Naama Ben-Dor
email: naamabd@gmail.com1

Einat Heyd-Metzuyanim

Adolescents' Endorsement of Narratives Regarding the Importance of Mathematics: A Dialogic Perspective

Keywords: Sociocultural narratives, mathematics identity, adolescents, immigrants

Sociocultural accounts of students' identity emphasize the endorsement of narratives about learning from "significant narrators". In contrast, psychological accounts highlight adolescences as a time of separation from parents and other authority figures. In this study, we attempt to reconcile these two views based on the Bakhtinian concepts of Authoritarian discourse and Internally Persuasive discourse. The study examined the mathematical identity narratives of five adolescents, 2 nd generation to former Soviet Union immigrants in Israel, participating in a STEM afterschool activity. The findings focus on three students' endorsement of one narrative, "math is important", in both individual interviews and a focus group. The analysis showed that all participants endorsed the narrative and related it to their designated identities. However, participants differed in the level of endorsement of this narrative, ranging from 'full' endorsement to 'oppositional' endorsement.

Rationale and Goal

In recent years, students' mathematical identities have attracted increasing attention [START_REF] Darragh | Identity research in mathematics education[END_REF][START_REF] Radovic | Towards conceptual coherence in the research on mathematics learner identity: a systematic review of the literature[END_REF]. Many of these studies focus on adolescence as a critical time for the development of such mathematical identities [START_REF] Black | Developing a 'leading identity': the relationship between students' mathematical identities and their career and higher education aspirations[END_REF][START_REF] Heyd-Metzuyanim | Vicious cycles of identifying and mathematizing: a case study of the development of mathematical failure[END_REF]. Yet historically, adolescence has been treated very differently by psychological and sociocultural (or sociological) accounts of human development. Sociocultural accounts, especially those leaning on structural sociological theories which theorize individuals as products of social structure, assume that the construction of identity narratives draws on narratives authored by "significant others" (e.g. [START_REF] Sfard | Telling identities: in search of an analytic tool for investigating learning as a culturally shaped activity[END_REF]. In contrast, psychological theories highlight adolescence as a time where individuals construct their unique identity in separation from the adults around them (e.g. [START_REF] Blos | The second individuation process of adolescence[END_REF]. This disparity, between studies that point to adolescence as a time of separation vs. studies assuming the adoption of socioculturally based narratives into one's identity, invites a closer examination of the endorsement and separation processes of students' mathematical identity. Hence, this report aims to investigate this duality of endorsement and separation in the case of three adolescents who belong to a minority community of former Soviet Union immigrants in Israel.

Theoretical framework

The concept of identity has been notoriously ambiguous and difficult to define [START_REF] Radovic | Towards conceptual coherence in the research on mathematics learner identity: a systematic review of the literature[END_REF][START_REF] Sfard | Telling identities: in search of an analytic tool for investigating learning as a culturally shaped activity[END_REF]). To answer this ambiguity, [START_REF] Sfard | Telling identities: in search of an analytic tool for investigating learning as a culturally shaped activity[END_REF] suggested an operational definition of identity as "narratives about individuals that are reifying, endorsable and significant " (p. 16). They differentiated between actual (or current) identities, namely stories about the current state of affairs, and designated identitiesstories of what is expected to be the case in the future, stating that learning resides in the gap between current and designated identities. Black and colleagues (2010) pointed to the importance of vocational aspirations and their connection to mathematics in relation to adolescents' designated identities. Sfard and Prusak also distinguished between 1 st person (1 st P) identity stories (narrative told by the author about him/herself), 2 nd P and 3 rd P stories (narratives told by others about the protagonist, to him/her or to others respectively). They, as well as Heyd-Metzuyanim (2015), showed how 3 rd P identity stories of significant narrators turn over time into 1 st P identity stories.

Despite the operational clarity of [START_REF] Sfard | Telling identities: in search of an analytic tool for investigating learning as a culturally shaped activity[END_REF] definition, this definition is unclear regarding the status of narratives about mathematics and learning mathematics, such as "math is hard", "math is about calculations" or "math is important". Such narratives, often related to in the literature as "beliefs about mathematics", play a crucial role in shaping students' relationship with mathematics [START_REF] Törner | Beliefs-no longer a hidden variable, but in spite of this: where are we now and where are we going[END_REF]. For this matter, we incorporate into our conceptual framework Holland and her colleagues' (1998) concept of the space of authorship. [START_REF] Bakhtin | The dialogic imagination: four essays[END_REF] suggested that a human being is always in a state of addressing different sociocultural voices and authoring herself and others. Holland and colleagues (1998) define this space of authorship as a "broad venue, where social languages meet, generically and accentually, semantically and indexically, freighted with the valences of power, position, and privilege" (p. 191). Drawing on these ideas, we conceptualize the space of authoring mathematics identity as a dynamic arena containing narratives relating to mathematics, learning mathematics and the person as a mathematics learner.

In order to reconcile the "endorsement vs. separation" conflict between psychological and sociological views of adolescence as a time of identity development, we draw on another Bakhtinian concept: the process of ideological becoming. [START_REF] Bakhtin | The dialogic imagination: four essays[END_REF] conceptualized, from a discursive point of view, the process whereby individuals selectively assimilate others' words. This process begins, according to Bakhtin, with a separation between authoritarian discourse and internally persuasive discourse. Authoritarian discourse mostly belongs to parents, teachers, adults and other authoritative social figures. Bakhtin described this discourse as demanding unconditional allegiance and permitting no play with its borders, no arguing or dialogue. However, the words of others, according to Bakhtin, do not have to remain isolated and static; they can be internally persuasive and become "half-ours and half-someone else 's" (p. 345). In this process the other's narratives are questioned and challenged. [START_REF] Morson | The process of ideological becoming[END_REF] defined these dialogically tested narratives as internally persuasive. Our research question in this study was thus: to what extent are adolescent students' narratives in the Space of Authoring their Mathematical Identity, internally persuasive?

Methodology

The study included five 8th grade students (13-14 years old) that participated in a weekly STEMoriented afterschool activity at the Technion (hereafter "the Activity"). They all studied in the same school. The study was a pilot study aiming to build conceptual and methodological tools for a future larger study on students' identity in relation to dialogic instruction in mathematics. Hence, we chose this specific population because the students were participating in a STEM activity held in our university and could be easily accessed. The students' age corresponded with the age of our future study's population. Participation in the study was voluntary, and anonymity and confidentiality were guaranteed through an informed consent letter signed by the participants and their parents. Only four cases were analyzed. The four participants were born in Israel, 2nd generation of former Soviet Union (hereafter: 'Russian') immigrants; the fifth participant was a recent immigrant from Russia. This difference between her and the other participants became known during the data collection stage. Due to her limited ability to communicate in Hebrew, as well as our wish to maintain a cohesive framework of participants who are 2 nd generation to Russian immigrants, her data was not analyzed in the study.

The study included semi-structured individual interviews with four participants that preceded one focus group interview with all five participants, all recorded and transcribed. The interviews were conducted by the first author in a separate room. Each individual interview took approximately 45 minutes and included questions intended to elicit current 1 st and 3 rd current identity and designated identity narratives such as "what subjects are important for you?", "what kind of student are you?", "how would your parents answer that same question?", and "what would you like to be when you grow up?". Since our focus was the dialogic aspect of the endorsement of narratives, the interviewer questioned the participants on the sources of their narratives (e.g. "who says that math is important?") and asked for justifications (e.g. "why is it important?"). The focus group lasted 90 minutes. It began with a discussion of important subjects in school and ended with discussion of participants' wishes for the future. The rationale for holding a focus group in addition to interviews was double. First, there were participants who did not want to participate in the interview but agreed to participate in the focus group. Second, we wanted to observe processes of identity construction and endorsement of narratives from students' space of authorship in a more natural, "conversational" situation, where the voices of different students could meet and be negotiated.

Data analysis consisted of a two-stage analysis. The first stage included searching the whole interview dataset for common endorsed narratives belonging to the participants' Space of Authoring Mathematics Identity as well as current and designated identity narratives and their related narratives. Here, we encountered a difficulty with relation to narratives indicating "separations", since often, narratives authored as "truths" about the world have lost their sociocultural origin. For example, "I am a student" is a narrative that rarely contains evidence of its sociocultural source, although at a certain point in time, the author must have learned from others that this narrative identifies him. For this methodological reason, we chose in our analysis to focus only on those narratives that included indications of sociocultural sources. For example, "when I was young, my father told me that art is just a hobby" is a narrative ("art is a hobby") where the identified sociocultural source is the "father". A narrative was categorized as endorsed if the participant voiced the narrative, at least in part, in her own voiced or agreed with the narrative voiced by others (interviewer or peers).

The second stage analysis included a close-up examination of the dialogical aspect of the narrative's endorsement. Narratives which were voiced as solely coming from an authoritative voice (for example "my parents say that…") were categorized as Authoritative. Signs of separation of the participant's voice from the voice of the source were derived from positioning statements that included the opinion or thoughts of the "I" in relation to the narrative. For example, "my mother thinks art is a hobby, but I would like to pursue it". If signs of separation were identified, we further searched for signs of internally persuasive endorsement. These can appear in the form of providing justifications for the narrative, such as "art is a hobby because you cannot earn money with it". We further examined those justifications as "sub-narratives" along the same line to determine if those sub-narratives were themselves internally persuasive. Endorsed narratives, along with their justifications and sub-narratives were compared and related to students' designated identities. For reliability, we employed a consensual coding process based on both authors' mutual agreement.

Results

For the purpose of this report, out of the four cases, we chose the three cases of Benny, Sonia and Denis, which could best exemplify the different levels of endorsement in relation to the same narrative, "math is important". The forth participant showed a level of endorsement similar to Benny. The overall results included the examination of other endorsed narratives related to the participants' designated identities such as "art is just a hobby" and "I need these studies". The narrative, "math is important" was stated by all students during the interviews as a response to the interviewer question "which (school) subjects are important?"

Benny's casefull endorsement

Benny participated only in the focus group, since at the time of individual interviews, he did not yet volunteer to participate in the study. In the focus group, after several students talked about the importance of different subjects, the interviewer (1 st author) turned to Benny and asked: "Benny, what do you think, because I think you also said math (is important)". To this, Benny answered: It's very important, like if you want to get accepted for the good jobs that you have regular hours and all that ... you must understand mathematics well, otherwise you get stuck in hard jobs where you break your back to get a few shekels.

Notably, these claims were authored in his own voice, as facts about the world, even though some of them (such as those relating to jobs with "good hours" in contrast to jobs "where you break your back") most probably were authored by adults around him. We base this interpretation on our acquaintance with students of Benny's age and social position who rarely have direct access to different jobs' "hours", their "regularity" and the exact physical activities that are entailed by them. Although we interpret these narratives as sourcing from the sociocultural world, the way they are authored by Benny completely hides their sociocultural origin. By that, Benny shows full separation from the sociocultural origins of this narrative; it is not related by him in any way to what his parents or other authority figures say.

In relation to his designated identity, Benny talked about his wish to become a surgeon and said he would have liked to study the subject of surgery in school and in the Activity. When asked what in the Activity was relevant to his wish to become a surgeon, he asserted: "math is always relevant" and elaborated this claim with the following justifications:

In many ways (math is important) ... measurements... say you are doing a cardiac surgery, then it is important, say, how long you have between transferring the heart, between the body from which you are taking out the heart until he gets another heart and how long it... a lot to calculate... We see here that Benny links his designated identity as a surgeon to the narrative "math is important" and describes specific future situations as a surgeon in which he could find himself using math. Although some of the relations Benny made between the job of a surgeon and mathematics do not actually fit the place of mathematics in the surgery world, it is clear that Benny uses his conceptions of mathematics (as mostly calculations) to link to what he imagines the world of surgery to be. Since there is no mention of anyone who told Benny about this relation of surgery and mathematics, and since the connections themselves are a bit constrained and indicative of Benny's own yet limited grasp of what mathematics (and surgery) is all about, we categorize the endorsement of this narrative as internally persuasive. In no place does Benny offer conflicting narratives to the narrative "math is important" (neither in his own voice nor in revoicing someone else's voice), thus strengthening our identification of his discourse about the importance of mathematics as full endorsement.

Sonia's caseconflictual endorsement

In her interview, Sonia reported liking best the subject of art and sports. Despite that, and similarly to the other participants, she claimed both in her individual interview and in the focus group, that math and English are the most important subjects studied at school, thus providing evidence of endorsement of the narrative.

When the interviewer asked her why it is so, Sonia replied: First, (they 1 ) say it is most important, and also, like, it's needed…it's also a very very important part of the matriculation exams…(they) always say that most important are math and English... I also think English is important.

The interviewer then asked again why math is important, and the following exchange took place: 1 In literal translation the word "they" would be omitted. The omitting of "they" is often used to avoid the passive voice in Hebrew.

(implied
In line 47, Sonia specifies the identity of the source to be her parents and "almost all people". There are clear signs of separation between her own voice ("I still don't understand why math" [47]) and her parents' voice ("The parents say" [47]). We thus see Sonia's endorsement of the narrative "math is important" as conflictual. Although she claimed not to have internally persuasive reasons for the narrative and even challenged the interviewer to provide her with justifications, Sonia did initially author the narrative as her own (in answering the interviewer's question which subject was most important at school), as well as provided a reason: engineering (49). The mentioning of engineering as a reason for the importance of mathematics was related to Sonia's designated identity. She said: "I already have one idea for what I want to go study… software engineer". When asked why she wants to study software engineering, she said:

…I really like computer games and I want to do software for computer games, and also actually in my opinion I think that the fact that I like drawing and with all the thoughts and fantasy, it will help me in computer games to invent something interesting.

Although in her interview Sonia mentioned the plan to be a software engineer as her own idea, two weeks later, in the focus group interview, she said, "My mother wanted me to be a software engineer. For that I need five math units2 …" Despite the identification of her mother as the source of her future plans, Sonia displayed in her justifications an internally persuasive endorsement of her reason for studying mathematics (studying engineering). Thus, like Benny, Sonia links her designated identity (as a software engineer) to the endorsement of the narrative. However, she also poses conflicting narratives doubting this narrative and confirming its authoritative nature. To summarize, Sonia's endorsement of the narrative "math is important" is double-voiced and conflictual, showing signs of internally persuasive discourse together with authoritarian discourse.

Denis's case -oppositional endorsement

In his individual interview Denis reported he liked computer games and sports, and that he wants to have "two degrees" and study in the "biggest university". Both in the individual interview and in the focus group interview Denis listed exact sciences (including math) as most important. When the interviewer asked why, Denis cited his mother "because she always has this thing with numbers", thus identifying his mother as the source for the narrative. In the focus group, when the subject of why math is important was raised, Denis said: a. Because the parents force you; b. Because they came from the Soviet Union and for them math is in the blood; c. Because we are in Israel and (you) don't earn much in Israel

This claim was consistent with Denis's claim in his individual interview that 'Russian' parents try harder to teach their children math at home: because they want the children to come out smart in the country... and Israel for them is not an easy country, not a lot of money, problems, they want their children to come out lawyers and doctors… geniuses... He then added that, as their (the 'Russians') child, you need to "make as much effort as possible to impress the parents at the end of the year to show them the report card". Examining the endorsement of the narrative "math is important", it is clear that Denis separates his own voice from the identified source by positioning himself as being coerced to study hard and earn high grades because of his immigrant community. Thus, implicit in Denis words, is a narrative that in fact opposes the narrative "math is important", a narrative that could be explicated as there are no relevant reasons for me to study mathematics aside from pleasing my parents. The justifications to the narrative provided by Denis are all related to the source (his mother and other 'Russians') and thus cannot be categorized as internally persuasive. The links Denis makes between designated identities and the endorsement of the narrative are all related to 3 rd P designated identities, that is, what others expect him to become. We thus label Denis's endorsement as oppositional.

Discussion

This study sheds light on the process by which narratives in the Space of Authoring Mathematics Identities, originating in sociocultural voices, are endorsed and incorporated into adolescent students' 1 st P identities. In addition, it highlights the connection of narratives such as "math is important" to students' designated identities. We see this in the fact that Denis, whose endorsement of the narrative was oppositional, connected the "math is important" narrative to his future ability to earn money without specifying how this narrative would be relevant to his designated identity. In contrast, Sonia and Benny (whose wishes were to become a surgeon and a software engineer) related the importance of mathematics specifically to their future desired careers. This link between narratives in the Space of Authoring Mathematics Identity and designated identities reinforces Black and colleagues' (2010) suggestion that motives, derived from students' aspiration, may play a crucial role in shaping their relationship with mathematics. In other words, although narratives about the importance of mathematics circulate around students, the answer to whether these narratives would turn from authoritative discourse to internally persuasive discourse may lie in the connection of these narratives to students' plans and aspirations for the future.

Rather than assuming a relatively straightforward adoption of narratives authored in the sociocultural sphere as some structural sociological accounts do, the present study examined the dialogic process of endorsement, based on the Bakhtinian concepts of authoritative discourse and internally persuasive discourse. Although in all three cases reported herein there were indications of sociocultural sources to the narrative "math is important", these cases showed different degrees of separation of the adolescents' voice from the voices of significant narrators around them. This exemplifies the process by which narratives authored by significant others are not simply "facsimiled", in the words of [START_REF] Holland | Identity and agency in cultural worlds[END_REF] onto their children's identity. Moreover, the endorsement of the narrative by the three adolescents was different in subtle yet noticeable ways. The analysis revealed three levels of endorsement of the narrative: full endorsement, conflictual endorsement, and oppositional endorsement. Full endorsement was established by signs of an internally persuasive discourse regarding the narrative; conflictual endorsement was established by evidence of both authoritative discourse as well as internally persuasive discourse; and oppositional endorsement was established by signs of an authoritative discourse. These findings suggest that students' endorsement of narratives in the space of authorship is a complex process. The fact that a student states for example that "math is important", and even cites his parents or author authority figures as the source of this narrative, does not provide the whole picture regarding the endorsement of the narrative and the separation of the adolescent from the authoritative discourse.

Although we present a small-scale study and our report focuses on one narrative only, our findings foreground the need to use dialogical concepts and tools when examining students' mathematical identities. Furthermore, the theoretical approach we adopted in this study goes some way in mediating between psychological approaches emphasizing adolescence as a critical time of separation from authoritative figures, and sociocultural approaches signifying the importance of narratives surrounding the students. By this, the study contributes to bridging the gap between studies taking a psychological approach to identity and studies taking a sociological, discursive take on this concept.

The matriculation exams in Israel have different level of "units". The most advanced level is "5 units".

Acknowledgements

This research was partially supported by ISF grant 2699/17.


Wendy Ann Forbes
email: wendyannforbes@gmail.com

Joyce Mgombelo

Sarah Gannon

Chantal Buteau

Undergraduate students' mindsets in a computer programming mathematical learning environment

Keywords: Mindset, University mathematics education, Programming, Identity

This paper focuses on how undergraduate students' mindsets are enacted and constructed in a programming and mathematics learning environment. This research draws from year one of a fiveyear study entitled "Educating for the 21st Century: Post-graduate Students Learning Progmatics (Computer Programming for Mathematical Investigation, Simulation, and Real-world Modeling), which addresses the need to empower students within the STEM field. A narrative approach is employed to present findings from two students, Sydney and Jim (pseudonyms), followed by a discussion on their enacted mindsets during a first-year 'progmatics' course.

Introduction

Jobs in the STEM field are on the increase. A major percentage of those jobs require computer science-related knowledge (CBC News, 2015). However, students in undergraduate computer science programs often feel defeated, having little confidence in their ability because of obstacles they encounter that are inherent to programming [START_REF] Cutts | Manipulating mindset to positively influence introductory programming performance[END_REF][START_REF] Murphy | Dangers of a fixed mindset: Implications of self-theories research for computer science education[END_REF]2008). Moreover, many mathematics students avoid engaging in mathematical tasks that require reasoning, exhibiting phobias and anxiety in extreme cases, because they perceive these tasks as difficult (Boaler, 2016). The pressing need to educate students in STEM, combined with students' often negative responses to learning programing and mathematics, necessitate inquiry of how students perceive their intelligence and abilities as they learn these disciplines. This paper draws from year one of a five-year study funded by the Canadian Social Sciences and Humanities Research Council (SSHRC) entitled "Educating for the 21st Century: Post-graduate Students Learning Progmatics (Computer Programming for Mathematical Investigation, Simulation, and Real-world Modeling). This study addresses the need to empower students within the STEM field and to better understand the complexities involved when students learn to appropriate tools such as 'progmatics'. The study focuses on the question of how mathematics students come to appropriate programming as a computational thinking instrument in the context of three 'progmatics'-focused undergraduate mathematics courses, called Mathematics Integrated with Computers and Applications (MICA). This course sequence teaches students the fundamentals of programming for conducting mathematical explorations and applications. Analysis of data gathered within this study provides a suitable opportunity to investigate students' mindsets as they grapple with the complexities of a 'progmatics' learning environment. Specifically, this paper will focus on how students' mindsets are enacted and constructed while learning 'progmatics' during the first MICA course. To achieve this aim, the narratives of two students, Sydney and Jim (pseudonyms), will be explored.

Mindset refers to the brain's potential to formulate perceptions, affecting attitude and achievement. Mindset was first introduced in educational research in the 1920s but in recent years, Carol Dweck's work on oppositional theories of growth and fixed mindsets has popularized mindset [START_REF] Popan | Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages, and innovative teaching[END_REF]). Dweck's notion of growth mindset has been applied in various contexts. [START_REF] Cutts | Manipulating mindset to positively influence introductory programming performance[END_REF] assert that the barrage of obstacles faced by students during their undergraduate programs promote fixed mindset beliefs. Furthermore, [START_REF] Murphy | Dangers of a fixed mindset: Implications of self-theories research for computer science education[END_REF] contend that while selftheories are applicable in all disciplines, the way students perceive their abilities may be more significant in computer science education, particularly in computer programming, due to inherent challenges such as contending with elusive and puzzling syntax and runtime errors. Based on these arguments regarding the significance of self-theories to mathematics and programming, a growth mindset may be even more important for learning mathematics within a programming environment, given the added difficulties stemming from both disciplines. Our study adds to other mindset research focusing on mathematics or programming as it involves not only one aspect (programming or mathematics), but the combination of both disciplines.

Conceptual Framework

Our view of learning relies on [START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF] work on communities of practice. [START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF] contend that learning takes place relative to the context in which it is learned through legitimate peripheral participation, whereby newcomers become oldtimers by authentically taking part in a seamless process of gradually increasing responsibilities. The newcomer, starting at the periphery, may initially observe, but contributes within a community of practice almost immediately by doing small tasks which are progressively increased until he/she becomes an oldtimer. For [START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF], the process of becoming a legitimate member of a community involves not just change in knowledgeable skill, but also a change in identity. [START_REF] Wenger | Communities of Practice[END_REF] defines this identity as "a layering of events of participation and reification by which our experience and its social interpretation inform each other" (p. 151). This definition implies that a change in mindset results from a change in identity because individuals' perceptions of their own abilities is shaped as they experience their sociocultural environment. In this sense, individual stories and narrative approach are relevant in inquiry on one's experienced mindsets. Storying or constructing narratives are common means utilized by humans to make sense of the myriad of complex experiences they encounter; contributing to the formation of their identity [START_REF] Mcadams | Personal narratives and the life story[END_REF], and by extension, their mindset. A narrative approach is a growing trend in research relating to identity and, in general, educational research [START_REF] Mcadams | Personal narratives and the life story[END_REF]. [START_REF] Dweck | Even geniuses work hard[END_REF] describes two mindsets, fixed or growth, that influence student performance in two distinct ways. Students with a fixed mindset perceive their abilities as stable traits, believing that they have a set capacity to be successful that cannot be changed. Students with a growth mindset, however, understand that with effort and perseverance, their abilities can be improved. [START_REF] Solomon | Not belonging? What makes a functional learner identity in undergraduate mathematics[END_REF] asserts that a students' mathematical mindset plays a key role in developing their identity. Mindset is also a critical factor in determining a student's attitude towards learning and their level of achievement (Boaler, 2016;[START_REF] Dweck | Carol Dweck revisits the "growth mindset[END_REF][START_REF] Murphy | Dangers of a fixed mindset: Implications of self-theories research for computer science education[END_REF]. [START_REF] Dweck | Even geniuses work hard[END_REF] claims that in general, individuals with a fixed mindset avoid challenges, give up easily, do not value effort, dismiss positive feedback, and are threatened by others' success. In contrast, individuals with growth mindset flourish on challenges, remain persistent despite setbacks, value effort as a path to mastery, use criticisms to improve learning, and use others' success as motivation and a source of valuable lessons. Furthermore, [START_REF] Murphy | Dangers of a fixed mindset: Implications of self-theories research for computer science education[END_REF] note that as it relates to challenges, students with fixed mindsets focus on performance goals by opting for easier tasks as their ultimate objective is to display their ability. In contrast, students with growth mindsets focused on learning goals and are not deterred by difficult tasks or by making mistakes, in fact, they will seek out opportunities for challenge [START_REF] Dweck | Even geniuses work hard[END_REF]. Importantly, a growth mindset goes beyond effort, referring to the extent to which students use innovative strategies and seek help when they are stuck [START_REF] Dweck | Carol Dweck revisits the "growth mindset[END_REF].

Methodology

Throughout the duration of the larger five-year study, a mixed-methodology approach will be utilized. An iterative design approach (Plomp & Nieveen, 2013) will be employed to refine and develop the research tools for yearly data collection and analysis. The study employs a naturalistic case study approach to examine how students' instrumental genesis of programming for mathematics develops in the MICA course sequence. This research takes places in the mathematics department of a university in Ontario, Canada. The present paper draws from data gathered in year one of the larger study, where six participants in the MICA I course were recruited voluntarily. Data gathered included each participant's four 'progmatics' projects, including both their program (called exploratory objects (EOs)) and assignment report, and semi-structured individual interviews that were conducted as a follow up to each assignment. Interview prompts were informed by a model of a student's developmental process in designing, programming and using a mathematics EO [START_REF] Buteau | Student development process of designing and implementing exploratory and learning objects[END_REF]. Data also included online post-laboratory session reflections, where after each of the ten weekly two-hour MICA lab sessions, participants recorded reflections on their learning during the lab as prompted by guiding questions. Finally, all participants completed an online questionnaire before beginning the MICA I course, followed by individual interviews where participants were asked to elaborate on their questionnaire responses. The purpose of the questionnaire and follow-up interview was to uncover baseline information about participants' background experiences in learning mathematics with technology, as well as their early sentiments towards the MICA I course.

Analysis of the study's qualitative data followed [START_REF] Creswell | Educational research: Planning, conducting, and evaluating quantitative and qualitative research[END_REF] general principles of qualitative data analysis: preparing and organizing data, exploring data, and describing and developing themes from the data. To begin the analysis, codes were developed according to categories informed by the theoretical framework [START_REF] Buteau | Computational thinking in university mathematics education: A theoretical framework[END_REF] and related literature, with additional codes emerging during the analysis process. Each participant's qualitative data was coded individually by two researchers, who then jointly completed a thematic analysis of the data. Themes were consolidated among the six participants' analyses, leading to the development of sixteen overall themes. These themes were further regrouped into five meta-themes. In this paper, we focus on the meta-theme of identity and its subthemes of affect and students' perceptions of learning mathematics.

Findings and Discussion

In this section, we present and discuss findings from two participants, Jim and Sydney (pseudonyms).. We present two narratives of the participants' individual enactment of mindsets upon entering and during the MICA I course, followed by a comparative discussion.

Sydney's Story

With no prior programming experience, Sydney approached the MICA I course with feelings of nervousness and apprehension. In the first MICA I labs, her anxious sentiments were somewhat alleviated through interactions with her instructor, helping her to grasp basic programming concepts. She faced some challenges in the first assignment (EO1), where students were tasked with creating a program to explore a mathematical conjecture of their choosing. Sydney posed a conjecture that was a minor modification of one previously covered in the MICA I labs. She kept her program relatively simple, ensuring that she did not go "beyond [her] limits" during the coding process. The greatest obstacle she faced in completing this task was debugging her code, but with the help of a more experienced peer she created a functioning program. Sydney felt relieved that she was able to successfully complete the assignment, but was still hesitant about her programming ability and hoped that the next assignment would be easier.

Sydney experienced varying levels of success and confidence throughout the remainder of the MICA I course. The second assignment, involving the design of a program about RSA encryption, presented many challenges for Sydney. She was frustrated that she was unable to create a functioning program, compounded by the fact that she did not have enough time to seek help from a peer or an instructor before the submission deadline. Sydney felt discouraged by this assignment, claiming that she was unsure she could achieve success in the MICA program. By consulting her notes from lab and collaborating with a peer, she was able to complete the final assignments successfully and gained some confidence in her programming ability. However, Sydney ended the course hesitant about MICA II, feeling significant doubts toward her ability to learn the new programming material.

Jim's Story

Jim began the MICA I course with a general disposition of curiosity and excitement. Though Jim had limited formal experience in programming, he had been raised by a mother employed in the computer science field and was often given the opportunity to experiment with technologies throughout his childhood. This informal understanding of programming helped Jim to feel excited at the opportunity to formally learn to code. Jim's openness towards learning programming was also reflected in his greater attitude towards learning mathematics. He believes that individuals must keep an open mind towards their mathematical abilities to avoid feeling prematurely defeated and be able to persevere through challenges.

For EO1, Jim first developed a conjecture by creatively representing prime numbers and exploring patterns amongst them. Unfortunately, Jim was unable to pursue this ambitious idea within the scope of the course and his basic programming knowledge, eventually settling for a doable yet challenging alternative conjecture, that of Pólya conjecture suggested by the course instructor. However, he expressed the desire to follow up on his conjecture, feeling confident that he could develop a program to explore it when he has greater knowledge of programming. Once his program was complete, he was pleased to see it running, but stated that he always had faith it would work. Jim encountered challenges while completing the remaining assignments and coursework but did not seem bothered by these setbacks. When Jim was limited by his beginner level knowledge of programming, he would conduct research or try to find an alternative method of solving his problems. He ended the course understanding the advantages of working within a 'progmatics' learning environment, only feeling disappointed that he was not able to do more. [START_REF] Dweck | Carol Dweck revisits the "growth mindset[END_REF] notes that mindset is a critical factor in determining students' attitudes towards learning and their level of achievement. Analysis of both Sydney's and Jim's accounts of their experiences provide evidence of their mindset upon entering and throughout the course. The following quotes, taken from Sydney's and Jim's baseline questionnaires, provide insight into their initial sentiments and mindsets towards learning at the beginning of MICA I: Question: This MATH 1P40 [MICA I] course has a significant component of (computer) programming. This makes me feel…

Comparison between two enactments of mindsets

Sydney: Nervous

Jim: Very confident Question: Because:

Sydney: I barely have any knowledge in programming

Jim: If I'm finally going to get a chance to learn this stuff, I'm all for it. I've never had problems in the past, and if anything, I was annoyed that I wasn't being taught enough.

Here, Jim describes his confidence and excitement about entering the course, explaining how he looked forward to learning the programming content. In contrast, however, Sydney highlights her nervousness resulting from her lack of programming knowledge, suggesting her initial confidence level in the MICA program was low. Notably, although Jim also had limited programming knowledge, it did not cause him to feel this same apprehension. This could indicate that he was open to challenges, possessing characteristics of a growth mindset as Dweck ( 2010) asserts.

There are several notable elements regarding both students' enacted mindsets as they experienced "progmatics" over the semester, first in terms of how they approached challenges. Throughout the course, Sydney demonstrates a desire to avoid tasks that present significant challenge. For EO1, Sydney chose a conjecture that was quite similar to a conjecture covered in one of the lab sessions (see Sydney's Story). In her EO1 follow-up interview, she explains that she did not investigate a more creative conjecture in order to avoid going "beyond [her] limits" with the coding. She also expressed that she was looking forward to EO2 due to its more specific nature, stating, "I hope it's going to be easy". Later in the course, Sydney noted that she enjoyed the graphing component of her EO3 because "it wasn't that tough".

Unlike Sydney, Jim was not afraid of taking on challenges during his time in MICA I. When asked how he selected his conjecture for his EO1, Jim explained how he tried to develop a complex conjecture by himself through creatively exploring a variety of mathematical concepts. Though his mathematical ideas could not be explored with his startup programing knowledge, and he ultimately explored a more feasible (yet challenging) conjecture as advised by his professor, he expressed interest in pursuing his ideas in the future. In his EO1 follow-up interview, Jim reflected on his initial idea, stating, "I don't think it is as hard as everyone makes it out to be, it just requires a bit of a different way of thinking". Similarly, in Jim's final assignment, he again expressed the desire to go beyond his level when developing his question and was disappointed that he did not yet have the required knowledge of programming to do more with his project.

Students' approaches to challenges provide a strong indication of their mindset towards both learning and intelligence. The fixed mindset approach was consistently evident in Sydney's data, as her actions and perspective routinely demonstrate how she chose to explore simpler problems and favoured assignments that were seemingly easier. In contrast, Jim routinely attempted to challenge himself when developing conjectures and mathematical questions, demonstrating a growth mindset approach.

A second notable point about Sydney's and Jim's enacted mindsets is how they overcame the challenges they encountered and the effects of these challenges on their attitudes towards their abilities. While Sydney was able to overcome difficulties in her first assignment with the help of a peer, her EO2 presented much more significant issues. Because of external time constraints, Sydney was unable to seek help from a peer or the course instructor, and was frustrated when trying to get her program to work. Due to her inability to create a functioning program, Sydney doubts her ability to continue in the MICA program,stating,"if [my EO2 program] goes poorly I don't think I want to continue". This doubt persisted until the end of the semester, despite greater success in her EO3 and EO4 tasks. When asked how she felt about taking the MICA II course next year, she responds, "I'm still hesitant… about the new things we will learn and if I can manage to understand them".

Jim's approach to setbacks was markedly different from Sydney's. Throughout the course, it seemed that he used challenges to improve his strategies for future assignments instead of allowing them to diminish his confidence. Though he experienced setbacks in his EO2, Jim reflected on this process and how those challenges helped him improve. When he was asked in his EO3 follow-up interview if he would do anything differently, he expressed the following:

Jim: I feel like in assignment two… halfway through I had stopped planning and kind of regretted that afterwards, as you kind of get lost in the code but… for this [EO3] project I planned fairly well, I didn't get lost… I'm pretty happy with the way I did it.

In EO3, Jim had to do a fair amount of debugging to get his program to work, but he did not seem frustrated and used clear strategies to get his program to work. When asked about the obstacles he faced in EO3, Jim stated, "with computer programming, you accept that you are going to run into some bugs, which is why you test it at different stages". Jim maintained this attitude when completing his final assignment. When asked about debugging his program in the EO4 follow-up interview, he responded, "It didn't work for a bit. I went back and looked at it and realized that I had forgotten something, but nothing really significant". At the end of the course, Jim did not face the same doubts as Sydney towards his ability to be successful in MICA II. When asked about moving forward, he claimed, "I think it would be a reasonable course to take next year".

This approach to assignments and challenges is linked to two broader themes of mindset noted by research in a variety of contexts (Boaler, 2016;[START_REF] Murphy | Dangers of a fixed mindset: Implications of self-theories research for computer science education[END_REF][START_REF] Cutts | Manipulating mindset to positively influence introductory programming performance[END_REF], underscoring much of Dweck's work. These contrasting mindset themes are labelled as the helpless pattern and the mastery-oriented pattern. Students that view intelligence as fixed are more likely to demonstrate a helpless response to significant challenges; whereas students with a growth mindset display a mastery-oriented response to setbacks. Students who exhibit the helpless pattern are easily defeated when faced with challenges, measuring their capability against the obstacles and significantly doubting their ability to succeed when faced with a difficult task. Conversely, students who display mastery-oriented responses remain motivated through setbacks, persevering through the problem and overcoming it by applying innovative strategies [START_REF] Diener | An analysis of learned helplessness: Continuous changes in performance, strategy, and achievement cognitions following failure[END_REF]). Sydney's and Jim's responses to the challenges they face could give further insight into how these patterns are enacted within a 'progmatics' learning environment. Sydney's response to challenges shows aspects of the helpless orientation stemming from her fixed mindset, as she is quickly overcome with doubt and discouraged by setbacks. In contrast, Jim was not deterred by mistakes and setbacks, which he perceived as a natural part of programming. Jim's ability to persevere and flourish when faced with obstacles illustrates a mastery-oriented response in accordance with his growth mindset.

Conclusion

This paper highlighted the enactment of two students' mindsets in the context of an introductory 'progmatics'-focused undergraduate course. Because of the inherent challenges in learning programming and necessity of resilience in the field of mathematics, having a growth mindset holds particular importance in a computer programming mathematical learning environment such as MICA I. The assignments and reflections of two MICA I students, Sydney and Jim, were compared to illustrate their differing mindsets towards their intelligence as they engage with 'progmatics' through legitimate peripheral participation [START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF]. As a result of their differing mindsets, Sydney finished the course feeling doubtful towards her ability to be successful in MICA II, while Jim maintained his confidence and expressed the desire to further develop his programs.

Considering the important role played by programming and mathematics in STEM jobs, it is critical that students are encouraged to pursue and achieve success in these fields. Our work provides a beginning of how we can explore students' mindsets as they learn using programming for mathematics investigations and applications.


Eeva Haataja
email: eeva.haataja@helsinki.fi

Visajaani Salonen
email: visajaani.salonen@helsinki.fi

Anu Laine
email: anu.laine@helsinki.fi

Miika Toivanen
email: miika.toivanen@helsinki.fi

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

Teacher-student eye contacts during whole-class instructions and small-group scaffolding: A case study with multiple mobile gaze trackers

Keywords: classroom orchestration, multiple gaze tracking, mathematical problem solving, teacher-student interaction

Recent gaze tracking research have illustrated patterns of teachers' visual attention in the classroom. However, the reciprocity of teacher-student interaction needs to be explored using multiple mobile gaze tracking. This descriptive case study is our first step to chart the role of dyadic eye contacts in the teacher-student interaction during collaborative mathematical problem solving. Our results indicate that, during mathematical problem solving, teacher gaze at student faces and student gaze at teacher face vary between whole-class instruction and small-group scaffolding moments. As a conclusion, we suggest that forming dyadic eye contact requires optimal interactional, personal, and environmental states, and this method can offer us fruitful information on the micro-level processes of teacher-student interaction.

Introduction

The teachers' complex task of orchestrating the classes and lessons consists of interaction on various social levels [START_REF] Prieto | Orchestration load indicators and patterns: In-the-wild studies using mobile eye-tracking[END_REF]. In the classroom, a great deal of teacher's visual attention focuses on her students [START_REF] Mcintyre | Are you looking to teach? Cultural, temporal and dynamic insights into expert teacher gaze[END_REF]. The teacher's visual attention is affected by the social interaction in the classroom [START_REF] Prieto | Orchestration load indicators and patterns: In-the-wild studies using mobile eye-tracking[END_REF].

Recent research have illustrated the patterns and characteristics of teachers' visual attention during instructing the students (cf. Haataja et al., 2018a;McIntyre, Jarodzka, & Klasse, 2017;[START_REF] Prieto | Orchestration load indicators and patterns: In-the-wild studies using mobile eye-tracking[END_REF]. This study brings new dimensions to this discussion by using multiple gaze tracking between the teacher and the students in the context of collaborative problem solving. We claim that connecting teacher's gaze at student face and student's gaze at teacher face in authentic classroom contexts reveals us unexplored micro-level information on teacher-student interaction.

Classroom interaction as whole-class instruction and small-group scaffolding

Teacher actions of classroom orchestration vary from whole-class instructing to guiding an individual student [START_REF] Prieto | Orchestration load indicators and patterns: In-the-wild studies using mobile eye-tracking[END_REF]. In whole-class interaction, the teacher's visual attention distributes unevenly between the students (Dessus, Cosnefroy & Luengo, 2016). Gaze tracking research has shown that instructing the class as a whole and paying attention at students' faces increase the cognitive load of a teacher. Cognitive load decreases when the teacher focuses on single student's solutions or his own notes [START_REF] Prieto | Orchestration load indicators and patterns: In-the-wild studies using mobile eye-tracking[END_REF]. Cognitive load increases person's perseverance and experiences of positive emotions during the task [START_REF] Maranges | Comparing cognitive load and self-regulatory depletion: Effects on emotions and cognitions[END_REF].

In this study, we investigate two forms of teacher-student interaction during collaborative problem solving: whole-class instructing and small-group scaffolding. Scaffolding refers to the teacher's actions in a student group during a problem-solving learning process [START_REF] Wood | The role of tutoring in problem solving[END_REF]. The content and the direction of scaffolding influence the learning and the social interaction of the group. The teachers offer scaffolding on both mathematical contents and procedures and on peer interaction [START_REF] Akkus | Examining teachers' struggles as they attempt to implement dialogical interaction as part of promoting mathematical reasoning within their classrooms[END_REF][START_REF] Ding | Teacher interventions in cooperative-learning mathematics classes[END_REF]. Our previous analysis on the data of this study showed that the teacher's intention in scaffolding interaction directs her visual attention mostly between mathematical solutions, student faces and hands (Haataja et al., 2018a;Haataja, Garcia Moreno-Esteva, Toivanen, & Hannula, 2018b). The teacher's role in supporting beneficial collaboration and successful discourse within the class is crucial [START_REF] Cross | Creating optimal mathematics learning environments: Combining argumentation and writing to enhance achievement[END_REF].

Eye contact in teacher-student interaction

In social interaction, eye contact conveys warmth and communion on one hand, and agency and status on the other [START_REF] Mehrabian | Nonverbal communication[END_REF]. The occurrence of eye contact is found to be dependent of various variables. Other person's friendly response to eye contact initiative increases the experienced liking of her [START_REF] Frischen | Gaze Cueing of Attention: Visual Attention, Social Cognition, and Individual Differences[END_REF]. Teacher's eye contact, together with her open body position, use of personal examples, and humor form warmth that encourages students into participation and interaction [START_REF] Roberts | The impact of teacher immediacy on student participation: An objective cross-disciplinary examination[END_REF]). Teacher's gaze towards students, while asking questions and listening to them, increases their experience of close interpersonal relationship with the teacher (McIntyre et al., 2017).

The teacher can also address her authority in the classroom with direct gaze at student face [START_REF] Mehrabian | Nonverbal communication[END_REF]. With eye contact, the teacher can communicate that the students are in the locus of her attention and instructions [START_REF] Adams | Eye behavior[END_REF]McIntyre et al., 2017). Additionally, an eye contact can convey the teacher's emotional state to students [START_REF] Zeki | The importance of non-verbal communication in classroom management[END_REF].

The shorter the distance between the teacher and student is, the higher is the quality and amount of visual interaction between them. Beyond five metres, the visual interaction decreases significantly, as the distance to the source of information distracts the attention [START_REF] Cardellino | Classroom environments: an experiential analysis of the pupil-teacher visual interaction in Uruguay[END_REF].

Research questions

The aim of this case study was to explore the general tendencies of teacher-student eye contacts during mathematical problem solving. Thus, our research question were:

1. How do one-way face-targeted gazes and dyadic eye contacts between teachers and students distribute during collaborative problem-solving phase of a mathematics lesson? 2. Does this distribution differ between whole-class instruction and scaffolding interactions?

Methods

The participants of this study were two Finnish mathematics teachers and their 9 th grade classes. Both classes included 19 students. The first teacher, called by pseudonym Joanne, was 39 years old and had 14 years of teaching experience at the time of the data collection. In Joanne's class, the target group students were four girls. The second teacher, Fred, was 30 years old with three years of teaching experience. In his class, the target group consisted of four boys. Both teachers and their students had volunteered for participating in this research. Written consent for data collection was inquired from the school principals, all the students in the classes, and the teachers.

We collected the data in both schools in May 2017. To record the interaction in the classrooms, we used three video cameras and several microphones. The teachers' and target students' visual attention was recorded using self-made mobile gaze tracking devices consisting of eye cameras, a scene camera and goggles (for further information, see [START_REF] Toivanen | Probabilistic approach to robust wearable gaze tracking[END_REF]. The laptops, that recorded the gaze-tracking data, were located in backpacks to enable moving around the classroom for the participants. After the lessons, we conducted stimulated recall interviews with the teachers and target students. Unfortunately, one student gaze data and Joanne's interview were missing due to technical problems.

During the data collection lessons, the students worked on a geometrical problem task. The goal of the problem solving process was to find the shortest way to connect vertices of a square. For this report, we have analyzed the collaborative phases of the lessons. During these phases (16 minutes in Joanne's and 18 minutes in Fred's class), the students worked on the problem task collaboratively. They were encouraged to discuss their ideas and solutions in small groups. In both classes, the four target students worked together as a collaboration group. Both teachers walked around the classrooms scaffolding student groups one-by-one. Additionally, they gave whole class instructions to orchestrate the course of the lesson.

To analyze the gaze data, we used ELAN software. Using gaze targets and dwell times as coding units, we annotated the gaze data during the collaborative phases of lessons. Dwell time means the duration of one gaze at a certain target, (Holmqvist et al., 2011) such as a student face, and offers us information on the distribution of a person's visual attention. Further analysis was conducted on IBM SPSS and Microsoft Excel. The face-targeted gazes from the same lessons were synchronized and categorized into four categories. The categories were 1) Teacher gaze at student face, 2) Teacher-started dyadic eye contact, 3) Student-started dyadic eye contact, and 4) Student gaze at teacher face. We chose this analysis to explore the amount of reciprocal nonverbal interaction between teachers and students and these categories to chart the initiative roles of this interaction. We also compared the gaze data with the transcribed stationary video data from the lessons to form qualitative and quantitative descriptions. From the classroom videos, we separated the moments of teacher instructions to the whole class from moments of scaffolding with collaboration groups of two to four students. In this report, we analyze these variables of face-targeted gazes and teacher interaction descriptively.

Results

The teacher-student interaction differed between Joanne's and Fred's lessons despite the similar problem task and lesson structure. Joanne's students seemed more motivated and the atmosphere in class was warm and supportive throughout the lesson. Fred's students expressed more negative emotions, but also enthusiasm and joy of learning. The Figure 1 shows all the indiviual facetargeted gazes across the collaborative phase in Joanne and Fred's lessons. During collaborative problem solving phase, Joanne instructed the whole class more than Fred. In these moments, Joanne gave the students information and instructions on the progress of the lesson. During Joanne's problem solving phase, 63% (n = 149) of gazes occurred during whole-class instructions, while in Fred's class the proportion was only 4% (n = 13). While instructing, Joanne looked at her student's faces 88 times, even though the instruction moments covered only four minutes of the 16-minute-long collaboration phase.

During whole-class instructions, the target students looked at Joanne's face 20 times, and dyadic eye contacts between students and Joanne occurred 41 times, thrice with the target group. Fred gazed at his students' faces only five times and they looked at him twice without eye contact. Dyadic eye contacts occurred six times between a student and Fred during whole-class instructions but none of these was with a target student.

In Fred's class, the proportion of dyadic eye contacts was higher during the short moments if whole class instructions (46%, n = 6) than in Joanne's class (27%, n = 41). When giving whole-class instructions, Joanne spoke with loud voice and stood in front of the class. In these moments, the students' tendency to create or respond to dyadic eye contacts with the teacher decreased. In Fred's class, the interaction during collaborative problem solving phase happened almost completely in scaffolding interaction with student groups and he did not seek for capturing the attention of the whole class while the students were working on the task.

Figure 2 presents the distribution of four types of face-targeted gazes between the teacher and target students. Again, the gazes are categorized according to the direction between the teachers and students, and compared between interactional categories of whole-class instructions and scaffolding. While scaffolding other small-groups than the target group, Fred (74%, n = 131) scanned the students with one-way teacher gaze more often than Joanne (41%, n = 15) and Joanne had more dyadic eye contacts with students in other groups (54%, n = 20) than Fred did (23%, n = 41). In general, Joanne's students were more motivated and less frustrated that Fred's students and he had to be more critical in the interaction with them. In Joanne's classroom, the interaction throughout the problem solving process was warm and immediate, while Fred's lesson included also negatively loaded communication. However, Fred's class also included students who were concentrated in solving the challenging task, and hence not in need of the teacher's scaffolding.

In Fred's class, also the teacher's emotional state seemed to affect his gaze behavior. In the beginning of the collaborative phase, the target students joked at Fred, and they were not enthusiastic to solve the problem throughout the lesson. The uncertainty and pauses in Fred's speech in this moment indicated that this event annoyed or embarrassed him and he avoided eye contacts with these students. After the confusion, Fred told the students to concentrate and continued scaffolding successfully. During scaffolding, three moments occurred, during which the target students tried to create an eye contact with Fred unsuccessfully. They directed several gazes (n = 14, 8, and 8) at Fred's face without receiving any response. During these moments, Fred guided them with verbal interaction but directed his gaze at student papers. Interview with him indicates that this aversion was more of pedagogical than purely emotional origin, as Fred wanted to calm the students and behave as an authority.

Discussion

Orchestrating collaborative instruction and classroom interaction is a complex and challenging task that is affected by various different factors [START_REF] Prieto | Orchestration load indicators and patterns: In-the-wild studies using mobile eye-tracking[END_REF]. The educational research field has rapidly adopted mobile gaze tracking as a method of data collection in real learning contexts. The use of multiple gaze tracking opens unexplored aspects of micro-level teacher-student interaction.

In this paper, we have moved towards capturing the social nature of teaching and learning by using multiple gaze trackers simultaneously.

To summarize the results, the teachers' gaze behavior seemed to vary more depending the interactional state than the students' did. The students looked their teacher in the face both during whole-class instruction, and when the teachers were scaffolding the small-groups. This seems to be a result of teachers' social status and high agency in the classroom [START_REF] Mehrabian | Nonverbal communication[END_REF]. Even though collaborative problem solving is a student-centered learning method, the teacher's contribution in orchestrating the learning process is essential [START_REF] Cross | Creating optimal mathematics learning environments: Combining argumentation and writing to enhance achievement[END_REF]. The students need teacher's guidance, advice, and confirmation, which she can offer through eye contact and verbal instruction.

However, the teachers, especially Joanne, focused on her students' faces during whole-class instruction moments. In these moments, she looked at her students with numeral, short gazes. This finding is in line with the literature on teacher orchestration load [START_REF] Prieto | Orchestration load indicators and patterns: In-the-wild studies using mobile eye-tracking[END_REF] and expertise (McIntyre et al., 2018). When speaking in front of the class, regardless of the content of the instructions, Joanne concentrated in addressing her message through direct gazes at student faces.

In Joanne's class, the target group was located closer to the front of the classroom, while in Fred's class the target students were seated in the back. As [START_REF] Cardellino | Classroom environments: an experiential analysis of the pupil-teacher visual interaction in Uruguay[END_REF] suggest, this may affect to the count of eye contacts between the students and the teachers. In Fred's class, no dyadic eye contacts were formed between him and the target students when Fred was standing in the front. Additionally, Fred's target students were not as motivated in solving the task as Joanne's were, and he may have unconsciously directed his instructions to those students he knew were waiting for them.

When changing the interactional state from whole class to a collaboration group, we found general characteristics in the gaze behaviors of teachers and students. The teachers were in the locus of students' attention in all interactional moments. While in whole class interaction the teacher's attention distributes unevenly (Dessus et al., 2016), we interpret that in small-group scaffolding moments all kinds of students had an opportunity to be seen and heard. Despite their misbehavior, the target students in Fred's class pursued his attention. Fred seemed to avoid dyadic eye contact with them after the confusions to act as an authority and direct their attention to the learning task.

Naturally, this exploratory study has certain limitations. First, the missing data (one student gaze recording and one teacher interview) and the differences in the counts of gazes between the lessons hinder the comparative interpretations. However, this is a case study, and all the comparisons are qualitative by nature. Their purpose is to chart the phenomena for more statistical analysis in the future. For instance, in this report the student groups happened to be of same gender as their teachers. In this analysis, the attentional patterns were quite similar between the two lessons. In future, the effect of altering the gender structure of the participants might be relevant to investigate. The quality of the gaze and video data was very high, which helped us in making analysis and interpretations. Each recording was annotated by only one researcher. Nevertheless, the research group negotiated on controversial issues during the annotation and interpreted the data collaboratively.

This study shows that adding student gaze data to teacher gaze data reveals aspects that would not be possible to be explored with only one gaze tracker. For instance, students' unsuccessful attempts to create eye contacts with the teacher and vice versa are indicators of the complexity of teacherstudent verbal and nonverbal interaction. In future, we wish to examine more how teacher-student eye contacts are affected by aspects emerging in this analysis: the teacher and student personalities, methods of teacher-student interaction, and the physical learning environment.

Figure 1

 1 Figure 1 Face-targeted gazes during the collaborative problem solving sessions. The moments of whole-class instructions marked with blue squares and the phases of small-group scaffolding with green circles. The horizontal axis shows the beginning time of the gazes (seconds from the beginning of the lesson) with the frequency of 0.5 seconds and the vertical axis (1-4) the division into four categories: 1) teacher gaze at student face, 2) teacher-started dyadic eye contact, 3) student-started dyadic eye contact, and 4) student gaze at teacher face.

Figure 2

 2 Figure 2 The distribution of teacher and student face-targeted gazes during two collaborative problem solving session with teachers called Joanne and Fred.During scaffolding the target group, the eye contacts distributed similarly with both teachers. The observed four girls in Joanne's class and three boys in Fred's class gazed at their teachers more than other way around. While the teachers focused on one student's solutions or face at the time, the other three or two students often gazed at the teacher. Student gazes at teacher faces covered 44% (n = 22) of face-targeted gazes during scaffolding interaction in Joanne's class and 48% (n = 60) in Fred's class. The teachers seemed to have different intentions for the scaffolding interaction than the students. While the teachers' intention in the group varied between the problem-solving process and the motivational and emotional states of the students, students were willing to focus on the teachers as they arrived to the groups. The teachers' arriving to the group seemed to capture students' attention and pause their problem solving process. During verbal communication, the students often looked the teacher in the face even if the teacher was looking at and talking about the solutions.

Acknowledgments

We would like to thank Jessica Salminen-Saari and Tran Phuoc for their contribution in collecting, organizing and annotating the data. This research is a part of MathTrack project that is funded by the Academy of Finland (grant 297856).