Annette Rouleau
email: arouleau@sfu.ca

Natalia Ruiz
email: nruiz@dim.uchile.cl

Cristián Reyes
email: cristian.reyes@ciae.uchile.cl

Peter Liljedahl
email: liljedahl@sfu.ca²universidadde

²natalia Ruiz

First-person vicarious experiences as a mechanism for belief change

Keywords: Beliefs, teacher belief change, first-person vicarious experiences, problem solving

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Introduction

Underlying the heart of this study is the belief in the importance of problem solving in mathematics classrooms. Widely recognized as leading to a deeper understanding of key mathematical principles and content [START_REF] Pehkonen | On teaching problem solving in school mathematics[END_REF], problem solving has become an integral aspect of curricular reform [START_REF] Törner | Problem solving around the world: Summing up the state of the art[END_REF]. This has led to an emphasis not only on the teaching of problem solving but on the teaching of mathematics through problem solving [START_REF] Liljedahl | Problem solving in mathematics education[END_REF]). Yet ours is not a study of problem solving in mathematics. Rather, it is a study of belief changes that occurred because of the implementation of problem solving.

When the Chilean government, in response to deepening concerns over the low results of their students in international standardized mathematics assessments, decided to reform the country's mathematics curriculum, it was the belief in the importance of problem solving that provided direction. Problem solving was seen as integral and with that came the mandate that it be implemented in Chilean primary and secondary classrooms where, prior to this, it had been practically non-existent. (Felmer & Perdomo-Diaz, 2017). It was into this context that ARPA (Activando la Resolución de Problemas en las Aulas/Activating Problem Solving in Classrooms ) was born. Developed in response to the mathematics curriculum reform, ARPA is a research and development initiative that seeks to implement teacher professional development (PD) strategies that promote problem solving in an integrated classroom setting (Felmer & Perdomo-Diaz, 2017). Organized as a series of workshops held over the course of 10 months and led by trained monitors, ARPA is based on principles of teachers doing and reflecting. As such, its initial focus is the development of the teachers' own mathematical skills through collaborative problem-solving experiences that incorporate non-routine problems and randomized groupings (see [START_REF] Liljedahl | Building thinking classrooms: Conditions for problem solving[END_REF]. This leads to opportunities for the teachers to reflect on their own abilities, their mathematical knowledge, the strategies they used, and the emotions they experienced. This gradually moves to the teachers preparing and implementing analogous collaborative problem-solving activities in their classrooms for all their students. In our study we outline how this reform, predicated on the belief of the importance problem solving in mathematics, resulted in a cascading effect that was felt at the micro level of the classroom. In particular, we focus on the phenomena of belief changes experienced by two teachers as a result of the imposed implementation.

Framing Beliefs

Writing on beliefs requires treading on shifting sands. One of its challenges is the lack of consensus regarding an accepted definition [START_REF] Skott | Classroom practice and teachers' knowledge, beliefs and identity[END_REF]. Noting the difficulty in reaching a definition that is acceptable across all types of studies and disciplines, [START_REF] Mcleod | Synthesis -Beliefs and mathematics education: Implications for learning, teaching, and research[END_REF] suggest, "There is no single definition of the term "belief" that is correct and true, but several types of definitions that are illuminative in different situations" (p. 118). From [START_REF] Leatham | Viewing mathematics teachers' beliefs as sensible systems[END_REF] comes a view of beliefs as anything an individual regards to be true. His sensible system framework for understanding beliefs assumes "that what one believes influences what one does" (p. 92). With this framing of beliefs comes the understanding that beliefs cannot be directly observed but must be inferred. Following [START_REF] Leatham | Viewing mathematics teachers' beliefs as sensible systems[END_REF], for the purposes of this study, we call beliefs those things we just hold to be true and must be inferred from what we say and do.

The second area where writing on beliefs requires careful attention is the realm of stability of beliefs. As [START_REF] Liljedahl | Stability of beliefs in mathematics education: A critical analysis[END_REF] argue, "The field of mathematics education has assumed for too long that stability is an inherent and definable characteristic of beliefs" (p. 101). Their meta-study of beliefs in mathematics education found that stability has many meanings, "from difficult to change, to slow to change, to resistant to change" (p. 112). Further impacting our understanding of the stability of beliefs is [START_REF] Green | The activities of teaching[END_REF] stress on distinguishing what we believe and how we believe it. He suggests that beliefs that are held evidentially are less resistant to change than are beliefs held nonevidentially. By this he is referring to the basis on which the belief is held-with or without regards to evidence. For [START_REF] Green | The activities of teaching[END_REF], beliefs held on the basis of evidence or reason "can be rationally criticized and therefore can be modified in the light of further evidence or better reasons" (p. 48). For example, a teacher may hold an evidentiary belief that group work is untenable because she tried it unsuccessfully once. There is potential to change her belief by providing her with a positive experience with group work. Following [START_REF] Liljedahl | Stability of beliefs in mathematics education: A critical analysis[END_REF] and [START_REF] Green | The activities of teaching[END_REF], we propose a further understanding of beliefs as changeable truths that must be inferred from what one both says and does.

Models of Change in Belief and Change in Practice

Early models proposed that changes in beliefs would lead to change in practice (see Figure 1). Accordingly, teacher PD models focused on occasioning belief changes as a change in belief was seen as a necessary precursor to change in practice (see Philipou & Christou, 2002). However, this does not capture what happened in our phenomenon. As we shall detail later, what we suggest happened for our participants is that changes in practice led to changes in beliefs (see Figure 2). To better understand this phenomenon, we turn to [START_REF] Guskey | Staff development and the process of teacher change[END_REF] who, noting the abysmal success of PD programs, proposed an alternative model. Arguing that the previous models failed to take into account the process by which change in teachers typically occurs, he proposed a model in which the elements were reordered around student outcomes (see Figure 3). His model suggests that significant change in teachers' beliefs occurs primarily after they gain evidence of improvement in student learning, which for Guskey can include not only cognitive achievement but also elements such as behaviour or attendance. These improvements typically result from changes teachers have made in their classroom practices-a new instructional approach, the use of new resources, or simply a modification in teaching procedures. It is important to note that, for Guskey, it is not the PD per se, but the experience of successful implementation that changes teachers' beliefs. They believe it works because they have seen it work in their students, and that experience shapes their beliefs. Equally important is Guskey's view of change as gradual and difficult-new practices require incremental implementation.

Liljedahl ( 2016) also talks about this phenomenon of a change of practice leading to change of belief. He differs from [START_REF] Guskey | Staff development and the process of teacher change[END_REF] in that the participants are asked to implement a practice for which they have no personal certainty will work. In doing so, this method bypasses the classroom norms that frequently act as an impediment to change in practice [START_REF] Liljedahl | Building thinking classrooms: Conditions for problem solving[END_REF]. Liljedahl came to call this a first-person vicarious experience:

They are first person because they are living the lesson and observing the results created by their own hands. But the methods are not their own. There has been no time to assimilate them into their own repertoire of practice or into the schema of how they construct meaningful practice.

They simply experienced the methods as learners and then were asked to immediately implement them as teachers. As such, they experienced a different way in which their classroom could look and how their students could behave. (p. 384)

For example, in order to promote the practice of student collaboration, Liljedahl had to first bypass the classroom norm that the doing of mathematics was an individual pursuit. To that end, he had teachers work on problem-solving tasks in visibly random groups during a 90-minute PD session. At its conclusion, the participants were required to introduce visibly random grouping in their classrooms. The result was a significant uptake in the practice that Liljedahl attributes to two factors: (1) the ease of modelling it in a PD setting and (2) the teachers having personally experienced the impact on their own learning when visibly randomly grouped.

Similarly to [START_REF] Liljedahl | Building thinking classrooms: Conditions for problem solving[END_REF], the ARPA problem-solving initiative also incorporates [START_REF] Guskey | Staff development and the process of teacher change[END_REF] notion of changing practice to change beliefs-the teachers in our study were required to implement changes in practice which resulted in changes of beliefs. We were curious about the mechanism behind that change and this curiosity led to our research question: Can we understand their belief changes through the lenses of [START_REF] Guskey | Staff development and the process of teacher change[END_REF] and [START_REF] Liljedahl | Building thinking classrooms: Conditions for problem solving[END_REF]?

Methodological Considerations

While the difficulty in ascertaining beliefs is generally acknowledged [START_REF] Skott | Classroom practice and teachers' knowledge, beliefs and identity[END_REF], it is argued that qualitative studies have much to offer in the study of beliefs in that they offer a deep understanding of the ways in which people develop and change their beliefs [START_REF] Olafson | Qualitative approaches to studying teachers' beliefs[END_REF]. For the purposes of our study, talking with the participants and allowing them to tell their stories provided a rich description of the phenomenon and allowed us to respond to our research question regarding the nature of their changes. Note, ours is a small-scale study meant to document the occurrence of a phenomenon rather than its prevalence.

The source of our data was interviews with two participants: Luisa, a primary education teacher who was teaching in a Chilean public school at the time of the study and Josefa, a special education teacher who was assigned to a student in Luisa's fifth-grade classroom. These interviews were originally conducted as part of a larger project that studied the problem-solving experiences of a child with special needs. During their separate interviews, her teachers were also asked a series of questions intended to elicit their perceptions of changes experienced as a result of the problemsolving implementation. Ranging from 40 to 60 minutes in length, the interviews were audio recorded and transcribed in their entirety in Spanish. As this study is an international collaboration conducted by researchers from Canada and Chile, the transcripts were subsequently translated from Spanish to English by a researcher at the University of Chile and later reviewed by another researcher at the University. When first analyzing the interviews, we realized that not only was there a change in the student, but also in her teachers. So, guided by [START_REF] Guskey | Staff development and the process of teacher change[END_REF] and [START_REF] Liljedahl | Building thinking classrooms: Conditions for problem solving[END_REF], we reanalyzed the transcripts for evidence of changes in practice. For example, one teacher detailed the change in how she offered a student support in terms of 'before and after'. In keeping with our understanding of beliefs as inferable truths, we attended to those before and after descriptions to establish the teachers' beliefs prior to the change in practice and the resultant new belief.

Findings

In the following we present two situations, which best exemplify instances in which we noted that an imposed change in practice led to change in beliefs. Each analysis begins with a summation of the original belief and the circumstances surrounding its change.

Change in teachers' beliefs regarding integration in mathematics classrooms

Prior to participating in ARPA, Luisa and Josefa's students with special education needs were usually taught mathematics using individual materials prepared by Josefa, the special education teacher, in a resource room isolated from the classroom. While both supported integration in theory, they were uncertain whether it could be successful in their classroom. This is common among mathematics teachers with 80% (n = 228) holding positive beliefs regarding integration yet less than one-third believing they possessed useful philosophies or strategies to implement its practice (DeSimone & Parmer, 2006). For Josefa and Luisa, this changed when, as part of their commitment to the ARPA program, they prepared and delivered a multi-level problem-solving lesson. Their success at meeting the needs of students with varying levels of ability in this lesson changed their beliefs regarding their own ability to successfully integrate their students. Their theoretical belief in the importance of integration was now reflected in their practice where they moved to a full inclusion model.

One of the expectations of teachers participating in ARPA is preparing lessons incorporating the strategies they have been introduced to in ARPA. At the urging of their ARPA monitor, Luisa and Josefa decided to prepare a public problem-solving lesson. Unique to this lesson was that it was designed to be multi-grade and open to observation by the community. Teachers are sometimes hesitant to prepare materials for multi-level students as they believe the wide range of needs and abilities may be an obstacle to success [START_REF] Kiely | Teachers' beliefs about students with special needs and inclusion[END_REF]. Luisa and Josefa were no exception to this belief as we see in this excerpt from Josefa: "We were worried about the reactions of the girls to working with younger girls". Luisa was worried in particular about the students with special education needs and felt unsure whether they would be able to contribute effectively in this type of group dynamic. She was especially concerned for one girl saying, "Maybe she won't to be able to talk, because all the other girls do not know her, because they are from second and eighth grade". It was concerns such as these that had prevented both teachers from implementing the integration practices that they valued. It is not that the teachers opposed integration, rather they struggled to believe it could be effectively implemented, as Josefa states,

There is this integration and diversity discourse, theoretical stuff, but how do we face it? There is a lot of talk about how everybody is different, that we must work with our diversity, but what I used to do was to bring the girls [with special education needs] to the resources classroom. I'd take them out of the classroom and give them their individual material, that I created and used.

However, despite their uncertainty, they prepared a task in which students from grades 2, 5, and 8 all worked in visibly random groups within their grade levels on the same problem. Both teachers expressed satisfaction with the outcome of the public lesson with Josefa noting, "And, you know what? Surprisingly, it turned out very well". They had observed the students successfully working and learning together regardless of age or ability. Their general fears for the success of the students were eased by observing the students respond to each other as Josefa explains, "For example, when the fifth-grade girls were in front of the class explaining their solutions, the eighth-grade girls would listen and say: "Hey, I didn't think about that"". Their particular fears for their special education students were negated as they watched them be included in the groups and share their contributions. Although initially reluctant to implement this public lesson, its success impacted their beliefs about their own classroom practice with Josefa remarking, "If we can prepare a lesson for three different [grade] levels, shouldn't we be able to do that inside the classroom? I mean, it was empirically demonstrated that we can-when you talk about diversity in the classroom-it is obvious that we can do it".

Concerning the initial inconsistency between the professed beliefs of the teachers in our study and their actual practice, [START_REF] Leatham | Viewing mathematics teachers' beliefs as sensible systems[END_REF] notes that we must look deeper, "for we must have either misunderstood the implications of that belief, or some other belief took precedence in that particular situation" (p. 95). In our study, we found the latter-while both teachers believed that integration of students with special education needs was important, they also held a stronger belief that it only works in theory, which possibly prevented them from enacting their belief in practice. Their involvement with ARPA and its expectation that they integrate all students in lessons offered a firstperson vicarious experience that changed their belief regarding the difficulties of integration. This allowed for a match between their professed belief and their actual classroom practice.

Change in teachers' beliefs about low-performing mathematics students

We view the teachers' beliefs regarding integration as being contained within a cluster of beliefs that also comprised beliefs regarding low-performing students' ability to work successfully in collaboration on non-routine problems. Most simply put, they did not believe it was possible. They believed that low-performing and/or students with special needs would be unable to cope with the vagaries of group dynamics and have little to offer in the way of solutions for non-routine problems. Despite these beliefs, and with the encouragement of their ARPA monitor, they changed their mathematics practice to include all students in visibly random groupings that worked collaboratively on non-routine problems. The results surprised them. The students were not only accepted within their groups, they were viewed as contributing members. Observing the success of their students led both Josefa and Luisa to change their beliefs regarding their students' capabilities.

Luisa recalls that when the ARPA monitor suggested, "Those children who have more difficulties are the first to solve the problems", both her and Josefa's initial response was a disbelieving, "Yeah, hopefully". Similarly, they held low expectations for successful collaboration, particularly for Cristina, a student with special education needs. As Luisa explains, "Well, I thought her [Cristina's] participation and the acceptance of her classmates would be minimal". However, with little expectation of success, they implemented the non-routine problems and visibly random grouping recommended by their ARPA monitor. The teachers were pleased with the results. In particular, Josefa shares that Cristina, the student with special needs, "has improved a lot. She can solve a problematic situation and face it, read it and look for solutions. We suggest that it was the change in the teachers' beliefs regarding their students' ability to work on non-routine problems in visibly random groupings that allowed the change in the belief regarding integration to occur. The teachers believed in integration, but until now did not believe they could successfully implement it. To enact their beliefs, the teachers needed the necessary pedagogical knowledge of how to implement practices that support integration [START_REF] Buehl | The relationship between teachers' beliefs and teachers' practices[END_REF]. In speaking of the difference that visibly random grouping and non-routine problems made in the classroom, Josefa explains that "Those things [visibly random groups and non-routine problems] make the things you want to happen, happen". We can infer from this that, within their belief cluster, the successful implementation of integration practices is positioned as an overarching belief-it is the 'thing' they want to make happen. And it happened because ARPA's requirement that the teachers implement specific practices allowed them to observe positive results in their students. This changed their beliefs regarding students' abilities, which in turn allowed them to change their belief of their own ability to integrate. And as Josefa explains, "That was a process we had to go through. But it was very good for us".

Discussion and Conclusion

Although not all imposed changes in practice result in changes in beliefs, our participants implemented imposed practices that did change their beliefs. As our data was collected post-hoc, we did not directly observe the changes nor were we able to conduct pre-interviews. However, we were able to infer that, after being asked to prepare and teach a public problem-solving lesson, the teachers changed their beliefs regarding their ability to effectively integrate children with special education needs and their belief regarding the ability of low-performing students. Fitting with [START_REF] Guskey | Staff development and the process of teacher change[END_REF], a partial answer to our research question then, is that their beliefs changed after noting evidence of improvement in student learning. They believed it works because they saw it work and that experience shaped their beliefs. Not fitting with [START_REF] Guskey | Staff development and the process of teacher change[END_REF], however is how the belief change occurred. [START_REF] Guskey | Staff development and the process of teacher change[END_REF] views change as gradual and difficult with new practices requiring slow implementation. Our participants experienced something quite different. There was no gradual assimilation-they were required to change their practice even if they were uncertain about its effectiveness. They were given personal assurances that it would work from their ARPA monitors, but they were not given time to assimilate the methods into their own schemas. They had simply experienced the changes themselves as learners in the ARPA sessions where they were exposed to collaborative problem-solving and then been required to implement the new practice.

They had undergone a first-person vicarious experience [START_REF] Liljedahl | Building thinking classrooms: Conditions for problem solving[END_REF]. [START_REF] Leatham | Viewing mathematics teachers' beliefs as sensible systems[END_REF] reminds us that "The challenge for teacher education is not merely to influence what teachers believe-it is to influence how they believe it" (p. 100). To further answer our research question, the findings suggest that first-person vicarious experiences influence the howthe teachers now believe it experientially, which we argue, motivated the belief change.

Additionally, the PD aspect is vital as our participants needed to feel or see the changes themselves before their beliefs changed. They still had no certainty it would work in a different context, but they had experienced first-hand that it could. Our findings also suggest that first-person vicarious experiences are effective at altering evidential beliefs-those beliefs that are based on evidence or reason [START_REF] Green | The activities of teaching[END_REF]. For example, the teachers believed that integration only worked in theory because of their previous unsuccessful attempts, which only served to reinforce the belief. [START_REF] Buehl | The relationship between teachers' beliefs and teachers' practices[END_REF] would point out quite rightly that the teachers were lacking the pedagogical tools to enact their belief in integration. However, if this was all that was missing, teacher practice would be easy to change. We suggest that they were also lacking a first-person vicarious experience that provided the necessary evidence to change the belief. For our teachers, the success of their public lesson gave them new evidence that they could effectively integrate students and resulted in a change of beliefs.

  Figure 1Figure2

Figure 3

 3 Figure 3

  She doesn't sit still, she tries to solve it". Additionally, Cristina's peers valued her contributions and were willing to work with her in groups too. As Josefa notes, "Working in groups is like a second-nature to her and she can work in any group". Likewise, Luisa recalls the other students saying "'Look, she [Cristina] did it'". She goes on to add that "although Cristina wouldn't say "Girls, let's do it this way", she was making small contributions that were useful for what they were doing as a group". Similarly, the teachers observed the success of other low-performing students, like Daniela, who solved a difficult problem. As Josefa explains, She[Daniela] was present the class when we were teaching the problem of Theresa's floor tiles. Nobody in the class could find the solution. And it is a class where all the girls think they are very good at math. They are very competitive and this girl, Daniela, raises her hand and answers the problem. Luisa and I look at each other surprised.The teachers came to realize that low-performing students did not need simpler problems, nor special resources, but that they could solve challenging problems. As Josefa notes, And many times, we had simplifications, we'd plan a simple problem, or with more concrete support, or we'd plan problems with smaller numbers. And we realized it wasn't necessary. Do you get it? That helped us to realize that the students were more capable than we thought. So, you have a different disposition. And according to that, you start giving different support too, things that maybe you didn't even plan, but you go with the flow and you realize that they can do it.


Stanislaw Schukajlow
email: schukajlow@uni-muenster.de

Judith Blomberg

Johanna Rellensmann

I am scared to make a drawing. Students' anxiety and its relation to the use of drawings, modelling, and gender

Keywords: Anxiety, Emotions, Strategy, Drawing, Modelling problems

Emotions are important for students' learning and achievement. In the present paper, we report on a study of ninth-and tenth-graders (N=194) in which we investigated the relations between anxiety about strategy use, learning outcomes, and gender. We found that anxiety about making a drawing to solve a modelling problem was higher in female than in male students. Further, anxiety about strategy use was negatively related to different indicators of learning outcomes: strategic knowledge about drawing, number of drawings generated spontaneously while solving modelling problems, and modelling performance. Moreover, after we controlled for intra-mathematical performance, strategic knowledge about drawing, and gender, anxiety about self-generated drawing was negatively related to the use of drawings and modelling performance. This finding indicates the importance of anxiety about strategy use for learning and performance.

Introduction

Emotions are important for the learning of mathematics, mathematical achievement, a person's future career, and human well-being [START_REF] Schukajlow | Emotions and motivation in mathematics education: theoretical considerations and empirical contributions[END_REF]. One of the best-researched emotions is students' anxiety about mathematics. Students have been found to experience test anxiety often, and it can negatively influence students' test performance. However, much less is known about the impact of anxiety on students' strategies. In the present paper, we present a study carried out in the framework of the research project "Visualization while solving modelling problems" (ViMo). The main aim of the ViMo project is to investigate the strategy of learner-generated drawing and its relation to affective and cognitive learning factors. In the present study, we explored anxiety about making a drawing (anxiety about self-generated drawing) and its importance for students' learning outcomes. Our results contribute to clarifying the role of anxiety in strategy use, which was recently identified as an important research gap [START_REF] Ramirez | Math anxiety: Past research, promising interventions, and a new interpretation framework[END_REF].

Anxiety, drawing strategy, modelling, and gender

Anxiety as an achievement emotion

Definitions of emotions have emerged from different paradigms, such as the Darwinian, Freudian, or cognitive-psychological traditions [START_REF] Hannula | Emotions in Problem Solving[END_REF]. Emotions are typically defined as phenomena that include affective, cognitive, physiological, motivational, and expressive components. The components of anxiety are uneasiness and nervous feelings (affective component), worry (cognitive), avoidance motivation (motivational), anxious facial expressions (expressive), and peripheral physiological activation (physiological) [START_REF] Pekrun | The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice[END_REF]. In the control-value theory of achievement emotions, anxiety is considered to be a prospective emotion that is related to learning outcomes such as mathematical performance [START_REF] Pekrun | The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice[END_REF]. Control and value appraisals are assumed to be important for the emerging of emotions. Students feel test anxiety if they ascribe a high value to an exam and perceive themselves as not able to avoid failure on the exam. Similar to other affective constructs, anxiety has state and trait components that refer to the temporal stability of this emotion [START_REF] Schukajlow | Emotions and motivation in mathematics education: theoretical considerations and empirical contributions[END_REF]: If students are anxious while solving a specific problem, this is an emotional state; if they are disposed to often being anxious while solving a problem, this is an emotional trait. Another important characteristic of emotions is their object, which varies from more general objects (e.g. the learning of mathematics) to specific objects (e.g. a mathematical problem) [START_REF] Schukajlow | Emotions and motivation in mathematics education: theoretical considerations and empirical contributions[END_REF]. Mathematical activities (e.g. strategy use) can also serve as objects for emotions.

Self-generated drawing and modelling

Self-generated drawing describes the process and the product of generating an illustration that corresponds to the objects and relations described in a problem [START_REF] Rellensmann | Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students' mathematical modelling performance[END_REF], and it has been identified as an important strategy for problem solving [START_REF] Hembree | Experiments and relational studies in problem solving: a meta-analysis[END_REF]. Strategic knowledge makes part of the static knowledge component of metacognition as opposed to the dynamic process component of metacognition, which includes the application of strategies (strategy use). In a prior study, we investigated the importance of strategic knowledge about drawing for solving modelling problems. Modelling problems are problems with a connection to reality, whose solutions require demanding transfer processes between reality and mathematics [START_REF] Niss | Introduction[END_REF]. Modelling performance can be clearly distinguished from students' ability to solve problems without a connection to reality, called intra-mathematical performance. Students' strategic knowledge about drawing comprises students' views on the characteristics of a drawing that fits a given problem. This knowledge is positively related to students' performance in modelling problems that can be solved by using the Pythagorean theorem [START_REF] Rellensmann | Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students' mathematical modelling performance[END_REF]. Another important predictor of student performance is their strategy use. Students who spontaneously apply a drawing strategy were found to demonstrate higher performance in mathematics than students who did not apply the strategy [START_REF] Hembree | Experiments and relational studies in problem solving: a meta-analysis[END_REF]. However, this finding was not always confirmed when students were asked to construct a drawing. One reason why students do not always spontaneously generate drawings might be their affective perceptions of this strategy such as anxiety about making a drawing to solve a problem.

Anxiety, gender, and performance-related outcomes

Anxiety accompanies solitary problem solving processes [START_REF] Debellis | Affect and meta-affect in mathematical problem solving: A representational perspective[END_REF], and because working individually is important for gaining new knowledge, different contextual factors have been accessed to explore anxiety in mathematics. One such factor is students' gender. Most studies on anxiety have revealed that female students report a higher level of anxiety in mathematics than male students do (for an overview, see [START_REF] Ramirez | Math anxiety: Past research, promising interventions, and a new interpretation framework[END_REF]. To account for this finding, researchers have primarily discussed two reasons that are both related to stereotypes about gender and mathematics. One explanation refers to the hypothesis that female students report their true level of mathematical anxiety because they do not feel bad about having negative feelings toward mathematics, whereas male students try to repress their anxiety because of the stereotype that males have to be good at mathematics. Another explanation of the higher level of anxiety in women refers to the stereotype that women are worse at math than men. The crucial role of stereotypes for gender differences in anxiety is supported by the findings that gender differences disappear when students report their realtime (state) anxiety before, during, and after an exam, and they do not have time to reflect on their anxiety [START_REF] Goetz | Do girls really experience more anxiety in mathematics[END_REF]. Although these findings refer to anxiety about mathematics in general and not anxiety about the use of strategies for solving mathematical problems, we expect similar gender differences for anxiety about self-generated drawing while solving modelling problems.

Anxiety about the use of a specific strategy might be one factor that prevents students from applying this strategy while solving a problem. If a student fears failure when it comes to drawing, he or she might not try to generate a drawing, might solve the problem by using another less effective strategy, or might even give up at the very beginning. The relation between anxiety and students' strategies has rarely been investigated yet. We know that students' negative emotions (including anxiety) seem to impede their use of creative strategies [START_REF] Pekrun | The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice[END_REF] such as self-generated drawing. The use of inappropriate strategies is considered to be one factor that can explain why students with high levels of anxiety show poor mathematical performance in primary school [START_REF] Ramirez | On the relationship between math anxiety and math achievement in early elementary school: the role of problem solving strategies[END_REF]. However, the negative relation between anxiety and performance depends on students' cognitive abilities and is stronger for students with high working memory capacity. This finding indicates that anxiety about self-generated drawing might be negatively related to students' use of this strategy while solving modelling problems. A similar relation can be expected for strategic knowledge about drawing because students with high levels of anxiety about drawing tend to practice this strategy only rarely and do not have the opportunity to acquire advanced knowledge about this strategy. The relation between anxiety and modelling performance might be negative because anxious students do not make drawings to solve modelling problems. Further, students with high levels of anxiety might perform worse in modelling because their working memory might be overloaded with negative feelings, and they cannot focus on problem solving to the same degree as students who do not feel anxious.

Research questions

On the basis of prior research, we investigated the following research questions:

1. Does anxiety about self-generated drawing differ for female and male students? We hypothesized higher anxiety for female students. 2. Is anxiety about self-generated drawing related to strategic knowledge about drawing, use of drawings, and modelling performance? We expected a negative correlation between anxiety and these learning outcomes. 3. Is anxiety about self-generated drawing related to the use of drawings and modelling performance after strategic knowledge about drawing, intra-mathematical performance, and gender are controlled for? We expected that the relation would remain negative even after the strategic factor, the achievement factor, and gender were controlled for.

Method Sample and procedure

Two hundred twenty German ninth-and tenth-graders from ten classes in middle-and hightrack schools participated in the presents study (mean age 14.9 (SD = 0.64), 109 female students). On the first day of the study, among other questionnaires, students filled out a questionnaire on mathematical anxiety, age, and gender and took a test on strategic knowledge about drawing. On the second day, they worked on a modelling test and an intramathematical test. Students were not instructed to make a drawing. Their spontaneous use of drawings was coded by analyzing students' solutions.

Measures

Anxiety about self-generated drawing was assessed with a new Likert scale that we adapted from the anxiety scale from the Achievement Emotions Questionnaire [START_REF] Pekrun | Measuring emotions in students' learning and performance: The Achievement Emotions Questionnaire (AEQ)[END_REF]) by focusing on the items from the drawing strategy. It consisted of five items that ranged from 1 (not at all true) to 5 (completely true) and addressed emotional trait. Example: "When I make a drawing for a difficult word problem, I am very nervous." The reliability (Cronbach's ) was .860.

The strategic knowledge about drawing scale was developed and validated in a prior study (Rellensmann, Schukajlow, & Leopold, under review). It comprised eight real-world problems, each of which was followed by an item concerning situational drawings and an item concerning mathematical drawings (see Figure 1). On the 16-item test, students were asked to evaluate the utility of the drawings provided for each problem by comparing the drawings with regard to their helpfulness in solving the problem. Each item consisted of three drawings (a correct and complete drawing, a correct but incomplete drawing, and an incorrect drawing) that students rated on the scale from 1 (not helpful at all) to 5 (very helpful). The scores for each item ranged from 0 to 3, as each item offer 3 comparisons. The number of points depended on the sequence of drawings concerning their usefulness for solving the task. For example, if students identified that the correct and complete drawing was more helpful than the correct but incomplete drawing and that the correct and incomplete drawing was more helpful than the incorrect drawing, they were given 3 points (Cronbach's .762).

The modelling performance test comprised eight problems that could be solved by applying the Pythagorean theorem (for an example, see [START_REF] Schukajlow | Effects of prompting multiple solutions for modelling problems on students' performance[END_REF]. Students' solutions were scored by two raters on a scale ranging from 2 (correct problem solution) to 0 (incorrect solution resulting from an incorrect mathematical model or a missing solution). The inter-rater reliability (Cohen's κ) was >.81 for all modelling problems. Cronbach's reliability was .772.

Students' use of the drawing strategy was assessed via the number of drawings they constructed while solving the eight problems on the modelling test. If students made a drawing for a problem, they received a score of 1; if they did not make a drawing, they received a score of 0.

The intra-mathematical test comprised ten items on applying the Pythagorean theorem or solving quadratic equations (e.g., x 2 = 3.8 2 -2.5 2 ). They received 1 point for the correct solution and 0 points for an incorrect or missing solution. Cronbach's Reliability was .760.

Statistical analysis

To analyze the relations between the constructs, we used Pearson product moment correlations and partial correlations. Because students worked on the tests on two different days, some data were missing (3% and 8%). Thus, we applied listwise deletion for our analysis. This means that we excluded students from the analysis if they missed the first or second test session, and we performed the analysis on 194 students.

Results

First, we analyzed the differences in anxiety for female and male students. As expected on the basis of prior research, students' level of anxiety about self-generated drawing was significantly higher for female students (M = 2.12, SD = 0.96) than for male students (M = 1.67, SD = 0.63). About 11% of the female students and 18% of the male students denied feeling anxiety about drawing for all 5 statements. The medium effect size (Cohen's d = .56) indicates that women feel anxiety more often than men if they make a drawing while solving a demanding modelling problem.

The second research question was about the relation between anxiety about making a drawing and learning outcomes. The analysis of Pearson correlations revealed that students' anxiety about making a drawing was negatively related to all constructs measured in the present study (see Table 1). The effect size for the relation between anxiety about making a drawing and modelling was medium in size, and the effect sizes for the correlations between anxiety, use of drawings, and intra-mathematical performance were weak. Note. a p: one-tailed. SKD: strategic knowledge about drawing, UD: Use of drawings, IM: intramathematical performance, MOD: Modelling performance.

Table 1: Correlations between anxiety and the learning-and achievement-related measures

Students who felt anxious about making a drawing had lower levels of strategic knowledge about drawing, applied this strategy less often while solving modelling problems, and demonstrated poorer modelling performance.

The third research question focused on the analysis of the relation between anxiety about drawing, strategy use, and modelling performance after strategic knowledge about drawing, use of drawings, intra-mathematical performance, and gender were controlled for. The partial correlations confirmed a significant relation between anxiety and the use of drawings (r = -.161, p = -.026) and between anxiety and modelling performance (r = -.236, p < .001). The relations between anxiety about drawing and the use of drawings and modelling performance go beyond the contribution of the strategic factor (SKD), the cognitive factor (IM), and gender.

Discussion

In the present study, we explored the role of students' anxiety about self-generated drawing.

Our analyses confirmed the gender differences for anxiety about self-generated drawings that were found for mathematical anxiety in prior studies. Although mean values for anxiety about self-generated drawing were low, our results indicate that more than nine tenths of female students and more than one sixth of male students might be worried, feel uneasiness, or feel nervous while making a drawing for a difficult modelling problem. Giving prior findings about the importance of spontaneous drawings for problem solving [START_REF] Hembree | Experiments and relational studies in problem solving: a meta-analysis[END_REF] and the importance of drawings for solving modelling problems that can be solved by applying the Pythagorean theorem [START_REF] Rellensmann | Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students' mathematical modelling performance[END_REF], anxiety about using drawings seems to be an important construct that should be investigated in future studies. Moreover, female students might be disadvantaged while solving modelling problems because of their anxiety about using advanced strategies such as self-generated drawing. The main reason for gender differences in anxiety is considered to be social stereotypes that occur in the trait measures of anxiety. In future studies, it will be interesting to assess anxiety about self-generated drawing as a state, directly before or during problem solving, in order to clarify the stability of gender differences.

The analysis of the relations between anxiety about self-generated drawing and learning outcomes was the second main point of the present study. We hypothesized two ways in which anxiety about self-generated drawing can affect learning outcomes. First, students with high anxiety about making self-generated drawings might avoid using this strategy during problem solving because of their fear of failure, and they might look for other ways to solve the problem. The results of our analysis confirmed this hypothesis because we found a negative relation between anxiety and the number of drawings used while solving modelling problems. Moreover, a negative relation between anxiety and strategic knowledge about drawing indicates that students with high anxiety did not know the important characteristic features of good drawings. One reason for this negative relation might be that anxious students make drawings less frequently, and thus, they do not have many opportunities for improving this strategy. The relation between anxiety and the use of drawings while solving modelling problems remained negative after strategic knowledge about drawing, intramathematical performance, and gender were controlled for, indicating that affective constructs such as anxiety might influence the use of the strategies beyond strategic knowledge, general mathematical abilities, or gender. Second, anxiety is considered to have a negative influence on performance because it diminishes cognitive resources (e.g. working memory) while problem solving. If students are anxious during problem solving, they cannot focus on the entire problem and perform worse than students who do not feel this emotion. We found a negative correlation between anxiety about drawing and modelling performance that remained significant after strategic knowledge about drawing and intra-mathematical performance were controlled for. This finding might indicate the specific relevance of anxiety about self-generated drawing for students' modelling performance.

In sum, our findings revealed the importance of overcoming anxiety about the use of strategies. In the case of drawing, we suggest that students should learn how to apply this strategy in the classroom and to reflect on differences between more and less helpful drawings. Engagement with drawings should begin in primary school because anxiety about mathematics emerges in the early school years, and decreases performance in secondary school [START_REF] Ramirez | Math anxiety: Past research, promising interventions, and a new interpretation framework[END_REF].

Finally, we would like to point out one limitation of the present study. Although anxiety, students' drawings, and modelling performance were ordered along the timeline, we cannot draw conclusions about causal effects of anxiety on the use of drawings or on modelling performance. Prior research has revealed some evidence of reciprocal causation between emotions and achievement [START_REF] Putwain | Reciprocal relations between students' academic enjoyment, boredom, and achievement over time[END_REF]. In mathematics, however, the relation between emotions and performance has yet to be clarified. For example, students' prior performance in modelling did not influence enjoyment during lessons, whereas enjoyment during lessons affected modelling performance after mathematics lessons [START_REF] Schukajlow | The power of emotions: Can enjoyment and boredom explain the impact of individual preconditions and teaching methods on interest and performance in mathematics[END_REF]. We interpret this result as evidence of possible effects of anxiety about drawing on performance in modelling. In future studies, we suggest to pay more attention to the interaction of affective, strategical and cognitive factors while solving mathematical problems.

Figure 1 :

 1 Figure 1: An item from the knowledge about drawing scale (Rellensmann et al., under review)

Acknowledgments

The present study was funded by the German Research Foundation [Deutsche Forschungsgemeinschaft]. Grant SCHU 2629/3-1.


Ute Sproesser
email: sproesser@ph-heidelberg.de

Markus Vogel
email: vogel@ph-heidelberg.de

Tobias Doerfler
email: doerfler@ph-heidelberg.de

Andreas Eichler
email: eichler@mathematik.uni-kassel.de

Do students perceive mathematics and the mathematical subdomain of functions differently with regard to their self-concept and interest?

Keywords: Confirmative factor analysis, self-concept, interest, mathematics, functions

Motivational variables such as self-concept and interest accompany learning processes and hence are considered as relevant target variables of schooling. In order to evaluate effects of a teacher training on their students' domain-specific self-concept and interest, established mathematicsrelated scales for these variables were adapted to the domain of functions and administered in a pre-, post-, follow-up-design to a sample of 1177 secondary school students. Results of confirmative factor analyses in Mplus indicate that the scales mathematics-and functions-related self-concept and interest are empirically separable despite relatively large latent correlations for the functionsrelated scales in the pre-test. Together with satisfying values of Cronbach's alpha, this finding implies that the instrument can be used to reliably assess students' functions-related self-concept and interest and hence fulfills the requirements for further analysis steps.

Introduction

The project ProfiL9 of Heidelberg University of Education evaluates the effects of a teacher training about learning difficulties related to elementary functions (e.g., Sproesser, Vogel, Doerfler, & Eichler, 2017). The field of functions was chosen because of its relevance within mathematics (education) and its difficulty for many students (e. g. [START_REF] Selden | Research perspectives on conceptions of function: Summary and overview. The concept of function: Aspects of epistemology and pedagogy[END_REF]. Effects of this teacher training are to be evaluated both at the teacher and the student level. Beyond investigating students' competency development related to elementary functions in the course of the corresponding teaching unit, this study focuses on changes in students' self-concept and interest. In the context of assessing effects of teacher trainings at the student level, Yoon and colleagues (2007) state that the corresponding instruments have to meet high scientific standards. Hence, a reliable and valid instrument is mandatory. In order to get insight in particular cause-effect-relations, such a test instrument should target variables directly related to the subject of the teacher training, in the present case related to the mathematical subdomain of functions. This paper presents results of a confirmatory factor analysis showing that self-concept and interest for mathematics in general and functions in particular are empirically separable within our sample. Therefore, in the next section, we will introduce the theoretical background related to self-concept and interest. Afterwards, we will outline the methods of this study and finally present and discuss our findings.

Theoretical Background

Motivational dispositions such as self-concept and interest nowadays are considered to be relevant for learning (e.g., [START_REF] Schiefele | Interest, learning, and motivation[END_REF][START_REF] Valentine | The relation between self-beliefs and academic achievement: A meta-analytic review[END_REF] as they accompany learning processes and affect (the development of) students' achievement in and beyond school (e.g., [START_REF] Jacobs | Changes in children's self-competence and values: Gender and domain differences across grades one through twelve[END_REF]. Therefore, they represent important target variables also of mathematics schooling (e.g., [START_REF] Pekrun | Schülermerkmale im Fach Mathematik[END_REF]). In the following, the motivational variables self-concept and interest will be described in more detail. An individual's beliefs about the own abilities in a particular academic or non-academic domain are subsumed as self-concept (e.g., [START_REF] Bong | Academic self-concept and self-efficacy: How different are they really?[END_REF]. Among other self-belief constructs domain-specific self-concept can be considered as ranging between the more general self-esteem, i.e., an "emotional attitude towards the self" (ibid., p. 11) and the more particular self-efficacy expressing a conviction of how to perform in a concrete task. Self-concept is considered to be relatively stable and based on prior (achievement) experience in the corresponding domain (ibid.) on the one hand. On the other hand, there is a reciprocal relationship between self-concept and achievement: achievement in a certain domain is clearly influenced by prior self-concept levels (e.g., [START_REF] Calsyn | Self-concept of ability and perceived evaluation of others: Cause or effect of academic achievement[END_REF]. An example for domain-specific (academic) self-concept is mathematics-related self-concept, i.e., a person's confidence in his or her abilities in mathematics. Such domain-specific self-concept was found to be positively correlated to achievement in the corresponding domain (e.g., [START_REF] Baumert | Unterrichtsgestaltung, verständnisvolles Lernen und multiple Zielerreichung im Mathematik-und Physikunterricht der gymnasialen Oberstufe[END_REF], becausefor instancea positive selfconcept can support success-oriented behaviors like practicing [START_REF] Valentine | The relation between self-beliefs and academic achievement: A meta-analytic review[END_REF]. Because of the domain-specificity of self-concept, e.g. [START_REF] Marsh | Academic self-concept: Beyond the dustbowl[END_REF] recommend that researchers should measure this variable as closely connected as possible to the domain focused in the corresponding research.

According to Schiefele (1991, p. 302) individual interest is a "relatively enduring preference for certain topics, subject areas, or activities". Hencesimilarly to self-conceptindividual interest is domain-specific and it makes sense to assess it closely related to the corresponding domain. Moreover, Schiefele (ibid.) states that an interested person attributes a personal relevance to his or her target of interest (i.e., value-related valences) and that interest usually is accompanied by positive feelings such as involvement and enjoyment (i.e., feeling-related valences). Therefore, an interested person engages in an activity or subject for its own sake and makes efforts in order to learn about it. Various studies show that interest and achievement are correlated (e.g., [START_REF] Baumert | Unterrichtsgestaltung, verständnisvolles Lernen und multiple Zielerreichung im Mathematik-und Physikunterricht der gymnasialen Oberstufe[END_REF][START_REF] Helmke | Bedingungsfaktoren schulischer Leistungen[END_REF]. Schiefele (ibid.) gives several reasons for this relationship: For instance, interested students are more likely to employ meaning-oriented learning activities and they use more elaborated learning strategies. In the present study, we focus on the relatively stable (individual) interest for the domain of mathematics and functions as opposed to the more fluctuating situational interest (e.g., [START_REF] Hidi | The four-phase model of interest development[END_REF]) that won't be considered here.

In order to investigate certain cause-effect-relations, it makes sense to measure self-concept and interest particularly related to the focused domain (cf. [START_REF] Bong | Academic self-concept and self-efficacy: How different are they really?[END_REF][START_REF] Marsh | Academic self-concept: Beyond the dustbowl[END_REF][START_REF] Schiefele | Interest, learning, and motivation[END_REF]. This consideration does also reflect the finding that empirical correlations, e.g., between self-concept and achievement are found to be larger when these variables are measured as domain-specific as possible [START_REF] Hansford | The relationship between self and achievement / performance measures[END_REF]. As stated above, self-concept and interest differ between certain school subjects such as Mathematics and English. In this sense, it can be assumed that students can also hold different perceptions of, e.g., particular mathematical subdomains such as functions with regard to these motivational variables. However, this assumption still has to be verified empirically. Considering all combinatorial possibilities, the following alternatives appear to be particularly plausible: As mathematics and its subdomain of functions as well as self-concept and interest are closely connected, it is possible that students do not differentiate between these variables at all. A second possibility is that they discriminate between self-concept and interest but do not perceive mathematics distinct from functions as the latter represents a subdomain of mathematics. A third possibility is that students perceive mathematics distinct from functions but do not discriminate between self-concept and interest within each domain. A fourth possibility is that students discriminate between self-concept and interest within mathematics because they have long-time experiences with this subject. However, and due to less experience with the domain of functions, self-concept and interest related to this domain could be not distinguishable for them. The latter possibility of students being not able to distinguish between self-concept and interest related to functions particularly applies for the data collection before the teaching unit of functions when students were not familiar with this domain.

Research questions

Concerning the domain of mathematics, motivational variables have been examined by various studies including PISA and TIMSS; they document for instance considerable correlations between self-concept, interest and achievement (e.g., [START_REF] Baumert | Unterrichtsgestaltung, verständnisvolles Lernen und multiple Zielerreichung im Mathematik-und Physikunterricht der gymnasialen Oberstufe[END_REF][START_REF] Pekrun | Schülermerkmale im Fach Mathematik[END_REF]. However, to our best knowledge these variables have not been investigated specifically related to the mathematical subdomain of (elementary) functions and hence there are no approved instruments to reliably measure them. Taking into account that self-concept and interest are both domainspecific, it is reasonable to assess these variables related to the particular domain of functions (cf. [START_REF] Marsh | Academic self-concept: Beyond the dustbowl[END_REF] for investigating the above assumption that students can hold different perceptions of mathematics and functions with regard to their self-concept and interest. In this study, we investigate the following research questions:

 Are mathematics-and functions-related self-concept and interest empirically separable?  How reliable do the scales measure these variables?

Methods

Sample and context of the study

In this study, data from 1177 German students (45.6 % female) from 53 classes in 32 German secondary schools was analyzed. Due to missing values the concrete number of subjects varies for the different analyses and hence is indicated in the corresponding part of the Results Section. The classes were recruited among teachers that participated in the teacher training mentioned above as well as from schools that were in contact with the project contributors. In both cases, the participation of the classes and the individuals were voluntary. The students were aged between 11 and 17 years (M = 13.20, SD = 0.94). The large range of age is due to the fact that in Germany the teaching unit of linear functions is taught in different grades (mainly in grade 7 and 8) depending on the level of academic performance of the corresponding class or school, respectively. In all the participating classes, the assessment was done before (pre-test) and after (post-and follow-up-test) the teaching unit of linear functions as it focuses on the development of functions-related competence and motivational variables in the course of this teaching unit. As many students were not familiar with functions at the pre-test, they were told that the tasks focusing on interpreting graphs in the preceding competency test were examples for functions tasks. Moreover, they were given the option not to answer the functions-related items if they did not feel comfortable responding to them. In each class a regular teacher of the corresponding school as well as a researcher (the first author or a specifically trained assistant) were present during the testing.

Measures

Motivational variables referring to mathematics and functions were assessed using 5-point Likert scales. Mathematics-related self-concept (6 items) and interest (3 items) were measured through scales established by Pekrun and colleagues (e.g., 2007). In order to tap students' self-concept (6 items) and interest (3 items) related to functions, the mathematics-related items were adapted to the mathematical subdomain of functions. More information about the scales, e.g., sample items are displayed in Table 2 in the Results Section.

Analyses

The confirmatory factor analysis (research question 1) was conducted with the software Mplus 7.3 with type = complex which accounts for nested data in school classes. The implemented Full Information Maximum Likelihood (FIML) procedure estimates missing values except for cases with missing on all variables. The robust maximum likelihood (MLR) estimator was used which corrects for non-normality in the measures. For each factor, the regression weight of one item was fixed to 1.0 by default. In order to assess global model fit, we refer to the comparative fit index (CFI), the root mean squared error of approximation (RMSEA) and the standardized root mean squared residual (SRMR). [START_REF] Hu | Cutoff criteria for fit indexes in covariance structure analysis. Conventional criteria versus new alternatives[END_REF] suggest using a combination of cutoff values such as CFI ≥ 0.95, RMSEA ≤ 0.06 and SRMR ≤ 0.08 for a good model fit. As the χ² statistic is samplesize dependent the χ²/df ratio was additionally determined. According to [START_REF] Bollen | Testing structural equation models[END_REF] the model fits the data sufficiently when this ratio does not exceed 5. The different (nested) models are furthermore compared by the sample-size adjusted Bayesian Information Criteria (adj. BIC). Values for Cronbach's alpha (research question 2) were determined with SPSS 25. The structure, factor loadings and correlations of the full Model E are displayed in Figure 1 for the pre-test. Similar results were found for the post-and follow-up-test. The rather low factor loading of Item 2 referring to mathematics-related interest is comparable, respectively slightly higher for the post-(.59) and follow-up-test (.63). Analyzing Model E without this item did not lead to better fit indices. Although the four scales mathematics-and functions-related self-concept and interest turned out to be separable, they expose latent correlations of .44 to .88. The correlation between functions-related self-concept and interest is lower for the post-and follow-up-test (both .80).

Table 2 shows Cronbach's alpha in pre-, post-and follow-up-test (fut) for mathematics-and functions-related self-concept and interest ranging from .81 to .94. All in all, the values for the functions-related scales are comparable to the corresponding mathematics-related counterparts.

Discussion

The purpose of this study is to investigate whether student's self-concept and interest related to mathematics and functions are empirically separable. If students actually perceive mathematics and functions differently with regard to their self-concept and interest, it appears to be reasonable to assess these motivational variables as domain-specific as possible. Doing so is not only recommended by [START_REF] Marsh | Academic self-concept: Beyond the dustbowl[END_REF] but also enables investigating particular cause-effectrelations of the teacher training under evaluation in the project ProfiL9.

Although single global fit indices show slight deviations from the recommended cutoff values, they all favor the full Model E that differentiates between the four scales mathematics-and functionsrelated self-concept and interest. This finding is also supported for pre-, post-and follow-up-test by model comparisons based on the adjusted BIC. Moreover, the magnitude of the (significant) factor loadings as well as good values of Cronbach' alpha suggest that the latent variables are reliably measured by the corresponding items. In particular Cronbach's alpha is comparable or even better for the adapted scales than for the established ones. This means that the students of our sample perceive mathematics and functions differently from a motivational point of view and hence that it is reasonable to evaluate self-concept and interest with the adapted functions-related scales. In this sense, we will use these function-related scales in order to investigate particular cause-effectrelations in further analysis steps of our project. Developing and using domain-specific instruments appears to be applicable also for other studies investigating the effects of domain-specific interventions on motivational variables such as self-concept and interest.

At the pre-test, students were not familiar with functions. Therefore, it could have been assumed that the distinction between mathematics and functions and in particular between functions-related self-concept and interest would not have been perceptible for them (see Model D). However, the factor analysis favors the four-factor Model E already for the pre-test. Hence, students appear to differentiate between these variables even before they participated in the teaching unit of functions.

In any case, latent correlations of about .6 emphasize the relatedness between mathematics and functions. The magnitude of the correlations between self-concept and interest within each domain are in line with the literature and underline the proximity of these variables. The particularly high correlation between functions-related self-concept and interest found in the pre-test decreases in post-and follow-up-test to a level comparable to the mathematics-related counterparts. This is in line with our expectation that at the beginning of the teaching unit of functions students do not differentiate between these variables to the same extent as they do afterwards.

Although the presented results justify the use of the adapted instrument for further analyses such as investigating the development of student's functions-related self-concept and interest, the following limitations should not be disregarded. Some of the global fit indices still show potential for improvement and single factor loadings were lower than they should be. As the scale of mathematics-related interest only contains 3 items and as it proved in other research, we favor to maintain Item 2. However, we will further analyze if a model can be found that provides better fit indices. Concerning future analyses, the instrument will be used to examine how functions-related self-concept and interest develop for students with / without teacher participation in the teacher training mentioned above. In these analyses, covariates such as age or gender will also be introduced in order to monitor the effectiveness of the teacher training in more detail.

Figure 1 :

 1 Figure 1: Factorial structure and parameters of Model E (pre-test)

Table 2 : Reliabilities and sample-items of mathematics-and functions-related self-concept and interest

 2

	Construct	Number	Sample item	Cronbach's	Cronbach's	Cronbach's
		of items		Alpha pre	Alpha post	Alpha fut
	Self-concept	6	Understanding mathematics is	.92	.92	.93
	(mathematics)		easy for me.	(N = 811)	(N = 878)	(N = 674)
	Interest	3	Doing mathematics is one of my	.82	.81	.83
	(mathematics)		favorite activities.	(N = 831)	(N = 892)	(N = 683)
	Self-concept	6	The domain of functions is easy	.94	.94	.94
	(functions)		for me to understand.	(N = 675)	(N = 863)	(N = 677)
	Interest	3	I like best working on functions.	.83	.88	.89
	(functions)			(N = 745)	(N = 895)	(N = 701)

Acknowledgment

This study is supported by the Ministry of Science, Research and the Arts in Baden-Wuerttemberg (Germany) and by research funds from Heidelberg University of Education.

Results

Research question 1 investigates whether the mathematics-and functions-related scales were empirically separable within our sample. For this purpose, we specified several models with the items for mathematics-and functions-related self-concept and interest as stated above: Model A integrates all 18 items to a single factor. Model B assigns the items to the two factors self-concept (for mathematics and functions) versus interest (for mathematics and functions). Model C differentiates between mathematics (self-concept and interest) and functions (self-concept and interest). Model D contains three factors, namely self-concept for mathematics, interest for mathematics and a combining factor for functions (self-concept and interest). The full Model E incorporates the four factors mathematics-related self-concept, mathematics-related interest, functions-related self-concept and functions-related interest. Details about the fit of these models are displayed in Table 1 for the pre-, post-and follow-up-test. Concerning all times of testing, global fit indices (χ²/df, CFI, SRMR, RMSEA) indicate except for slight deviations that the full Model E fits the data reasonably whereas the other models do not meet most of the recommended thresholds (see Analysis Section). Similarly, model comparisons based on the adjusted BIC are in favor of Model E.


Hanna Viitala
email: hanna.viitala.edu@gmail.com

Low achiever's mathematical thinking: The case study of Maya

Keywords: View of mathematics, problem solving, metacognition, affect, meta-affect

In this paper, one low achiever's, Maya's, mathematical thinking is reported. Mathematical thinking is studied through problem solving and view of mathematics. The results indicate that Maya has severe gaps in mathematical knowledge and problem solving. Furthermore, affect hinders her learning and activities in mathematics. Despite the challenges, Maya wants to learn mathematics and succeeds in giving an answer to every mathematical problem in the study. In addition to further support in mathematics and affect, she could benefit from learning to use different representations in mathematics and picturing the tasks in real life.

Introduction

Finnish pupils' success in mathematics has been acknowledged in international studies such as PISA and TIMSS (see e.g. [START_REF] Välijärvi | Osaaminen kestävällä perustalla: Suomen PISA-tulosten kehitys vuosina 2000-2009[END_REF] and national assessments show that Finnish pupils' performance is at a satisfactory level (e.g. [START_REF] Hirvonen | Onko laskutaito laskussa? Matematiikan oppimistulokset peruskoulun päättövaiheessa 2011[END_REF]. However, the most recent studies indicate that Finnish pupils' performance is descending (ibid.; [START_REF] Rautopuro | Hyödyllinen pakkolasku. Matematiikan oppimistulokset peruskoulun päättövaiheessa 2012 [Useful forced landing[END_REF], showing also the diminishing number of high achievers and the growing share of low achievers in Finland [START_REF] Kupari | Matematiikan osaamisen taustatekijät [The background for mathematical performance[END_REF]. Furthermore, pupils' affect towards mathematics is alarmingly low at the end of lower secondary school [START_REF] Tuohilampi | Matematiikkaan liittyvien asenteiden kehitys sekä asenteiden ja osaamisen välinen vuorovaikutus 3., 6. ja 9. luokalla [The development of mathematics related attitudes and the interplay between attitudes and performance in 3rd, 6th and 9th grades[END_REF].

To understand the recent development, we need to know more about Finnish pupils' mathematical thinking. What is it like at the end of comprehensive school? Previously, Viitala (2017a) has reported on four Finnish high achieving pupils' mathematical thinking. These pupils all liked mathematics, were successful problem solvers and enjoyed learning mathematics. However, a deeper look into their problem solving and view of mathematics revealed very different skills and competences in mathematics. Nevertheless, whatever their strengths and weaknesses were, their strengths compensated their weaknesses in mathematics. How is the situation different with low achievers? What is it that separates low achievers from high achievers? In addition to differences in mathematical knowledge, what might be the possible reasons for the low performance in mathematics?

In this paper, one low achiever's, Maya's, mathematical thinking is reported. The data is collected through problem solving and view of mathematics. The purpose is to study Maya's strengths and weaknesses in mathematics that go beyond her mathematical knowledge. The purpose is to answer the question 'What characterizes Maya's mathematical thinking at the end of lower secondary school?'

Theoretical framework

Following the recent theories on affect, mathematical thinking can be viewed through two temporally different aspects; state and trait (cf. [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF]2012). While thinking is always situational and contextual, a state, it is also influenced by more stable constructs such as beliefs and attitudes, traits (cf. ibid.). These two temporally different sources of data together can reveal the dynamic and complex thinking processes that pupils go through while thinking mathematically (for an example, see e.g. [START_REF] Viitala | A tool for understanding pupils' mathematical thinking[END_REF].

In this study, the state aspect is studied through problem solving. The pupils' actions and explanations during problem solving are interpreted as visible signs and expressions of their mathematical thinking. In addition to the mathematical knowledge and heuristics in problem solving, pupils' problem-solving behaviour is influenced by their metacognition, affect and metaaffect that occur in a problem-solving situation (cf. [START_REF] Schoenfeld | Learning to think mathematically: Problem solving, metacognition and sense making in mathematics[END_REF].

The successful application of problem-solving activities at the correct moment is a result of metacognitive skilfulness (e.g. [START_REF] Schoenfeld | What's all the fuss about metacognition[END_REF][START_REF] Flavell | Metacognition and cognitive monitoring: A new area of cognitivedevelopmental inquiry[END_REF]. These skills include for instance monitoring one's actions and directing resources in a problem-solving situation [START_REF] Schoenfeld | What's all the fuss about metacognition[END_REF]. Affective state influences problem-solving activities for instance through the feeling of confidence, and meta-affect transforms individuals' feelings [START_REF] Debellis | Affect and meta-affect in mathematical problem solving: A representational perspective[END_REF] directing pupils' problem-solving behaviour [START_REF] Carlson | The cyclic nature of problem solving: An emergent multidimensional problem-solving framework[END_REF].

Problem-solving processes can show patterns of thought that can be interpreted as signs of more stable ways of thinking. These patterns, together with pupils' view of mathematics, represent the traits in this study. Pupils' view of mathematics is studied through five components: mathematics as a science and as a school subject, oneself as a learner and user of mathematics, learning mathematics, teaching mathematics [START_REF] Pehkonen | Pupils' view of mathematics: Initial report for an international comparison project[END_REF], and view of mathematical thinking.

The structure of pupils' view of mathematics is derived from belief-research that has traditionally been studied as traits (see e.g. Op't Eynde, [START_REF] Oecd ; Eynde | Assessing scientific, reading and mathematical literacy: A framework for PISA[END_REF]. However, in separation from many earlier studies, in this study pupils' view of mathematics is considered as psychological phenomenon that is a mixture of cognitive, motivational and emotional processes [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF]2012). In addition to beliefs, these processes include also for instance attitudes, values, feelings and motivation (ibid.; thus the word 'view', see [START_REF] Rösken | Dimensions of students' views of themselves as learners of mathematics[END_REF].

Methods

Data collection

At the time of data collection, Maya was 15 years old. She was in 9th and final year of Finnish comprehensive school. She was a low achiever in mathematics (mathematics grades between 5 and 6 on a whole number scale of 4 to 10).

The data was collected in three cycles over the course of three months. In each cycle, 1-2 mathematical problems (PISA tasks) were solved in an ordinary classroom situation. In class, Maya solved the tasks individually, but she could talk about the tasks with her friends and ask help from the teacher. In the third cycle, Maya was absent from class and solved the problems in the interview. Maya was video recorded while she solved the tasks and her solutions on paper were collected.

Here is an example of a PISA task used in data collection: 'This problem is about planning the best route for a holiday. Figures A and B show a map of the area and the distances between towns [in a table]. […] Calculate the shortest distance by road between Nuben and Kado.' (OECD, 2006, pp. 77-78; for a part of the task, see also Figure 1).

Maya was interviewed three times, on the same day or on the next day after each of the problemsolving lessons. The interviews were semi-structured and contained two parts. The first part concentrated on Maya's view of mathematics. It followed the categorisation introduced in theoretical framework: background, mathematics as a school subject and as a science, oneself as a learner of mathematics, learning mathematics, teaching mathematics (following [START_REF] Pehkonen | Pupils' view of mathematics: Initial report for an international comparison project[END_REF]cf. Op't Eynde et al., 2002), and view of mathematical thinking (for example questions, see Table 1). The second part concentrated on problem solving. The data from class was used as stimuli when Maya's problem-solving processes were discussed. She was asked to explain her thinking and actions during the problem-solving situation. Also supporting questions were asked (e.g. What are you thinking now? Why are you doing so?).

Theme Example question

Finally, in each of the three interviews, Maya was asked to assess her confidence before, during and after solving the problems, as well as to assess her confidence in school mathematics using a 10 cm line segment (scale from 'I couldn't do it at all' to 'I could do it perfectly'). All interviews were video recorded.

Data analysis

The analysis was divided into two sections: problem solving (state) and view of mathematics (trait). First, problem-solving processes were analysed through four problem-solving phases: orienting, planning, executing and checking [START_REF] Carlson | The cyclic nature of problem solving: An emergent multidimensional problem-solving framework[END_REF]. Then, the results were complemented with the metacognitive activities of orienting, planning, evaluating and elaborating (see examples in van der Stel, Veenman, Deelen, & Haenen, 2010), affect (cognitive, emotional and motivational states and traits; [START_REF] Hannula | The structure and dynamics of affect in mathematical thinking and learning[END_REF]2012) and meta-affect [START_REF] Debellis | Affect and meta-affect in mathematical problem solving: A representational perspective[END_REF] emerging in the problem-solving processes. Finally, Maya's confidence to solve the problems were analysed.

Second, Maya's view of mathematics was analysed following the themes of data collection [START_REF] Pehkonen | Pupils' view of mathematics: Initial report for an international comparison project[END_REF]. After data condensation, a pupil profile was created. Pupil profile is a short description of the pupil that is used as background information. It is based on Maya's mathematics grade, motivation to learn mathematics, and ability, success, difficulty, and enjoyment of mathematics also known as the core of herself as a learner of mathematics [START_REF] Rösken | Dimensions of students' views of themselves as learners of mathematics[END_REF].

In the end, the results from problem solving (state) and view of mathematics (trait) were compared and combined to get a holistic view of Maya's mathematical thinking.

Results

Pupil profile: Maya is a low-achieving, very unsure and quite anxious mathematics learner. She does not value mathematics as important, but would still like to learn it. Her low level of mathematical knowledge and skills go hand in hand with her low level of affect and meta-affective skills in mathematics.

The most prominent result of the study is the low level of affect that Maya has in and towards mathematics. When we begin discussing about mathematics, Maya starts by saying that she is very bad at mathematics, has always been. She then continues to express her low level of affect throughout the study: She explains that she has no idea what mathematics is as a science because she is very bad at mathematics; Mathematics is not important because she does not really know mathematics; She does not feel confident in mathematics; She feels anxious in mathematics classes and especially so in tests; She feels that she has never succeeded in a mathematics test; Or simply, mathematics is too difficult for her.

Maya has great difficulties also in solving problems. She felt stressed in problem-solving situations and the tasks felt difficult. Before she could start solving the problems, she needed confirmation from the teacher (or from the researcher in the interviews) that she had understood the problem correctly. Unfortunately, Maya is not initiative in asking help in class. She feels afraid of making mistakes, she is afraid that the teacher asks questions or gives critique, and she is afraid that her classmates notice her problems. This mismatch between the need for help and not asking for it, and consequently staying stuck, is just one example of the low level of meta-affective skills that Maya has in mathematics. Fortunately, though, when the teacher approaches her, she usually gets the help she needs to continue with the problem.

In addition to the low affect and meta-affective skills in mathematics, Maya's mathematical skills are quite limited. For example, in the problem introduced above, Maya had difficulties in reading the table and in adding. First, she did not understand what the numbers in the table represent before the teacher explained the table to her. Even then, she took the distances she needed from the first cell next to the town names (500 next to Nuben and 850 above Kado; see Figure 1). Maya explained that she did not understand the table until the interview.

After getting the distances she needed from the table, Maya had to add the numbers together. She explained that she could not add the numbers (500 and 850) mentally, so she calculated them on paper (for a reproduction of the calculation, see Figure 1). Maya's calculation was a mixture of adding and multiplying rules and the result was incorrect. In the interview, Maya explained that she took the first and last two numbers as the result and ignored the third and fourth numbers. Even though the method seems absurd, there is something positive: she had a realistic view of the magnitude of the answer.

There are severe problems in Maya's knowledge base and affect. Metacognitive actions are almost non-existent, also meta-affective actions, and she needs very much and very concrete help to solve the problems. However, there too is something positive: She does want to learn mathematics. She recognizes that for her own learning, she needs to follow teaching in mathematics classes and ask questions. She has noticed that when the teacher relates new knowledge to old knowledge, it helps her to learn mathematics. Even though she has emotional obstacles in communicating with her teacher, she seeks and gets more mathematical help from their special needs teacher. She emphasizes the importance of doing homework alone and undisturbed. And, she recognizes that she needs mathematics in future studies and jobs.

In mathematics learning, Maya needs help in developing her conceptual and procedural skills, and support to overcome many emotional issues. However, in connection to problem solving, there is one concrete thing that rose above other issues. When Maya got stuck in problem solving, she understood the problem best with a new representation. Moving the task to real life or drawing a picture was enough to help Maya to understand the problem. Moving between different representations in mathematics could help her to improve her problem-solving skills and her conceptual knowledge.

Discussion

This paper aimed at answering the question 'What characterizes Maya's mathematical thinking at the end of lower secondary school?' Maya is best characterized through the low level of mathematical knowledge and skills, and through the low level of affect towards mathematics. More specifically, the very negative view of herself in mathematics seem to hinder her learning and solving mathematical problems. This negative trait connected with the negative state reactions to the problems and problem-solving situations, as well as nearly non-existent meta-affective skills, almost prevented her to solve the problems in the study. The low level of mathematical and metacognitive skills support the situation.

In addition to the low level of mathematical knowledge, the low level of meta-affective and metacognitive skills seem to be what separates Maya most from the high-achieving students in the research project (see the high-achieving students' results in Viitala, 2017a). Being unsure and having negative affect towards mathematics are features that students in all achievement levels experience. In fact, the most recent national studies show that Finnish pupils' affect towards mathematics is unnecessary negative at the end of comprehensive school [START_REF] Tuohilampi | Matematiikkaan liittyvien asenteiden kehitys sekä asenteiden ja osaamisen välinen vuorovaikutus 3., 6. ja 9. luokalla [The development of mathematics related attitudes and the interplay between attitudes and performance in 3rd, 6th and 9th grades[END_REF][START_REF] Hirvonen | Onko laskutaito laskussa? Matematiikan oppimistulokset peruskoulun päättövaiheessa 2011[END_REF]. However, while Maya seem to paralyse when experiencing difficulties in solving mathematical tasks, the high-achievers use their meta-affective and metacognitive skills to overcome them (see e.g. Viitala, 2017a). What all students in the research project have in common, though, is the positive feeling they get when they succeed in mathematics, and the will to learn mathematics, on their own levels and on their own terms.

The national studies also show that pupils start school with positive attitudes, but their attitudes towards mathematics as well as self-efficacy beliefs decrease consistently throughout comprehensive school, and pupils' anxiety in mathematics increase in lower secondary school [START_REF] Tuohilampi | Matematiikkaan liittyvien asenteiden kehitys sekä asenteiden ja osaamisen välinen vuorovaikutus 3., 6. ja 9. luokalla [The development of mathematics related attitudes and the interplay between attitudes and performance in 3rd, 6th and 9th grades[END_REF][START_REF] Metsämuuronen | Changes in achievement in and attitude toward mathematics of the Finnish children from grade 0 to 9: A longitudinal study[END_REF]). Maya's affect fits well with the results of Finnish large-scale studies at the end of comprehensive school. However, the results indicate that the situation has not been positive even at the beginning of her school years. For instance, Maya states that she has always been bad at mathematics and she cannot remember ever succeeding in a mathematics test. Having a different mathematics teacher in every grade might also been a factor in Maya's learning of mathematics: While Maya is shy and very unsure of herself in mathematics, it takes time for the teacher to get to know her thinking so that the teacher could have the best tools to help Maya to develop mathematically.

So, what can be done to help pupils like Maya first to create and then to maintain more positive attitude towards mathematics, and to build mathematical knowledge and skills at their own rate? One solution could be to increase the amount of problem solving in mathematics classes. Even one monthly problem-solving lesson has shown to slow down the decreasing trend in girls' attitudes towards mathematics [START_REF] Tuohilampi | The restricted yet crucial impact of an intervention on pupil's mathematics-related affect[END_REF]. Problem solving is also an effective tool for the teacher to learn about pupils' mathematical knowledge and thinking. Additionally, referring to real-life situation helped Maya to solve problems. Problem solving and applying mathematics to real-life situations are written as learning goals also in the Finnish curriculum (see [START_REF] Fnbe | Perusopetuksen opetussuunnitelman perusteet 2014 [National core curriculum for basic education 2014[END_REF].

One way of supporting Maya's growth in mathematics could also be setting long-term goals for her learning. In Maya's case, the long-term goal could be referring to real life and learning to use different representations in mathematics and problem solving (cf. [START_REF] Heinze | Flexible and adaptive use of strategies and representations in mathematics education[END_REF]. Setting and evaluating the long-term goals in learning discussions, using written selfevaluation and supporting the development through formative assessment in mathematics classes, have shown promising results for students learning in a small pilot study implemented during spring semester 2018 (see more in [START_REF] Viitala | Using formative assessment to support student development in lower secondary school mathematics[END_REF]. However, how and if these activities help pupils in a longer period of time, is a matter of a long-term continuation study with a larger number of pupils.

Figure 1 :

 1 Figure 1: A partial caption of the task 'Holiday' (OECD, 2006, p. 77) and Maya's interview notes

Table 1 : Interview themes and example questions

 1

	Background	Tell me about your family.
	Mathematics	Does mathematics exist outside of school? Where?
	Oneself and mathematics	Is mathematics important to you?
	Mathematics learning	How do you learn mathematics?
	Teaching mathematics	Does teaching matter to your learning? How?
	View of mathematical thinking How do you recognise mathematical thinking?