text:
Karin Elisabeth

Sørlie Street
email: karin.street@stx.ox.ac.uk

Lars-Erik Malmberg
email: lars-erik.malmberg@education.ox.ac.uk

Gabriel J Stylianides
email: gabriel.stylianides@education.ox.ac.uk

Self-efficacy and mathematics performance: Reciprocal relationships

Keywords: Self-efficacy, performance, reciprocal effects, levels of difficulty, test taking facets. Thematic Working Group 08

Several studies have looked at either the effects of students ' Self-Efficacy Expectations (SEE) on their mathematics performance results, or the effect of previous mathematics performances on students' SEE. Few studies have tested the theoretically proposed reciprocal relationship between mathematics SEE and performance in mathematics. Furthermore, previous studies have not included levels of difficulty, although this is an integral part of the definition of SEE. This study applied a new measure of SEE, which included both test taking facets and levels of perceived difficulty, to investigate their relationships with students' performance on a national test in mathematics. Two models gave very good fit to the data, and supported the reciprocal effects model. These models provided estimates of the relationships between students' test performance and their level and facet-specific SEE, respectively.

Introduction

Self-efficacy expectations (SEE) are important because extensive research indicates they are related to student learning and performance [START_REF] Zimmerman | Self-efficacy: An essential motive to learn[END_REF]. Studies have demonstrated the effect of mathematics SEE on performance results (e.g. [START_REF] Pajares | Self-efficacy beliefs and mathematical problem-solving of gifted students[END_REF] and vice versa (e.g. [START_REF] Pampaka | Measuring mathematics self-efficacy as a learning outcome[END_REF], but the proposed reciprocal relationship between SEE and performance [START_REF] Bandura | A longitudinal analysis of the relationship between mathematics-related affect and achievement in Finland[END_REF] has received little attention. Investigating this relationship empirically is important in order to understand the relationship between SEE and performance in mathematics, and the process by which SEE may influence performance in mathematics and vice versa. Few studies have investigated reciprocal effects between SEE and mathematics performance with longitudinal data, and none that we know of in relation to national tests in Norway. [START_REF] Bandura | A longitudinal analysis of the relationship between mathematics-related affect and achievement in Finland[END_REF] argued SEE vary according to three dimensions, but we know of no studies that have included SEE level of difficulty to investigate reciprocal effects with mathematics. As we argued in [START_REF] Street | Level, strength, and facet-specific selfefficacy in mathematics test performance[END_REF], including level of SEE is important as it is an integral part of the definition of SEE [START_REF] Bandura | A longitudinal analysis of the relationship between mathematics-related affect and achievement in Finland[END_REF]. We aimed to address this research gap by testing a reciprocal effects model including students' level and facet-specific SEE, and their scores on a Norwegian national test. To this aim, we applied a recently developed measure of mathematics SEE that includes four facets of test taking as well as three levels of perceived difficulty. We included students' SEE responses and test scores as they progressed from grade 8 to grade 9.

Theoretical background

Self-efficacy expectations (SEE) are individuals' judgments about perceived capability to perform on future tasks [START_REF] Bandura | A longitudinal analysis of the relationship between mathematics-related affect and achievement in Finland[END_REF], for instance students' expectations they are able to carry out the various facets involved with taking a mathematics test. Examples are problem-solving skills such as solving a certain number of problems or solving tasks of a certain challenge, or skills in self-regulation such as concentrating for a length of time or persevering through difficult problems. SEE affect future performance through mediating processes, influencing students' behaviors and motivations. SEE may influence individuals' tendencies to approach learning tasks, their effort and persistence while engaged in such tasks, as well as their self-regulatory processes [START_REF] Zimmerman | Self-efficacy: An essential motive to learn[END_REF]. SEE have been found to predict mathematical problem-solving after controlling for factors such as cognitive ability, mathematics Grade Point Average, anxiety, and gender [START_REF] Pajares | Self-efficacy beliefs and mathematical problem-solving of gifted students[END_REF]. There is also empirical support for the influence of past performances on SEE (e.g. [START_REF] Pampaka | Measuring mathematics self-efficacy as a learning outcome[END_REF]. According to [START_REF] Bandura | A longitudinal analysis of the relationship between mathematics-related affect and achievement in Finland[END_REF], SEE are formed through four sources, where mastery experiences is the strongest source. Mastery experiences stem from individuals' appraisals of previous performance situations, for instance their experiences from a previous, similar test. Importantly, previous experiences do not influence SEE directly, but are interpreted and made sense of by the individual [START_REF] Usher | Sources of middle school students ' self-efficacy in mathematics :A Qualitative investigation[END_REF].

A mutually reinforcing pattern of influence between self-beliefs and academic achievement is supported for several types of self-beliefs [START_REF] Valentine | The relation between self-beliefs and academic achievement: A meta-analytic review[END_REF]. Studies have, for instance, investigated reciprocal effects between self-concept and achievement [START_REF] Marsh | Academic self-concept, interest, grades, and standardized test scores: Reciprocal effects models of causal ordering[END_REF], however only few studies have investigated reciprocal effects between SEE and mathematics performance. [START_REF] Williams | Self-efficacy and performance in mathematics: Reciprocal determinism in 33 nations[END_REF] proposed a reciprocal effects model, which fit the data well for PISA results from 26 out of 33 countries. They argued reciprocal effects might be a fundamental psychological process, however the data from Norway was excluded from the analyses due to little variation in the grade-level variable. Hannula et al. (Hannula, Bofah, Tuohilampi, & Metsämuuronen, 2014) investigated the dominant direction of effect between mathematics performance and SEE, as students progressed from grade 3 to grade 9 in Finland. Their results supported a reciprocal effects model, where the effect from performance on SEE (.30) was somewhat stronger than the effect from SEE on performance (.26). Both measures were also relatively stable over time.

Measures of SEE have typically included SEE strength and SEE specificity (also called generality). While Bandura proposed a third dimension of SEE (1997), only few studies have included level of difficulty. Students' SEE may vary between different facets of mathematics, and in regards to different perceived levels of difficulty. In relation to the task 'naming uses for common objects' Locke et al. [START_REF] Locke | Effect of self-efficacy, goals, and task strategies on task performance[END_REF] found that SEE for tasks of medium to high difficulty were most predictive of subsequent task performance.

Theoretical model and research questions

In line with theory and empirical findings discussed, we propose a theoretical model (see Figure 1), where SEE and mathematics performance are reciprocally related. We included Norwegian students' SEE responses and subsequent test scores to investigate this model. In the model, students' SEE in grade 8 and 9 are related to their performance in mathematics in the same year. Furthermore, students' mathematics performances in grade 8 predict their SEE in grade 9. Both SEE and mathematics performance are relatively stable (mathematics performance the most stable of the two), with the grade 8 constructs predicting scores in grade 9. We further propose level and facet-specific SEE have differential relationships with mathematics performance.

Methodology Participants

The participants were 95 students (44 female) in Norwegian secondary school who completed selfreport questionnaires and took national tests in mathematics at the beginning of grades 8 and 9 (13 and 14 years old). The participants were part of a larger sample, selected for cross-sectional investigations (see [START_REF] Sørlie | Regionale forskjeller i mestringsforventninger[END_REF]. Included in the study were schools where students had performed above and below what might be expected on national tests, considering measures of socioeconomic-status. For a detailed explanation of this strategy see Langfelt (2015).

Measures

The Self-efficacy Gradations of Difficulty Questionnaire was applied to measure SEE at the two time points (see [START_REF] Street | Level, strength, and facet-specific selfefficacy in mathematics test performance[END_REF], for a detailed analysis). This is a recently developed multidimensional measure of mathematics SEE, that includes four test taking facets (facet-specific SEE) related to problem solving (complete a number of problems, solve tasks of a certain challenge) and selfregulation (concentrate, not give up), and three levels (easy, medium, and hard) of perceived difficulty (level of SEE). Each of the 14 items in the measure are related to one test taking facet and one level of difficulty within each facet (see Figure 2 for an example facet, "concentrate", with three levels of difficulty). For each item, students are asked to indicate their confidence (strength of SEE) on an 11point scale from 0 "not at all certain" to 10 "completely certain". The structural validity of this measure was tested [START_REF] Street | Level, strength, and facet-specific selfefficacy in mathematics test performance[END_REF], and the resulting best-fit measurement model included three latent (unobserved) level constructs, with correlated uniquenesses (correlated error terms) specified for each of the four facets.

Thematic Working Group 08 The performance measure was raw scores from national tests in numeracy (Norwegian Directorate for Education, 2016). "Numeracy" is similar to what researchers generally refer to as mathematics. Norwegian students sit these tests at the start of the school year in grades 5, 8 and 9. Students in grades 8 and 9 sit the same test, which involves 58 problems, scored as either correct (1) or incorrect (0). We used unique identifiers to link the national test scores with the questionnaire responses.

Specification of empirical models

Our modeling choices were informed by the previously established factor structure of our measure, as well as our theoretical model. [START_REF] Street | Level, strength, and facet-specific selfefficacy in mathematics test performance[END_REF] found that the best-fit models were those that accounted for the multidimensional nature of SEE, through correlated latent constructs and correlated uniquenesses. In order to estimate the relationships between latent constructs in our models we used item parceling technique. Parceling involves aggregating (taking the sum or average) two or more items to manufacture an indicator of a construct (Little, Cunningham, Shahar, & Widaman, 2002, p. 152). While our measure of SEE is multidimensional, the factor structure has been tested and established in a previous study. Item parcels were formed to achieve "clean" latent constructs from each year, through aggregating items in such a way that the secondary loading was spread across parcels [START_REF] Little | To parcel or not to parcel: Exploring the question, weighing the merits[END_REF]. For example, the "easy" parcel contains all items related to the easy latent construct, across four facets. Similarly, the "concentrate" parcel includes all items related to the concentrate facet, across three levels of difficulty. As an example, the "concentrate" parcel in Figure 2 is created through summing the scores of the three items included.

Two reciprocal effects models were specified. Our hypothesized model (Model 1) includes national test scores and item parcels for SEE levels, representing the best-fit model from the previous study. The alternative model (Model 2) includes national test scores and item parcels for facet-specific SEE. In both models, SEE constructs in grade 8 are related to test scores in grade 9. Similarly, SEE constructs in grade 9 are related to test scores in grade 9. Correlations are specified rather than regression paths (see [START_REF] Street | Level, strength, and facet-specific selfefficacy in mathematics test performance[END_REF]. SEE constructs in grade 8 are intercorrelated, as are SEE constructs in grade 9. Furthermore, corresponding SEE constructs are related across the two years (e.g. SEE easy in grade 8 predicts SEE easy in grade 9). Test scores in grade 8 predict test scores in grade 9. Finally, the reciprocal relationships tested are the path from test scores in grade 8 on SEE constructs in grade 9, and the paths from SEE constructs in grade 8 on test scores in grade 9.

Analyses

We analysed the data with structural equation modeling, using the maximum likelihood estimator in Mplus (version 7.31 for Mac: [START_REF] Muthén | Mplus User`s Guide[END_REF]. We used fit indices recommended by previous studies (e.g. [START_REF] Morin | Doubly latent multilevel analyses of classroom climate: An illustration[END_REF][START_REF] Schermelleh-Engel | Evaluating the fit of structural equation models : Tests of significance and descriptive goodness-of-fit measures[END_REF], specifically we used the chi square (χ 2 /df =<3 acceptable), the Root Mean Square Error of Approximation (RMSEA <.08 acceptable), the Standardized Root Mean Square Residual (SRMR<.10 acceptable), the Comparative Fit Index (CFI) and the Tucker-Lewis index (TLI) (CFI/TLI > .90 acceptable). To assess improvement in fit between models, we used the following cutoffs: RMSEA (.015), CFI (.010), SRMR (.030).

Results

Item cross-correlations for items from the same year were consistent with previous results from the same measure. The correlation matrix indicates items are related to two types of latent constructs; levels (easy, medium, and hard) and facets (no. of problems, solve tasks, concentrate, and not give up). Autocorrelations between items in grade 8 and the same item in grade 9 demonstrate two tendencies. First, autocorrelations are significant for all items, except for two cases, both associated with the easy level (no. of problems_easy, and concentrate_easy). Second, the magnitudes of the associations are consistently weaker for items associated with the easy level, than the medium and hard levels. To illustrate this: the strongest autocorrelation of the easy level items is .21, while the weakest autocorrelations of the medium and hard level items is .26 and .29, respectively. Thus, the cross-correlations for each of the years indicate support for a multidimensional construct, while the autocorrelations indicate that students' scores on the medium and hard level items were more stable across time, than students' scores on the easy level items. The non-significant autocorrelations between the grade 8 and grade 9 easy level items might be related to a lack of variability in scores. Most students were highly confident in relation to the easy level items. Results from the structural equation models are presented in Table 1. Both our proposed models resulted in excellent fit. The aim of the present study is not model comparison, but to test a reciprocal effects model, as well as to estimate the relationships between national test scores and level and facetspecific SEE, respectively. Accordingly, parameter estimates from both models are reported (see Figures 3 and4). As can be seen in Figure 3, results from Model 1 partially support a reciprocal effects model between levels of SEE and national test scores. Specifically, national test scores in grade 8 predict levels of SEE in grade 9 in relation to medium and hard, but not easy, tasks. The predicted relationship from SEE level constructs in grade 8 to national test scores in grade 9 is not supported. However, all three levels of SEE are associated with test scores the same year, including when previous test performance is controlled in grade 9. Two further relationships are nonsignificant: the path from SEE easy level in grade 8 to grade 9, and the path from national test scores in grade 8 to SEE easy level in grade 9.

Fit indices for confirmatory factor analyses

Model 2 results (see Figure 4) are generally consistent with results for Model 1. Test scores in grade 8 predict three facets of SEE in grade 9 (not "concentrate"), while the paths from SEE facet constructs in grade 8 to test scores in grade 9 are nonsignificant. However, facet-specific self-efficacy constructs are related to performance results in each year, including in grade 9 (not "solve tasks") when previous test scores are included in the model. Unlike the other facets, the magnitude of the relationship between "not give up" and test scores is very similar in grade 8 and 9. Similar to the easy level in Model 1, the relationships between "solve tasks" and "not give up" from grade 8 to grade 9 are not significant. Finally, we see that test scores are more stable over time than either level or facet-specific SEE.

Discussion

The current study lent empirical support to a reciprocal relationship between level and facet-specific SEE, and performance in mathematics (RQ1). It was demonstrated that students' SEE were associated with scores on a test in the immediate future, while national test scores predicted their SEE one year later. All SEE items in our study are explicitly linked with the immediately upcoming test, thus it is reasonable the SEE constructs are associated with performance results of the immediately upcoming tests, but not with the tests one year later. In two previous studies, measures of SEE and performance were collected over the course of a single day [START_REF] Locke | Effect of self-efficacy, goals, and task strategies on task performance[END_REF] or three years apart [START_REF] Bandura | A longitudinal analysis of the relationship between mathematics-related affect and achievement in Finland[END_REF]. Findings from both these studies indicated the dominant effect was from performance to SEE. The type of test, and whether it is considered important by the students, might influence the relationship between performance experiences and SEE. National tests are quite "talked about" in school in Norway, and students in grades 8 and 9 are likely becoming increasingly aware of upcoming exams (at the end of year 10). This might have provided a particularly memorable context for their test experiences. Furthermore (RQ2), differential relationships were found between national tests and easy, medium and hard levels of perceived difficulty. The associations with test scores were stronger for medium and hard levels of difficulty, which is in line with previous research [START_REF] Locke | Effect of self-efficacy, goals, and task strategies on task performance[END_REF]. The relationships with test performance also varied between different facets of test taking, where "not give up" provided the strongest association.

In our sample students' SEE for medium and hard level tasks had both a stable (SEE regressed on SEE) and a dynamic (SEE regressed on test scores) component, while these estimates differed for the different SEE facets. For instance, in regard to 'concentrate', grade 9 SEE were predicted by grade 8 SEE, while in regard to 'not give up', grade 9 SEE were predicted by grade 8 test scores. This indicates students drew on their previous test experiences to a larger degree when formulating their SEE for perseverance, as compared to concentrating. This implies it is important to consider how to support student perseverance, particularly after adverse test experiences.

The current findings are limited in that we could not control for other factors that may have influenced students' SEE and test performance. Also, while the measure of SEE provided interesting information, partitioning the effects of test taking facets and levels of perceived difficulty, further replications are needed with a larger sample, and in different cultural contexts.

Figure 1 :

 1 Figure 1: Theoretical model RQ1: What is the relationship between SEE and performance on national tests in mathematics in grade 8, and SEE and performance on national tests in mathematics in grade 9? RQ2: Do the relationships between SEE and performance on national test differ according to levels or facets of SEE?

Figure 2 :

 2 Figure 2: Example from the Self-Efficacy Gradations of Difficulty Questionnaire

Figure 3 :Figure 4 :

 34 Figure 3: Structural equation model, Model 1 (correlated levels model)

Table 1 : Results from structural equation models

 1

		Model	 2 	p	RMSEA CFI	TLI	SRMR
		Correlated levels model,	 2 (6)= 4.397
	1	reciprocal relationship with test	 2 /df=0.73	p=.623	.000	1.00	1.02	.048
		scores	
		Correlated facets model,	 2 (12)= 10.890
	2	reciprocal relationship with test	 2 /df=0.91	p=.538	.000	1.00	1.01	.061
		scores	

Proceedings of CERME10


domain: math shs

text:
Stanislaw Schukajlow
email: schukajlow@uni-muenster.de

Kay Achmetli

Multiple solutions for real-world problems, and students' enjoyment and boredom

Keywords: Emotion, affect, modelling, word problems, multiple solutions

. In the present study, we had students construct multiple solutions by applying multiple mathematical procedures to real-world problems, and we investigated effects on the enjoyment and boredom. 307 students were assigned to the experimental or control group. Students in the experimental group applied two mathematical procedures, and students in the control group applied one mathematical procedure to solve real-world problems. During the lessons, they were asked to report their enjoyment and boredom. Contrary to our expectations, the results revealed no effects of the intervention on students' enjoyment or boredom.

Introduction

Emotions are important for learning (Zan, Brown, Evans, & Hannula, 2006). Although students' academic emotions are prerequisites, mediators, and outcomes of the learning process in mathematics [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], they were neglected for decades. Thus, except for the emotion of anxiety, we do not know much about students' emotional development. Moreover, there is a lack of research on how teaching methods influence emotions. As there have been several calls for intervention studies, we decided to conduct a study that was aimed at clarifying the impact of constructing multiple solutions for real-world problems on cognitive and affective outcomes. We chose this teaching method and this kind of problem because constructing multiple solutions and solving real-world problems are emphasized in curricula in different countries. In the present paper, we taught students to construct multiple solutions by applying different mathematical procedures to solve real-world problems, and we investigated how this process affected enjoyment and boredom.

Theoretical framework and hypotheses

High-quality mathematics teaching implies that students should develop multiple solutions and compare these solutions in the classroom. Empirical evidence for the effects of constructing multiple solutions on cognitive outcomes comes from international comparative studies [START_REF] Hiebert | Teaching mathematics in seven countries[END_REF] and from experimental studies [START_REF] Levav-Waynberg | The role of multiple solution tasks in developing knowledge and creativity in geometry[END_REF][START_REF] Schukajlow | Effects of prompting multiple solutions for modelling problems on students' performance[END_REF]. However, the impact of constructing multiple solutions on affect is an open issue. For high-quality mathematics teaching, both cognitive and affective outcomes have to be taken into account. As we determined in the project MultiMa 1 (Multiple Solutions for Mathematics Teaching Oriented Toward Students' Self-regulation Learning), apart from students' achievements and strategies, there is also a need to consider their self-regulation, interest, motivation, and emotions.

Multiple solutions and real-world problems

Previous research on multiple solutions was conducted for the most part on intra-mathematical problems in different content areas such as geometry [START_REF] Levav-Waynberg | The role of multiple solution tasks in developing knowledge and creativity in geometry[END_REF] or early algebra [START_REF] Star | Flexibility in problem solving: The case of equation solving[END_REF]. Students' ability to solve real-world (or modelling) problems was not previously the focus of research on multiple solutions so far. Solving real-world problems first and foremost involves demanding transfer processes between reality and mathematics [START_REF] Niss | Introduction[END_REF]. As real-world problems often include vague conditions and allow students to construct different mathematical models and apply different mathematical procedures, we distinguished between three categories of multiple solutions (Schukajlow & Krug, 2014b). The first category of multiple solutions are typical of real-world problems with vague conditions. In solving this type of problem, students make different assumptions about vague conditions and therefore arrive at different outcomes or results. Another type of multiple-solution problem occurs as a result of applying different mathematical procedures or strategies, a process that typically leads to the same mathematical outcome. The third category combines the first two categories. In the current paper, we explored the effects of applying multiple mathematical procedures while solving real-world problems for the topic of linear functions. We would like to illustrate this type of multiple-solution problem with the sample problem "BahnCard" (cf. Figure 1), which was developed in the framework of the MultiMa Project. The problem solver is asked to read the problem "BahnCard," and identify the important values: price per year for each card and the amount of a round-trip journey that would be paid with each card. After mathematizing the problem, different mathematical procedures can be applied.

One mathematical procedure that can be applied is called "differences." In order to solve the "BahnCard" problem by using differences, students first have to calculate differences in the prices per year and for each round trip for owners of each card. Whereas the "BahnCard 50" is 181 € (= 240 € -59 €) more expensive than the "BahnCard 25," each round trip with the "BahnCard 25" is 25 € (= 50 € -25 €) more expensive than with the "BahnCard 50." The open question is how often Mr. Besser has to take a trip with the more expensive "BahnCard 50" until the cheaper prices for the journeys pay off. This is exactly after 7.24 (= 181 € ÷ 25 €) journeys per year. This result has to be rounded up, interpreted-for example, "For up to 7 journeys per year, the 'BahnCard 25' is cheaper"validated, and the recommendation has to be wrote down.

Another way to solve this problem is to apply a mathematical procedure "table." To apply this procedure, students must compare the costs for owners of the "BahnCard 25" and the "BahnCard 50" for different numbers of journeys per year (e.g. 1, 3, 6…). By performing this comparison systematically, they can identify that the "BahnCard 25" is cheaper for up to 7 journeys. If the owner makes 8 or more journeys, the "BahnCard 50" is preferable for him/her. Finally, students need to validate their result and write down their recommendation.

Enjoyment and boredom as achievement emotions

Emotions are typically defined as complex phenomena that include affective, cognitive, physiological, motivational, and expressive parts [START_REF] Pekrun | Introduction to emotions in education[END_REF]. In the academic context, researchers are interested in achievement emotions, which occur in learning settings and are related to epistemic processes. Research on emotions in mathematics education has emerged from different philosophical traditions [START_REF] Hannula | Emotions in problem solving[END_REF] and has categorized emotions according their value (positive or negative), level of activation (activated or deactivated), or other characteristics. For example, enjoyment is one of the positive activating emotions [START_REF] Pekrun | The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice[END_REF]. Students who enjoy problem solving are expected to report pleasant feelings. Moreover, when students enjoy mathematics, they feel activated excitement while working on a problem. The opposite behavioral and cognitive patterns are expected for the emotion of boredom. Boredom was suggested to be a negative deactivated emotion because boredom is accompanied by unpleasant feelings, and if students feel bored, they experience a state of deactivating relaxation. Following these considerations, a positive relation between enjoyment and performance and a negative relation between boredom and performance were hypothesized and confirmed in two empirical studies in the domain of mathematics [START_REF] Schukajlow | Is boredom important for students' performance[END_REF]Schukajlow & Krug, 2014a). Moreover, enjoyment but not boredom was found to predict students' performance in a longitudinal interventional study [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].

According to the control-value theory of achievement emotions [START_REF] Pekrun | The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice[END_REF], emotions are strongly determined by control and value appraisals, which arise in learning situations. In order for a positive emotion such as enjoyment to emerge, students should (1) perceive their problem solving activities as controllable and be confident that they can influence the learning situation and (2) ascribe the problem solving activities a high value. If students think that they do not have any influence over their problem solving activities, or if they view these activities as meaningless, negative emotions will emerge. For example, boredom arises if students ascribe a low value to their activities. The relation between boredom and control appraisals is complex and is proposed to be a curvilinear U-shape. This relation implies that boredom occurs when perceived control is very high (i.e. task demands are very low) or when perceived control is very low (i.e. task demands are very high). However, in the context of problem solving activities, students do not have to deal with routine tasks. Thus, a negative linear relation between control appraisals (e.g. assessed via students' performance or self-efficacy beliefs) and boredom was expected and confirmed in most empirical studies (e.g. [START_REF] Schukajlow | Is boredom important for students' performance[END_REF].

Enjoyment, boredom, and multiple solutions for real-world problems

On the basis of theoretical considerations from control-value theory, we expected to find that constructing multiple solutions would increase students' control appraisals when solving real-world problems. Higher appraisals should increase students' enjoyment and decrease their boredom. Positive effects of constructing multiple solutions on enjoyment and negative effects on boredom were confirmed in our previous study. Students who constructed more solutions enjoyed their classes more and were less bored [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]. In the current study, we sought to confirm these findings for the other type of multiple-solution problem and investigated the effects of applying multiple mathematical procedures for real-world problems on enjoyment and boredom.

Hypotheses

The hypotheses we addressed were: 1) Constructing multiple solutions by applying multiple mathematical procedures for real-world problems has a positive effect on students' enjoyment of mathematics; 2) Constructing multiple solutions by applying multiple mathematical procedures for real-world problems has a negative effect on boredom in mathematics.

Method Sample and procedure

Three hundred seven German ninth graders from four schools with three middle-track classes each (48.26% female; mean age=14.6 years) participated in the present study. Before and after the teaching unit, students were asked about their enjoyment and boredom. The teaching unit consisted of two sessions with two 45-minute long lessons each. Each of twelve classes was divided into two parts with the same number of students in each part in the way students' mathematical achievements did not differ between the parts. Further, the number of males and females was approximately the same in each part. Eight of twenty-four groups were randomly assigned to the one-solution condition "differences" (OS1), eight groups to the one-solution condition "table" (OS2), and eight to the multiple-solutions condition "differences + table" (MS), taking into account that in each school, there had to be the same number of groups assigned to each condition, and the students in each class had to be assigned to different conditions (more details about the procedure can be found in Achmetli, Schukajlow, & Rakoczy, manuscript submitted for publication). Each group was taught separately by one of six teachers (three female, age: 27 to 60) who participated in the present study. The teachers taught the same number of groups in each condition in order to minimize the differences between conditions that might result from the influence of teacher personality on students' learning. All of the teachers received instruction manuals that included the lesson plans, problems for the students, and the solutions to these problems.

Treatment

The three treatment conditions implemented in the present study (OS1, OS2, and MS) were based on the positively evaluated student-centered learning environment for teaching modelling problems [START_REF] Schukajlow | Scaffolding mathematical modelling with a solution plan[END_REF]. This student-centered learning environment was complemented by direct instruction at the beginning of the teaching unit. For the purpose of maintaining comparability between the conditions, the same order was implemented for all three treatment conditions. In the first lesson, the teacher demonstrated how real-world problems could be solved by applying one mathematical procedure (in the OS conditions) or multiple mathematical procedures (in the MS condition). In the three lessons that followed, the students solved real-world problems by applying the demonstrated procedures according to a special procedure for group work (alone, together, and alone), presented their solutions, and discussed these solutions with the whole group in the classroom. At the end of each lesson, the teacher summarized the key points of each treatment condition. In the multiple-solutions condition, the teacher encouraged the students, further, to compare and contrast the two mathematical procedures and the mathematical results.

Students first solved four similar tasks in the one-solution conditions and in the multiple-solutions condition. The only difference between these four problems was that students in the one-solution conditions were required to apply one mathematical procedure ("table" or "differences"), whereas students in the multiple-solutions condition were required to apply both mathematical procedures ("table" and "differences"). The sample problem "BahnCard," which was given in the one-solution conditions, is presented in Figure 1. In the multiple-solutions condition, the problems were modified by adding the following sentence: "Use two different mathematical procedures to solve this problem." As the discussion of the connection between mathematical procedures required additional time in the MS condition, one additional task was offered in each OS condition. Thus, in sum, students in the MS condition solved six and students in the OS conditions solved seven problems.

Measures

Enjoyment and boredom during the teaching unit were measured after the second and fourth lessons with a 5-point scale ranging from 1 (not at all true) to 5 (completely true). Both scales included three items each (see Table 1).

Scale Item

Enjoyment I enjoyed task processing. I was happy during task processing. Task processing was great fun for me.

Boredom

Task processing was boring. I got so bored during task processing that I had problems staying alert. I did not want to continue my work because it was so boring. The scales were adapted from the well-evaluated Achievement Emotions Questionnaire (Pekrun, Goetz, Frenzel, Barchfeld, & Perry, 2011). The Cronbach's alpha reliabilities were .80 and .79 for enjoyment and .81 and .83 for boredom for Sessions 1 and 2, respectively.

Treatment fidelity

To ensure the fidelity of the treatment, we videotaped the teaching unit, observed the lessons, and analyzed the students' solutions. The analysis confirmed the treatment fidelity (Achmetli et al., manuscript submitted for publication). For example, we found that students in all classes worked on the respective version of the problem (MS vs. OS) and all teachers implemented the intended methodical order in their lessons. More specifically, we found that students in the MS condition developed significantly more solutions than the students in the OS conditions (MS vs. OS1: effect size Cohen's d=4.97; MS vs. OS2: d=3.61).

Results

Preliminary results

In order to simplify the analysis of the effects, we combined the OS1 and OS2 conditions into one OS condition. Combining the two conditions did not influence the results significantly, as our statistical analysis did not show a difference at the 10% level of significance between the two OS conditions for motivational variables such as self-regulation [START_REF] Achmetli | Effects of prompting students to use multiple solution methods while solving real-world problems on students' self-regulation[END_REF] or interest (t(186) = 0.182; p = .856). Further, in order to ensure that the two conditions were comparable, we compared interest between the MS condition and the combined OS condition as this construct is closely connected to students' enjoyment and boredom [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]. The analysis of interest at pretest revealed no differences between the MS and OS conditions (MS: M = 2.39 (SD = .90), OS: M = 2.39 (SD = .96)). This result indicates that students' emotional prerequisites were similar in the MS and OS conditions.

Applying multiple mathematical procedures and students' enjoyment or boredom

We hypothesized that constructing multiple solutions by applying multiple mathematical procedures would increase students' enjoyment and decrease their boredom. We tested both hypotheses by computing t-tests. The crucial assumption when using a t-test is that the variances are equal in the two groups. Levene's test of equality of variances was significant for students' boredom measured after the second and third lessons, indicating that the assumption of equal variances in the two groups had been violated (F(280) = 4.022, p = .046; F(279) = 4.851, p = .028). Thus, we used the adjusted degrees of Against our expectations, students' enjoyment during the first and second sessions did not differ between the MS and OS conditions (first session t(280) = 0.75, p = .940, Cohen's d = 0.02; second session: t(279) = 0.297, p = .767, d = 0.04). Thus, the enjoyment of students who solved real-world problems by applying multiple mathematical procedures was similar to the enjoyment of students who applied one mathematical procedure.

We did not find support for the second hypothesis. Our analysis did not reveal benefits of constructing multiple solutions by applying multiple mathematical procedures for students' boredom during the first or second session (first session t(165) = 1.67, p = .097, Cohen's d = 0.22; second session: t(156) = 1.62, p = .108, d = 0.22). Moreover, there was a slight (but not significant) tendency for students in the multiple-solutions group to feel greater boredom than students in the one-solution condition.

Discussion

In this paper, we aimed to analyze how constructing multiple solutions by applying multiple mathematical procedures while solving real-world problems would affect students' emotions. On the basis of theoretical considerations from the control-value theory of achievement emotions [START_REF] Pekrun | The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice[END_REF] and prior research that found that developing multiple solutions had positive effects on students' enjoyment and negative effects on their boredom [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 expected positive effects of the treatment on enjoyment and negative effects on boredom during learning. However, our analyses did not confirm these hypotheses. Enjoyment and boredom in solving real-world problems did not differ between the multiple-solutions and one-solution conditions. Moreover, boredom was slightly lower in the one-solution condition compared with the multiple-solutions condition. One explanation for this finding might involve students' high control appraisals. In the previous study, the mean values for students' experience of competence, which can be taken as an indicator of students' control appraisals (sample item: "I felt confident about my knowledge of the topic today"; range from 1 to 5), were 3.85 and 3.65 in the MS and OS conditions, respectively (Schukajlow & Krug, 2014b). However, in the current study, the mean values for students' experience of competence were nearly one standard deviation higher and close to the theoretical maximum of 5 (Achmetli et al., manuscript submitted for publication). As noted in the control-value theory, if students' control appraisals are too high (or task demands are too low), they can have a negative influence on students' emotions. Thus, a future research question might involve asking whether posing more demanding real-world problems that require students to apply multiple mathematical procedures can increase students' positive emotions such as enjoyment and decrease their negative emotions such as boredom. Another research question that should be addressed in an experimental study is about the non-linear connection between control appraisals and students' emotions. This assumption of the control-value theory needs more empirical evidence from randomized studies. More specifically, the corvilinear U-shape relation between control appraisals and boredom should be addressed in future longitudinal studies. Further, it might be the case that the type of multiple-solution problem makes a difference. Whereas students enjoy making different assumptions about missing information, constructing different solutions, and comparing their results, this enjoyment might not hold when they apply different mathematical procedures. Similar effects (low level of boredom for the first type of multiple-solution problem, but no difference in boredom for the second type of multiple-solution problem) were also found for students' boredom.

Another explanation for no effects of the intervention on emotions might be that students in the multiple solution condition were not offered to chouse their favorite procedure during three of four lessons. More efforts are needed to clarify the role of multiple solutions for affective measures and more generally, with respect to the effects of teaching methods on students' affect.

Figure 1 :

 1 Figure 1: Real-world problem "BahnCard" (Achmetli, Schukajlow, & Krug, 2014)

Table 1 : Items used in the study to assess enjoyment and boredom

 1

  freedom, t-values, and p-values for students' boredom. The descriptive statistics are presented in Table 2.

		Enjoyment	Enjoyment	Boredom	Boredom
		first session	second session	first session	second session
		Mean (SD)	Mean (SD)	Mean (SD)	Mean (SD)
	MS	3.42 (0.92)	2.96 (0.91)	2.24 (1.05)	2.53 (1.16)
	OS	3.43 (0.88)	3.00 (0.96)	2.03 (0.90)	2.30 (1.00)

Table 2 : Means and standard deviations for enjoyment and boredom

 2

Proceedings of CERME10

1 This research was financially supported by a grant from the German Research Foundation (Deutsche Forschungsgemeinschaft) allocated to the fist author (SCHU 2629-2).

Thematic Working Group 08


text:
Elisabet Mellroth
email: elisabet.mellroth@kau.se

The suitability of rich learning tasks from a pupil perspective

Keywords: Mathematically gifted, mathematically promising, rich tasks, pupils' perception

The construction of tasks is important to challenge pupils, but the exploration of pupils' perceptions connected to their work with tasks, is rare. This paper presents the results of a study using a tool aimed at measuring pupils' perception of joy and interest connected to 'rich learning tasks' by comparing the views of mathematically promising pupils and others. Two tasks were pre-and postevaluated, the first by 139 and the second by 106 pupils from grade 4-9. The results indicate that the tool is suited for the exploration of pupils' views, especially as it can be deduced from the comparison that mathematically promising pupils perceived both tasks more positively than the other pupils, and that the non-identified pupils became more positive after working especially with one of the tasks.

Introduction

One of the main goals in research on mathematical giftedness is to identify and foster mathematically promising pupils [START_REF] Käpnick | Ein konzeptioneller Ansatz zur Kennzeichnung mathematisch begabter Kinder und Möglichkeiten ihrer Diagnostik und Förderung aus fachdidaktischer Perspektive[END_REF]. The construction of mathematical tasks is seen to be important for both purposes [START_REF] Fuchs | Mathe für kleine Asse. Empfehlungen zur Förderung mathematisch interessierter und begabter Kinder im 3. und 4. Schuljahr[END_REF][START_REF] Nolte | Challenging math problems for mathematically gifted children[END_REF]. It is a consensus that tasks suitable to identify and foster mathematically promising pupils should, for example, be challenging, open-ended, encourage creativity and engagement, and promote enjoyment [START_REF] Fuchs | Mathe für kleine Asse. Empfehlungen zur Förderung mathematisch interessierter und begabter Kinder im 3. und 4. Schuljahr[END_REF][START_REF] Nolte | Challenging math problems for mathematically gifted children[END_REF][START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. In Sweden there is no differentiation among students, every classroom is diverse and includes pupils of all abilities. Therefore, it is interesting to explore how the work with specific tasks are perceived by all pupils in the classroom. Of special interest is the perception of the mathematical promising pupils since in a diverse classroom there is a risk that they not are given opportunities to be challenged [START_REF] Leikin | Teachers' images of gifted students and the roles assigned to them in heterogeneous mathematics classes[END_REF]. A task suitable to implement in the whole class should offer a challenge to pupils at every level, which for example rich learning task are said to do [START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. However, it is rare that the assessment process of the tasks appropriateness is explored, especially from the pupils' perspective. This leads to the question how tasks aimed to support mathematically promising pupils can be evaluated by the pupils. Against the background of this question this paper presents a study aiming to explore a tool in development that investigates pupils' perception of joy and interest connected to specific tasks. This paper gives a theoretical background on tasks suitable to challenge mathematical promising pupils and other pupils. Further, the aspect of perceived joy and interest for pupils connected to work on mathematics is elaborated. The study and its results are presented and thereafter the tool used and the interpretation of the results are discussed.

Theoretical background

Pupils in a diverse classroom naturally have different levels of knowledge. [START_REF] Engström | Middle town mathematics III: Pupils are calculating[END_REF] showed that in Swedish classrooms the mathematical knowledge of the 15 percentage lowest achieving pupils in grade nine are on the level of a grade four pupil. Also in a mathematical classroom there is a mix of pupils, some are highly motivated while others lack motivation, some are high achieving and others are low achieving [START_REF] Boaler | How a detracked mathematics approach promoted respect, responsibility, and high achievement[END_REF]. All pupils should be given opportunities to learn and develop, and on task level there are ways to differentiate education to meet and challenge all pupils. One way is for example through the use of rich learning tasks, which also fulfills the criteria for tasks seen to be suitable to identify and foster mathematically promising pupils [START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. Because of the Swedish context with the diverse classroom the aim is to meet and develop all pupils, however, the mathematically promising pupils are of particular interest in this study. Therefore, it is important to elaborate on what is important in a task for a mathematically promising pupil as well as for pupils in general.

First, considering the mathematically promising, it is important to give them challenging tasks to help them develop according their mathematical potential (e.g. [START_REF] Benölken | Mathe für kleine Asse" -An enrichment project at the University of Münster[END_REF][START_REF] Koshy | Mathematically gifted and talented learners: Theory and practice[END_REF][START_REF] Nolte | Challenging math problems for mathematically gifted children[END_REF]. Open-ended tasks, like rich learning tasks, are examples of tasks known to be challenging for mathematically promising pupils [START_REF] Nolte | Challenging math problems for mathematically gifted children[END_REF][START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. In addition, the joy factor is stated as important in the development process for the mathematically promising [START_REF] Fuchs | Mathe für kleine Asse. Empfehlungen zur Förderung mathematisch interessierter und begabter Kinder im 3. und 4. Schuljahr[END_REF]. The importance of joy in working with mathematics is further consolidated by being strived for in activities aiming to support and foster the mathematically promising, such as for example math clubs [START_REF] Benölken | Mathe für kleine Asse" -An enrichment project at the University of Münster[END_REF].

Second, considering pupils in general, [START_REF] Taflin | Matematikproblem i skolan: För att skapa tillfällen till lärande[END_REF] states that it is important that pupils perceive the problem solving process of a task as positive, challenging, and that it stimulates their creativity. Taflin actually writes that if they do not perceive this, then it is better not to implement the tasks. As to the perspective of joy, [START_REF] Mellroth | High achiever! Always a high achiever?: A comparison of student achievements on mathematical tests with different aims and goals[END_REF] showed that tasks aiming to evoke joy make some pupils achieve highly, even though they do not achieve highly on traditional mathematics tests. In addition, to further strengthen that pupils' enjoyment in mathematics is important, [START_REF] Chen | Motivation and mathematics achievement: A comparative study of Asian-American, Caucasian-American, and East Asian high school students[END_REF] showed that positive attitudes and interest are significantly related to mathematical achievement. And the results of [START_REF] Skaalvik | Mathematics achievement and selfefficacy: Relations with motivation for mathematics[END_REF] show that pupils' self-efficacy in mathematics is positively and strongly related to intrinsic motivation which they directly connected to pupils' enjoyment when working with mathematics.

Based on the theory it can be assumed that pupils' perception of interest and their positive attitudes towards the task have effect on their motivation on working with the task. This is valid for both mathematically promising pupils and for others. Therefore, it is interesting and important to explore how pupils perceive working with specific tasks, especially by comparing promising and other pupils. A developed tool, easy to use, could help teachers choosing tasks that challenge and interest all pupils.

Aim

The aim of the presented study is to investigate how to identify mathematical tasks that can stimulate all pupils in a diverse classroom, including the mathematically promising. Utilizing a pupil perspective, which stresses the importance of pupil interest and joy when working with mathematics, the study provides a comparison of data from promising children and others.

The study is conducted in Sweden within the frame of a professional development program on mathematical promise for seven in-service teachers, teaching mathematics for pupils from grade 4 to 9 [START_REF] Mellroth | En forskningscirkel för lärare om särskild begåvning i matematik (A research circle for teachers on mathematical promise[END_REF].

Method

Two tasks that fulfilled the criteria of rich learning tasks were implemented in seven classrooms, i.e. all pupils in "regular" classes worked with the tasks aiming to solve them. The pupils went in grade 4 (age 10) to 9 (age 15), all grades covered. The tasks considered to be suitable to challenge all pupils, specifically mathematically promising, were chosen [START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF], see Figure 1 (Task 1) and Figure 2 (Task 2). In the first intervention Task 1 was implemented: 139 pupils responded on the evaluation of the task, among them 32 pupils were identified as mathematically promising1 . In the second intervention Task 2 was implemented: 106 pupils responded, among them were 20 pupils identified as mathematically promising. All pupils did the interventions in the same order i.e. Task 1 first and Task 2 second. The pupils involved in the two interventions all came from the same seven classes, 44 pupils did Task 1 but not Task 2 and 11 pupils did Task 2 but not Task 1. Therefore 95 pupils participated in both interventions, 20 of those were identified as mathematically promising. Since the suitability of the tasks in the classroom was of interest, evaluations from all participating pupils were used in the analysis for each intervention.

Within the frame of the professional development program a tool how to measure pupils' perceptions on interest and joy connected to working with specific tasks was developed. In the development process experts on motivation and attitudes in mathematics education, and in educational psychology were consulted. The tool resulted in a pre-evaluation that utilized an emoji-note, Figure 3, and a postevaluation, in which the emojis were changed to words, Figure 4. The reason for the change from emojis to words was to decrease the risk that pupils would chose the same emoji twice due to the short time, the time of one lesson, between the pre-and post evaluation.

To collect data each teacher presented a power point slide with a picture related to each task2 in their specific classes, without revealing the actual task. Before the task was handed out to the pupils, they were asked to mark how they felt about the task by choosing an emoji on a paper given to each one of them, see Figure 3. Thereafter the pupils were given time to work with the task.

Your teacher has presented a mathematical problem. Which emoji best matches your feeling about this problem? When the teacher ended the pupils work with the task, but before the task was discussed orally in the whole classroom, pupils were asked to evaluate the task again. This time by choosing words, see Figure 4.

What words best describes how you felt about the task while working with it?

Very interesting Interesting Neither or Uninteresting Very uninteresting Data from all classes were collected and summarized. For the summary process pupils identified by the teachers as mathematically promising were separated from the non-identified pupils. The evaluations, see Figure 3and Figure 4, were translated to numbers from 1 to 5, where 1 was the most positive evaluation and 5 the most negative. Thereafter, a descriptive analysis was conducted. For further details of the method see [START_REF] Mellroth | En forskningscirkel för lärare om särskild begåvning i matematik (A research circle for teachers on mathematical promise[END_REF].

Results

The results from each task are presented in Figure 5 and Figure 6: the graphs show the distribution of pupils' perception of the task before they started to work on it. Each bar in the graph is also split to show pupils change in perception of the task after they completed working with it. For example, in the left-hand graph in Figure 5, the bar on number 2 shows that 12 pupils, identified as mathematical promising, chose the second most positive emoji before they started to work on Task 1. Further, the same bar shows that of those 12 pupils, five gave the task a more negative judgement, two gave it a more positive judgement and five still gave them the second most positive judgement after they completed the work with the task. As both Figure 5 and Figure 6 show, through the concentration of the bars to the left, pupils identified as mathematically promising perceive both tasks more positively compared to the non-identified pupils before they started to work with the tasks. Considering Task 1, the two groups of pupils, identified and non-identified, did not differ much in how they changed their evaluation of the task after they completed it. 28 percent compared to 33 percent judged the task more positively after they completed it, 28 percent compared to 24 percent judged it more negatively, and 44 percent versus 43 percent judged it the same as before. As to Task 2, and the results of comparing identified and nonidentified pupils and how they changed their evaluation of this task after they completed it show: 15 percent compared to 33 percent gave a more positive judgement afterwards, 25 percent compared to 19 percent gave the task a more negative judgement afterwards and 60 percent compared to 49 percent did not change their judgement of the task.

Interpretations and discussion

The aim of the study was to investigate how to identify mathematical tasks that can stimulate all pupils in a diverse classroom, including the mathematically promising. The two tasks used in the study were chosen because they were rich learning tasks and said to be suitable to challenge all pupils, including the mathematically promising [START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. The positive evaluation given by especially the mathematically promising pupils were expected, therefore the results can be seen to verify the developed tool.

For Task 1 the results show that the majority of the mathematically promising pupils, before starting to work on it, evaluated it as more positive: 63 percent choose the most, or the second most positive emoji, compared to 40 percent of the non-identified pupils. For Task 2 the comparable percentages are 85 and 63 respectively. This indicates that the mathematically promising, especially, perceived the tasks interesting and joyful already before they knew the associated question. The results show that Task 2 has this effect to a higher extend for all pupils, also the non-identified. The post evaluation of Task 2 shows a relatively large shift to a more positive judgement of the task for the non-identified pupils, Figure 6 right graph. Altogether the results indicate that considering pupils' perception of joy and interest, Task 2 is suitable for all pupils in the diverse classroom, including mathematically promising pupils.

The results also indicate that Task 1 is not as suitable for all pupils. However, even if Task 1 is not as good as Task 2 according to the results, the mathematical promising pupils perceived it relatively positively before starting to work on it. In addition, just as many of them judged the task more negatively as those who judged it more positively afterwards. Also slightly more pupils of the nonidentified judged it more positively after the completed work compared to the number that judged it more negatively. Therefore, Task 1 might also be a suitable task in a diverse classroom even if it is not as good as Task 2.

According to the chosen frame for this study, tasks challenging and stimulating for mathematically promising pupils lead to that they feel joy and develop learning [START_REF] Fuchs | Mathe für kleine Asse. Empfehlungen zur Förderung mathematisch interessierter und begabter Kinder im 3. und 4. Schuljahr[END_REF][START_REF] Nolte | Challenging math problems for mathematically gifted children[END_REF][START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. The identified pupils positive evaluation of the tasks, especially Task 2, can be a sign of that they felt the tasks challenging and stimulating. The results for the promising pupils can also be interpreted as an indication of that the developed tool fulfills its purpose to measure pupils joy and interest in a rich learning task. Furthermore, it is indicated that the tool can grade the suitability of different tasks, concerning joy and interest, in this case Task 2 is perceived slightly more positive than Task 1.

It has been found that teachers rarely provide mathematically promising pupils with learning opportunities that benefit them in the diverse classroom [START_REF] Leikin | Teachers' images of gifted students and the roles assigned to them in heterogeneous mathematics classes[END_REF], and also that positive attitudes towards working with mathematics make pupils achieve better [START_REF] Chen | Motivation and mathematics achievement: A comparative study of Asian-American, Caucasian-American, and East Asian high school students[END_REF]. Based on this, the results show that the tasks might provide mathematical learning opportunities for all kind of pupils. Further development and verification of this tool can provide teachers with a simple way to find tasks that provide learning opportunities for all pupils in the diverse classroom, also for the mathematical promising.

Even if the simple tool has proven its use in principle, there are, of course, several limitations in this study, the investigation is simple and the tool used is not statistically verified. Nor does the investigation consider in depth how joy and interest is perceived by the pupils. In this study, pupils' motivation to work on a task is assumed to be connected to the perceived joy and interest. The teachers' evaluation of how the pupils worked with the implemented tasks is another important aspect, which this paper does not address. Within the frame of the professional development program the teachers observed and interviewed some pupils connected to their work with the investigated tasks; inclusion of this data would have strengthened the results [START_REF] Mellroth | En forskningscirkel för lärare om särskild begåvning i matematik (A research circle for teachers on mathematical promise[END_REF]. Also, to be able to compare different groups of pupils like for example mathematically promising and others (nonidentified), teachers need knowledge on how to identify the different groups. In this study the teachers who collected the data participated in a professional development program on mathematical promise, their knowledge on how to identify those pupils can be considered as relatively deep. But it is needed to highlight this for someone who wants to repeat the study.

If further research is done to develop and validate the tool used here, it could provide in-service teachers with an easy and quick way to evaluate the suitability of tasks from a pupil's perspective. In turn this might result in mathematically promising pupils being presented with tasks that help them to develop according to their potential. In addition, complex single-case studies might explore specific aspects of tasks that are assessed highly by the pupils applying the tool presented in this study.

Figure 1 :

 1 Figure 1: Example from Task 1, named Where am I? (Sheffield, 2003).

Figure 2 :

 2 Figure 2: Example from Task 2, named Field of dreams (Sheffield, 2003); The number in a circle, denotes the total number of students in all adjoining fields.

Figure 3 :

 3 Figure 3: Evaluation note before starting to work with the task, adapted from Mellroth et al. (2016).

Figure 4 :

 4 Figure 4: Evaluation note after completing working with the task (Mellroth et al., 2016).

Figure 5 :

 5 Figure 5: Pupils evaluation of Task 1, before starting their work on the task and after they completed their work (Figure adapted from Mellroth et al, 2016, p. 19).

Figure 6 :

 6 Figure 6: Pupils evaluation of Task 2 before starting their work on the task and after they completed their work (Figure adapted from Mellroth et al, 2016, p. 19).

Proceedings of CERME10

Selected through a synthesis of different tools, see[START_REF] Mellroth | En forskningscirkel för lärare om särskild begåvning i matematik (A research circle for teachers on mathematical promise[END_REF].Thematic Working Group 08Proceedings of CERME10

The process of choosing and analyzing the tasks are described in[START_REF] Mellroth | En forskningscirkel för lärare om särskild begåvning i matematik (A research circle for teachers on mathematical promise[END_REF] Thematic Working Group 08 Proceedings of CERME10

Acknowledgment

Thanks to all teachers who participated in the professional development program and who were the driving force to include pupils' perception in the investigation. Those teachers, Agneta Arwidsson, Katarina Holmberg, Annika Lindgren Persson, Charlotta Nätterdal, Lotta Perman, Sofia Sköld and Annika Thyberg, are also the co-authors to Mellroth et al. (2016).


text:
Peter Liljedahl
email: liljedahl@sfu.ca

Simon Fraser

On the edges of flow: Student engagement in problem solving

Keywords: Flow, engagement, perseverance, challenge, tolerance

Engagement in mathematical problem solving is an aspect of problem solving that is often overlooked in our efforts to improve students' problem solving abilities. In this paper I look at this construct through the lens of Csíkszentmihályi's theory of flow. Studying the problem solving habits of students within a problem solving environment designed to induce flow, I look specifically at student behavior when faced with an imbalance between their problem solving skills and the challenge of the task at hand. Results indicate that most students have perseverance in the face of challenge and tolerance in the face of the mundane, and use these as buffers while autonomously correcting the imbalance.

Flow

In the early 1970's Mihály Csíkszentmihályi became interested in studying, what he referred to as, the optimal experience (1990), "a state in which people are so involved in an activity that nothing else seems to matter; the experience is so enjoyable that people will continue to do it even at great cost, for the sheer sake of doing it." (Csíkszentmihályi, 1990, p.4) In his pursuit to understand the optimal experience, [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF] studied this phenomenon among musicians, artists, mathematicians, scientists, and athletes. Out of this research emerged a set of nine characteristics common to every such experience [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF])the first three of which are characteristics external to the doer, existing in the environment of the activity, and crucial to occasioning of the optimal experience. 1. There are clear goals every step of the way. 2. There is immediate feedback to one's actions. 3. There is a balance between challenges and skills.

The last of thesebalance between challenge and skillsis central to [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF] analysis of the optimal experience and comes into sharp focus when we consider the consequences of having an imbalance in this system. Csíkszentmihályi found that if the challenge of the activity far exceeds a person's ability they are likely to experience a feeling of frustration. Conversely, if their ability far exceeds the challenge offered by the activity they are apt to become bored. When there is a balance in this system a state of, what Csíkszentmihályi refers to as, flow is created (see fig. 1).

Flow is a powerful ways for us, as mathematics education researchers, to talk productively about the phenomenon of engagement in general, and the three aforementioned elements of flow gives us a way to think about the potential environments that occasion engagement in our classrooms in particular. Williams (2001) used Csíkszentmihályi's idea of flow and applied it to a specific instance of problem solving that she refers to as discovered complexity. Discovered complexity is a state that occurs when a problem solver, or a group of problem solvers, encounter complexities that were not evident at the onset of the task, is within their zone of proximal development [START_REF] Vygotsky | Mind in society: The development of higher psychological processes[END_REF], and occurs when the solver(s) "spontaneously formulate a question (intellectual challenge) that is resolved as they work with unfamiliar mathematical ideas" (p. 378). Such an encounter will capture, and hold, the engagement of the problem solver(s) in a way that satisfies the conditions of flow. What Williams' frame-work describes is the deep engagement that is sometimes observed in students working on a problem solving task during a single problem solving session.

Extending this work, I argued that engagement was an affective experience and used the notion of flow to look at situations of engagement extended over several days or weeks wherein students return to the same task, again and again, until a problem was solved [START_REF] Liljedahl | Learning elementary number theory through a chain of discovery: Pre-service teachers' encounters with pentominoes[END_REF]. The results of this work showed that although flow was present in each of the discrete problem solving encounters, what allowed the engagement to sustain itself across multiple encounters was a series of discovered complexities in each session linking together to form what I referred to as a chain of discovery.

More recently, I looked at the practices of two teachers through the lens of flow in general and their ability to set clear goals, provide instant feedback, and maintain a balance between challenge and skill in particular (Liljedahl, 2016a). From this a number of conclusions emerged. First, thinking about flow as existing in that balance between skill and challenge, as represented in figure 1, obfuscates the fact that this is not a static relationship. Flow is, in fact, a dynamic process. As students engage in an activity their skills will, invariably, improve. In order for these students to stay in flow the challenge of the task must similarly increase (see fig. 2). In a mathematics classroom, these timely increases of challenge often fall to the teacher. But this is not without obstacles. For example, if a student's skill increases either too quickly or too covertly for the teacher to notice that student may slip into a state of boredom (see fig. 3). Likewise, when the teacher does increase the challenge, if that increase is too great the student may become frustrated (see fig. 4). How teachers manage these situations of boredom and frustration is important. In Liljedahl (2016a) one of the teachers managed such situations synchronously, either giving hints or extensions to the class as a whole, usually after three groups finished or she got three of the same questions respectively. For most groups the timing of these hints and extensions was off, and not helpful in maintaining flow. The second teacher, however, managed these situations asynchronously, dealing with groups individually as they got stuck or completed a problem. Student engagement in the second teacher's class was visibly higher as he was maintaining flow through the constant and timely maintenance of the balance between ability and complexity for each group.

What I did not learn from this aforementioned research is how students cope with imbalance when the teacher does not provide help or extensions in a timely fashion. In the research reported here I look closely at exactly this phenomenon in general, and student autonomous actions and reactions in such moments of imbalance in particular.

Methodology

To get at this behavior I chose to observe students in a problem solving settings where student work was easily visible. To this end I strategically selected two senior high school classrooms belonging to two different teachers (Cameron and Charmaine), both of whom conducted their classrooms according to a teaching framework designed to shape their classroom into a space "that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together and constructing knowledge and understanding through activity and discussion" (Liljedahl, 2016b, p.364).

My earlier empirical work (Liljedahl, 2016b) on the design of such classrooms had emerged a collection of nine elements that offer a prescriptive framework to help teachers build such spaces. For the research presented here, five of these elements are particularly salient:

1. At the beginning of every class, students are assigned to a visibly random group (Liljedahl, 2016b[START_REF] Liljedahl | The affordances of using visibly random groups in a mathematics class-room[END_REF] of two to four students. 2. These groups work collaboratively to solve a number of problems (usually) right from the beginning of the lesson. 3. This work is done with groups working at vertical non-permanent surfaces such as whiteboards, blackboards, or windows (Liljedahl, 2016b). 4. Students' flow is occasioned and maintained through the teacher's judicious and timely use of hints and extensions (Liljedahl, 2016a(Liljedahl, , 2016b)). 5. At some point within this sequence of tasks the teacher brings the students together to debrief what they have been doingeither by going over one or more of the students' solutions or working through a new problem together with the class as a whole. This is timed so that every group is able to participate in discussion and benefit from the reification.

Taken together, both of these classrooms offered the affordances for me to easily observe students working within and an environment designed to occasion flow. The teachers were both managing engagement through the timely use of hints and extensions to maintain a balance between the challenge of an activity and the ability of each group. The student work was visible and there was enough autonomy afforded in the room that the students could take some kind of action when they found themselves in a situation where challenge and ability may be out of balance.

Data for this research were collected in Cameron's grade 12 Pre-calculus class and Charmaine's grade 11 pre-calculus class. Each class was visited five times over a seven week period in the middle of the second semester.

The data

Because the collection of video data creates such a narrow field of view, I instead used a variant of noticing (van Es, 2011) to scan the classrooms. [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF] characterizes flow as enjoyment, fluidity, and focus. These characteristics manifest themselves in the physicality of individuals and groups in flow and allows for the easy identification of flow and the absence of flow in a classroom. As per my research question, what I was looking for, then, were moments where an individual or a group was out of flow and where that individual or group was left to cope with this on their own. Once such a moment was identified I would focus in on that individual, or that group, taking detailed field notes and occasional photographs. When these moment seemed to wane I would conduct short, in-the-moment, interviews.

Csíkszentmihályi's theory of flow (1990) predicts that lack of flow is the result of a group of students' abilities exceeding the challenge of the task (see fig. 3) or the challenge of the task exceeding the abilities of a group (see fig. 4), resulting in the groups quitting, respectively, out of boredom or frustration. As such, flow served as the initial framework for analyzing the data. As it turns out, the theory was far from adequate for explaining all of the students' actions and reaction in the data. As such, I also used analytic deduction [START_REF] Patton | Qualitative research and evaluation methods[END_REF] to look more closely at students' actions and to group these actions into themes.

Results and analysis

From this analysis a series of six nuanced themes emerged, each marked by a different type of student action or reaction to being out of flow. In what follows I present cases exemplifying each of these themes as well as some general comments about similar cases.

When skills exceed challenge: The case of quitting

As mentioned, [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF] found that if a person's ability exceeded the challenge they are apt to become bored, and then quit out of this boredom. I found evidence of such behavior in Cameron's and Charmaine's classrooms.

Researcher I notice you are not working on the assigned questions. What's up?

Mikaela

We did some of them.

Researcher I saw that. I noticed that you did two very quickly. Took a little break from the math and then went back and did another one. I was sort of waiting to see if you would get back to it.

Allison

This stuff is easy. I'll finish it at home on my own.

Mikaela

It's actually too easy. I don't even think I will bother finishing it at home. Allison … Yeah. I probably won't either.

During the ten lessons I observed in Cameron's and Charmaine's classes I only managed to capture three other instances that I would say fall into the same categoryquitting because the students were bored by seemingly too easy a collection of tasks.

When skills exceed challenge: The case of seeking increased challenge

Quitting out of boredom was not the only reaction to a situation where the skills of a group or of an individual exceeded the challenge of the task at hand. Some students opted, instead, to autonomously seek increased challenge. To exemplify this I look at a case from Cameron's class captured while students were working at the whiteboards in randomly assigned groups. During this part of the lesson Cameron moved around the room helping groups that were stuck (or had made a mistake) and giving more challenging questions to groups that were done. Before a group would get his help or the next question, however, he engaged the group in conversation to assess where the group's thinking was. This took time and sometimes groups that were done were left waiting.

Researcher Researcher Do you realize that you are doing the problems out of sequence from the order your teacher is giving them?

Colton

Oh really? That's probably why some were so hard.

This was a very common reaction in both Cameron's and Charmaine's classrooms. Rather than wait for their teacher to give them the next questions groups were opting, instead, to move forward on their own by pulling the next question from groups that were ahead of them. This was facilitated by the visible nature of the work on the vertical surfaces.

When skills exceed challenge: The case of tolerance in the face of the mundane

An altogether different reaction to being tasked with doing easy and redundant questions is to just do themwithout quitting and without seeking to increase the challenge. I observed such behavior in the case of Jennifer, who always worked at her desk on her own at the end of Charmaine's lessons.

Researcher I have been watching you while I have been here. I notice that you always do a lot of questions. Can you tell me about that? Jennifer Yeah. I like to do a lot of questions. It's good practice. It's how I learn.

Researcher So, are you looking for harder and harder questions to challenge yourself? Jennifer Not really. I just do all of them. So, if the teacher asks us to do 4a, I will also do 4bc and d and so on.

Researcher Do you find them easy? Jennifer Yeah.

Researcher How many do you do?

Jennifer I just work the whole time at the end of class and then for maybe an hour at home. I came to call Jennifer's behavior tolerance for the mundane. In my time in Cameron's and Charmaine's classes I saw two other girls who I suspect were very much like Jennifer in their approach to learning and their tolerance for the mundane. These girls also worked alone in their desks in the last part of every lesson.

When challenge exceeds skills: The case of quitting

Csíkszentmihályi's framework (1990) predicts that sometimes students quit out of frustration. I found three cases of this in Cameron's classroomall near the beginning of class.

Researcher I have been watching your group for a bit and I notice that you aren't working?

Robert

We gave up. This question is stupid.

Katrina

We tried, but we weren't getting anywhere. So we gave up.

Researcher For this group the question they have been asked to solve exceeded their abilities and without any help from the teacher they gave up. Interestingly, the help they were seeking was not only to reduce the complexity of the task (understand the question), but also to increase their ability (remind us of what we have done in the past). In the ten lessons I observed, I only managed to capture four instances of a group giving up out of frustration.

When challenge exceeds skills: The case of seeking help

A much more common reaction to facing too great a challenge was for students to seek help. What this looked like, however, was much more subtle than simply asking the teacher for help.

Researcher I notice that you have been moving about the room a bit. Why? Michael Oh. We were just stuck so we went over there to get some ideas.

Researcher Did it help? Michael Oh yeah. We got it now.

This sort of behavior was endemic in both classrooms with too many occurrences for me to track. The vertical and visible work spaces facilitated the ability for groups to check their answers and get ideas. The random groups created the porosity [START_REF] Liljedahl | The affordances of using visibly random groups in a mathematics class-room[END_REF] that made the more active interactions and movement of ideas possible.

When challenge exceeds skills: The case of perseverance in the face of challenge

But not all groups sought help when they were stuck.

Researcher Question #5 was a tough one, huh?

Oliver Yeah, that one took us a while.

Connor

In the end it wasn't that hard though. We were just missing something.

Researcher Oh really. How did you figure it out?

Connor

We just kept at it and then we saw it.

Researcher I noticed that your teacher came over to help. Did she help you?

Oliver No, we wouldn't let her. We knew how to do it and we wanted to figure it out ourselves.

In all the lessons I observed I captured four instances where a group or an individual opted to not seek help, either from the teacher or the groups around them. I called this behavior perseverance in the face of challenge.

Discussion

The aforementioned six nuanced student reactions to being out of flow show that for different individuals and different groups the transitions from flow to boredom or frustration has variable immediacy. Some groups became bored or frustrated and quit. For these groups, [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF] original representation of flow holds (see fig. 1).

For others, this transition was not as abrupt. Jennifer showed a great tolerance for the mundane as she spent long periods of time within a space where her ability far exceeded the challenge posed by the tasks she was working on. Likewise, Connor and Oliver demonstrating great perseverance while working on a task that presented too great a challenge for her ability. Taken together, these two cases, and the cases like them, indicate that for some students the boundary between flow and boredom and frustration is not as thin as [START_REF] Csíkszentmihályi | Flow: The psychology of optimal experience[END_REF] theory of flow would imply and is buffered by tolerance and perseverance (see fig. 5).

Other students used this buffer to avoid frustration or boredom as they sought to correct the imbalance between skill and challenge that they were experiencing. Carl, Ameer, and Colton used the groups around them to check their own answers and to seek out more challenging tasks when they were done. Similarly, Michael's group used the groups around them to access help when they were stuck. These groups, and the groups and individuals like them, managed to autonomously maintain the balance between challenge and ability. When their ability was too great they autonomously sought to increase the challenge (see fig. 6) and when the challenge was becoming too great they autonomously sought to increase their ability or decrease the challenge (see fig. 7). The highly visible and collaborative environments created by the use of vertical non-permanent surfaces and visibly random groups were shown by the data to be instrumental in facilitating these autonomous actions.

Fig. 1 :

 1 Fig. 1: Graphical representation of the balance between challenge and skill

Fig

  Fig. 2: Balance as a dynamic process

Fig

  Fig. 5: Modified representation of flow

  So, I notice that you guys are now on question 5 and your teacher has not visited you once. How are you getting your questions?AmeerWe just look around and see what the next question is and do that one.

	Researcher What would your teacher say about that?
	Carl	Um … he'd probably want to check to see that we got the previous one before giving
		us the next one …
	Ameer	… but we are doing that.

Researcher Why don't you just wait for your teacher to get here and give you the next question?

Carl

We're on a roll. And sometimes we have to wait a long time.

  What do you think the problem is?

	Shannon	This question is too hard.
	Robert	… too hard. We don't get it.
	Katrina	And the teacher hasn't come over to help us.
	Researcher What kind of help are you looking for?
	Shannon	You know, a hint or something.
	Researcher What would a hint do for you?
	Shannon	Help us understand the question.
	Katrina	… or remind us a little bit about how to do it.

Proceedings of CERME10