three functions.

A2 Amal What's the first function?

A3 Asil 3x-2

A4 Sana [drew the first function in GeoGebra]

A5 Amal What's the second function?

A6 Asil 2x+3

A7 Sana [drew the second function in GeoGebra] Let's find the intersection points with x.

Excerpt 1: claiming the collaborator positioning

Excerpt 1 shows the claiming of the collaborator's positioning as connected with the behavioral aspect of the group's learning. This aspect is expressed by the students' action with the Geogebra software (A4, A7). However, this positioning also involves the meta-cognitive aspect. The utterances of Sana (A1, A7) are concerned with regulating the processes of the problem solution.

Leader's functioning

The leader's positioning in the middle-achieving groups was claimed by directing the learning of the group, as well as to advance this learning towards the solution of the mathematical problems and the sharing of the new mathematical ideas. Moreover, leaders in the middle-achieving groups claimed their positioning through carrying out different types of processes, mainly cognitive, metacognitive, meta-emotional, social and linguistic processes. Below, we elaborate on these processes.

The group leader's cognitive functioning was actualized through demonstrating knowledge during carrying out the mathematical activity. For example, Fairouz, a leader in one middle achieving group, argued that they only needed to know the intersection points of the two linear functions with the x axis in order to draw the resulting quadratic function.

The group leader's metacognitive functioning was actualized through asking questions during the group learning as means to decide upon the method of solving a problem. Moreover, the group leader's meta-emotional functioning was actualized through trying to change the negative mood of the group when encountering a difficulty. For example, Alaa, a leader in a middle-achieving group, tried to lessen the anxiety of group members by saying: "Don't worry. It's O.K. Sure we made a mistake. Let's read again our solution to find it".

The group leader's social functioning was actualized through answering other members' questions, asking questions and requesting actions from the group members to keep the group learning going.

Regarding the linguistic aspect of the leader's functioning, the leaders in the middle-achieving groups used the first person plural pronoun to talk about the mathematical actions that they needed to perform, which showed them as collaborators with the other members of the group. This indeed happened in the middle-achieving groups but not numerously (See for example excerpt 2).

The leader's functioning resulted in different emotions, but generally speaking this functioning resulted in frustration, when unable to find a way for solving a mathematical problem, enjoyment during the successful solution process, and content when finally solving the activity.

Difficulties in claiming the positions of leaders and collaborators

The members of the middle-achieving groups, due to the lack of appropriate previous knowledge in the subject matter and sometimes in GeoGebra manipulation, encountered difficulties in claiming the positions of leaders and collaborators during the process of the mathematical problem solving. This led to their experiencing some negative emotions. Moreover, the members of the middleachieving groups experienced calmness, anxiety and confusion in accordance with their leader.

Working with GeoGebra, the members of the middle-achieving groups encountered at the beginning difficulties related to working with a new technological tool, which could be related to the behavioral aspect of the group's learning. Excerpt 2 describes such a difficulty, where Salim, Noura and Fairouz wanted to draw the function (2x-9)(3x-4) in GeoGebra [B1], but found difficulty doing that due to not writing correctly the appropriate number of brackets [B2-B6].

B1 Fairouz

We should write 3x-4 multiplied by 2x-9.

[Noura started to write the expressions] B2 Salim Perhaps the brackets can be put afterwards, wait Noura, wait, it keeps moving.

B3 Fairouz

Write it from the beginning. Excerpt 2 shows some of the difficulties encountered by the middle-achieving groups, as a result of their behavioral functioning; specifically when working with a technological tool. Fairouz, in the interview, said they felt out-of-control and thus frustrated not being able to draw from the beginning the graph of the function f(x)= (2x-9)(3x-4) in GeoGebra. Salim pointed at the teacher's interference as supporting them in getting back control over their work with GeoGebra, which made them satisfied with their work on the mathematical problem.

Encountering difficulties in learning the new topic, not only influenced students' positioning and emotions, but also colored the linguistic aspect of their learning, especially their use of pronouns. This is the case in excerpt 3, where the difficulty is related to simplifying an algebraic expression.

C1 Siham

We want to draw the quadratic function y=(2x-9)(-x-4).

C2 Amin

Multiply first the brackets.

C3 Alaa (-x-4)(2x-9)= -2x 2 +9x-8x+36, Now we compute +9-8.

C11 Alaa

Write -2x 2 +x-36.

Excerpt 3: Having difficulty in simplifying an algebraic expression

Excerpt 3 shows that confronting difficulty constrained the group's sense of control and produced anxiety. In this situation, singular pronouns or no pronouns were used.

To claim the leader positioning in a middle-achieving group, the member was involved with different learning processes, as demonstrating knowledge, which was also reported in Evans, et al. (2006), but their functioning was distinguished from the other group members by performing metacognitive and meta-emotional processes, as reported in [START_REF] Daher | Discursive positionings and emotions in a small group's learning of geometric definitions[END_REF]. These processes helped plan, monitor, evaluate and take decisions regarding the group learning, especially in time of difficulty in arriving at the learning goals. Thus, these processes helped maintain the leader positioning [START_REF] Black | Thinking about thinking about leadership: Metacognitive ability and leader developmental readiness[END_REF], as they supported the leader in advancing the group learning.

In addition, the leader metacognitive functioning was actualized by asking questions as means to decide upon the method of solving a problem. This decision making could be looked at as a social process [START_REF] Vroom | Decision making as a social process[END_REF] with the goal to advance the group learning. Moreover, it seems that critical thinking skills, actualized in decision making, were needed to claim the leader's positioning. Furthermore, the goal of the leader meta-emotional functioning was to change the negative mood of the group when encountering a difficulty, which motivated the members' work [START_REF] Leithwood | How leadership influences student learning[END_REF]. So, we argue that the leader positioning was claimed by paying attention to different aspects of the group learning, especially the metacognitive and meta-emotional aspects. This leader's functioning resulted in different emotions related to the difficulty and success in performing the mathematical task, which could be associated with [START_REF] Goldin | Affective pathways and representation in mathematical problem solving[END_REF] emotional pathway, where generally frustration preceded enjoyment and enjoyment preceded content. This emotional pathway included the three dominant emotions in the groups' learning, i.e. no singular emotion was dominant but the emotional pathway was thus. Furthermore, the group members' emotional experience was influenced by that of the leader, which could be related to the emotional contagion suggested by [START_REF] Hatfield | Emotional contagion[END_REF], where there is tendency to converge emotionally with others. We say this is especially true in group learning when the other is the leader.

Students encountered sometimes difficulties in learning the new topic. This encounter, not only influenced students' positioning and emotions, but also their linguistic use of pronouns. This was expressed in their use of singular pronouns or no pronoun at all when getting anxious for not being able to proceed with the carrying out of the activity.

Future studies are needed to compare the positioning and emotions of different achievement-groups in solving mathematical problems. Furthermore, research is needed to verify the effect of prior positions of the group members on their current positioning, which the present research did not target.

Students' difficulties in working with GeoGebra and related emotions

  He takes the mouse from Noura who seemed annoyed by the act of Salim. Salim works on GeoGebra] O.K. Now write it again.

	B4 Salim No, no [B5 Noura [Noura puts her hand on her mouth with boredom]
	B6 Fairouz	[Fairouz worked on GeoGebra, then she said with annoyance] something wrong
		with the brackets.
	B7 Teacher	Don't get anxious. At the beginning, brackets are tricky. Everything will get O.K.
	Excerpt 2:

Proceedings of CERME10

Discussion and conclusions

Research of students' emotions in mathematics learning is growing (e.g., [START_REF] Antognazza | The flow of emotions in primary school problem solving[END_REF][START_REF] Daher | Learning mathematics in the mobile phone environment: Students' emotions[END_REF][START_REF] Hannula | Affect in mathematics educationexploring theoretical frameworks[END_REF]. The present research aimed at characterizing grade 9 students' positioning and emotions when learning algebra with technology. The research findings indicate that to claim the collaborator positioning, members of the middle-achieving group were involved with behavioral processes (working with GeoGebra), social processes (class discussions), as well as cognitive processes (reasoning). These processes helped them reach their learning goal, thus resulting in positive emotions. It could be said that collaboration was associated mainly with positive emotions as enjoyment and content, though negative emotions as anxiety were experienced when having difficulty in solving the mathematical problem; i.e. in arriving at the learning goal.


text:
Pietro Di Martino

Inés M A Gómez-Chacón

Peter Liljedahl

Introduction to the papers of TWG08: Affect and mathematical thinking

Introduction

This chapter introduces the contributions discussed during the working sessions of the TWG8 "Affect and mathematical thinking" in CERME 10.

In this edition our TWG has been enhanced by the inclusion in the group and the contribution in the discussion of the researchers from the TWG7 "Mathematical Potential, Creativity and Talent".

The quantitative data about the participation to our group confirms the interest toward affective issues in the field of Mathematics Education: 41 manuscripts were initially submitted to the groups, 26 were accepted for the discussion, and finally in these proceedings 24 papers and 2 posters are included.

Moreover, our group confirms its spirit of inclusion: 12 different countries were represented and 11 newcomers welcomed.

The papers (24) and the posters (2) were presented and collectively discussed in the first six sessions. Presenters had 10 minutes (5 minutes for posters) to introduce the key-ideas of their papers, then we developed a 10 minutes discussion. At the end of each session, 20 minutes were used to discuss the main aspects emerged in the section.

In the fourth section, one hour has been devoted to the presentation of the ERME Chapter about affect: the draft of the chapter was been sent to all participants before the conference, and each participants had to share two questions, comments or criticisms about this draft. The collected comments are been the thread of the discussion related to the chapter.

In the seventh and last session, we discussed the main themes emerged during our previous work, developing the structure of the report for the final day.

The issues emerged in TWG8 at CERME10 and the related discussion

The analysis of the affective focus of the papers discussed in our TWG reflects the current diversity of interests and approaches inherent in the field of affect research. Many different affective constructs emerged : beliefs, motivation, values, emotions, needs, relaxed, memory, aesthetic, confidence, meta-affect, identity, self-efficacy, meaning, motivation, values, images, views, flow, perseverance, tolerance, interest. It is important to underline as many of these constructs are clearly related, sometimesand this is a well-known critical aspect in the affective researchdifferent labels are used to indicate the same constructs, and vice versa. Sometimes the same term is used in a very different meaning by different researchers. As we will underline later, this communicative issue is particularly critical for the emotions.

In our discussions, we underlined the lack of conceptual clarity again, and called researchers to use a clear definition in their studies and to label the constructs appropriate.

Despite this variety, we recognized five recurrent and crucial dimensions involved in the discussion of our group: students, problem solving, self-concept, emotions and context.

In the following, we briefly discuss some aspects related to these five dimensions.

Students

Enriched by the contribution of researchers from the TWG7, we have long debated about the so defined "achievement problem", discussing around the following clearly related questions:

 Looking at school transitions: what are the effects of these transitions on school achievement? How and why do the parameters of the school achievement change dramatically from a school level to the next?  What is the relationship between school achievement and mathematical talent?  What is the distinction between low and high achievers in mathematics apart from the grade?

We also argued as it would be interesting to develop research around the above themes, looking at students who are studying mathematics in different contexts (for example modelling, IT environment, etc.).

Problem solving

There is a known and long tradition of research about affective construct and problem solving. This kind of research it is particularly important for our field, since it shows two crucial aspects: the strict relationship between affect and cognition (problem solving activities surely involve cognitive aspects but also strong emotional reactions); the relevance and peculiarity of the research on affect in the specific context of mathematics: indeed problem solving is one of the main activity for mathematicians.

As usual, one of the main issues is how teachers can create the context in order to develop the appropriate mathematical activities and environment for positive affect, increasing motivation and also performance. In several discussions within our group, it emerges as problem solving is not only an essential activity for developing the mathematical competence, but it has also the potential to draw attention and to motivate students, becausein some senseproblem solving is one of the beautiful side of math (the eminent mathematician Ennio De Giorgi used to say: a nice problem, even if you do not solve it, accompanies you).

Many papers presented in our group stressed the relevance of two aspects in order to take advantage of the affective potential of problem solving.

The first one is related to the setting: here setting is understood in a broad meaning, on the one hand we mentioned the classroom climatea positive climate is needed to develop significant problem solving activities, in particular it appears crucial to not identify problem solving activities and assessmentand the context (for example the spatial organization of the classroom). On the other hand, we mentioned the organization of the problem solving activities with the promotion of collaboration and discussion among students.

The second one is related to the choice of the mathematical problem. It is confirmed as only cognitive demanding (and not routine) problems can foster students' engagement, but, above all, can challenge students shifting their attention from products to processes. In particular, it is also crucial the monitoring of the possible imbalance between skill and challenge that student can experience during problem solving. In this setting, problem solving can be a tool to involve highachievers, but also low-achievers, in mathematical activities and increase positive affects towards mathematics.

Self-concept

The fact that students/teachers self-concept strongly affects their choices, the effort they devoted and their perseverance in doing some specific activity is one of the fundamental points in our field. Therefore, it is not a surprise that many discussions in our group focus around self-concept and related constructs, such as: self-efficacy, self-perception, self-regulation, identity, personal meaning.

In particular, three aspects related to the self-concept have been analysed and discussed in this edition of the TWG8:

 How students' self-concept influences their interpretation of the mathematical environment;

 How the context affects the self-concept, in particular it emerges the idea that context provides available identities;

 The connection between the self and the emotions (in the context of mathematics education).

Self has seen as a filter for interpreting experiences.

Emotions

Obviously the study and the discussion around emotions is a must in the group of affect.

In particular, we discussed some crucial dichotomy related to the concept of emotion:

 Emotions sometimes are a cause for some didactical outcomes, but sometimes are a symptom. In the first case, we see positive emotions as an educational goal, in the latter emotions are an indicator;

 Emotions have a double nature: more rapidly changing state-aspect and more stable traitaspect. The study of state, as opposed to trait, is necessary to give a more detailed description of emotional experiences in the mathematical teaching-learning process;

 Emotions can be the cause for opposite pathways during problem solving, or more in general mathematical activities (go on vs. give up).

As usual, we also discussed how deal with two classical critical issues in the research about emotions: one related to the observation and the other related to the communication.

Emotions, in truth as many other constructs, are not directly observable, therefore we never observe emotions, but we infer them from some indicators. Sometimes we collect information about emotions through self-reports, andalready at this stagecommunication issues intervene; studies developed in different countries stress the level of emotional illiteracy of a large part of the population: it is difficult to reflect about emotions, and it is more difficult to have the dictionary to communicate them.

The communication problem involves the field of the research not only for the difficulties related to the emotional illiteracy, but also for the internal communication (the communication between researchers): as a matter of facts, the labels for emotions have different shades in every language, and sometimes they are associated to a different meaning.

Context

Affective issues are mainly social for their nature: as we have seen, context plays a crucial role in the development of affective reactions. For this reason, context is always been one of the leading actors in discussion within TWG8.

A still open problem is the exact definition of context: what is the context? We are convinced that the context is dynamic in nature (it depends on the group/individuals), and this dynamicity makes more complex to circumscribe it.

In the group discussion was observed that in the context of emotion, the term 'culture' needs to be 'unpacked' and broached not from the assumption that cultures are uniform, but rather from more dynamic conceits such as put forward by identity theories.

Some studies in this edition focus on the difficulties related to the transition from a context to another one (for example in the school transition), analyzing how and how much these contexttransitions change affect.

Proceedings of CERME10