Pre-service primary teachers’ shame experiences during their schooling time: characteristics and effects on their subject-choices at university

Emotions play an essential role in pre-service teachers’ competence development, particularly in mathematics. However, the emotion of shame in mathematics has been largely neglected so far. This article deals with shameful experiences of pre-service primary school teachers during their mathematical education at school and the various effects of shame on their decision to study mathematics as a subject at university. The research consists of a qualitative and a quantitative study with 311 prospective primary school teachers who responded to a survey about their experiences of shame in mathematics at school when they were students. Results of the qualitative study emphasize the different experiences in mathematics during the school years and reveal the characteristics of these situations, for example, social exposure or competition games. In the quantitative study, pre-service primary teachers’ subject choice was analyzed in relation to their experienced shame in mathematics at school. Results reveal that shame experienced at school has effects on the initial choice in favor of mathematics at university. Implications for primary teacher education are finally discussed.

Emotions consist of affective, cognitive, physiological, and motivational components (Pekrun et al., 2018) and can range from being pleasant to unpleasant emotions (Feldman Barrett & Russell, 1998). According to Hannula (2012), emotions can be conceptualized as an immediate emotional reaction within a situation (emotional state) and also as a situation-independent emotional disposition (emotional trait). Theories such as control-value theory (Pekrun & Perry, 2014) and attribution theory (Weiner & Kukla, 1970) suggest that the development of emotions can be explained by evaluations (appraisals) of situations. For instance, some studies revealed that experiences teachers had during their own schooling could influence these evaluations and, thus, their current emotions in a particular domain such as mathematics (Bekdemir, 2010;Brady & Bowd, 2005;Coppola et al., 2013;Scott, 2005).
Emotions associated with mathematics have been of interest for several years as they are important for the learning and teaching of mathematics (Hannula, 2012;Schukajlow et al., 2017). Although the importance of shame for mathematics teachers has repeatedly been emphasized (Bibby, 2002;Frenzel, 2014;Panagi, 2013), this emotion plays almost no role in current educational research. For the significance of shame in educational contexts, both the population of pre-service primary school teachers and the subject of mathematics seem to be particularly relevant (Bibby, 2002). Accordingly, Panagi (2013) assumed that shame-inducing situations in mathematics during school education are highly relevant for the professionalization of pre-service teachers, as they may influence choosing or avoiding mathematics as a subject in teacher education for primary school. However, this aspect has not been examined so far. The current study raises the question of how pre-service teachers' shame experiences at school can be characterized and how these experiences of shame affect their choice of mathematics as a subject during their university teacher education.

Phenomenology, causes, and effects of shame
Shame is defined as an unpleasant and activating emotion that can be described by different components (affective, cognitive, physiological, motivational;Pivetti et al., 2016). Shameful situations are affectively described as distressing, painful, and embarrassing, often accompanied by feelings of insufficiency. The embarrassed person feels humiliated or disgraced. In the foreground are self-referential pejorative cognitions that evaluate the person in a less differentiated and global way (e.g., "I'm not right the way I am."). Cognitions of shameful feelings are always self-referential (Lewis, 2003). Physiologically, shame manifests itself in a loss of muscle tension or blushing (Dickerson et al., 2001). Shame has a motivational effect, resulting in a tendency to restore, avoid (withdrawal), endure, or overcompensate (e.g., aggression; Ashby et al., 2006;Stuewig et al., 2015).
As for emotions in general, the experience of shame can be described in terms of frequency and intensity (Bailen et al., 2019;Diener et al., 1985). Shame, understood as an emotional disposition, develops through the interplay of biological, psychological, and social factors across the personal life span (Orth et al., 2010). Also, shame can be seen as an acquired emotion (Lewis et al., 1992). Consequently, the experience of shame is significantly determined by social experiences and social standards (Lewis, 2003). For example, situations that promote social comparison or involve social exposure may enhance shameful experiences (Smith et al., 2002). Studies have revealed that shame increases from early childhood to adolescence, decreases in middle adulthood and increases again in old age (Orth et al., 2010). In primary school, children are more and more able to differentiate emotions and also to identify shame (Seidner et al., 1988). Around the age of ten, children tend to relate shame to their achievement (Ferguson et al., 1991). As unpleasant emotions increase from primary school to secondary school due to increasing achievement requirements (Pekrun et al., 2018;Vierhaus et al., 2016), one could assume that shame experiences also increase from primary to secondary school. In view of these findings, it could be fruitful to differentiate between developmental stages when examining an individual's levels of shame.
As mentioned earlier, emotions do not result because of the situations per se. Moreover, emotions such as shame are conceptualized as emotional reactions to appraisals regarding these situations (Tracy & Robins, 2006). Both attribution theory and control-value theory explain how shame is experienced due to specific appraisals. According to attribution theory, shame represents an emotional reaction to the global negative evaluation of one's own performance, for example, after a failure (Lewis et al., 1992;Russell & McAuley, 1986). In terms of control-value theory, individuals experience shame when they believe that they do not have enough resources (e.g., knowledge) in a valuable domain (e.g., mathematics; Pekrun & Perry, 2014;Turner & Schallert, 2001). However, appraisals are affected by general beliefs which are formed through recurrent experiences. For example, a parental style using frequent punishments can cause shame already at pre-school age (Alessandri & Lewis, 1993). These parental evaluations may form an individual's general belief system about one's own performance. From the perspective of social learning theory (Bandura, 1977), it can be assumed that emotional experiences may be triggered by both experiences which individuals had by themselves or by observing others experiencing this emotion. This assumption has already been validated for the experience of shame (Schmader & Lickel, 2006). Additionally, studies revealed that shame can also be experienced by persons although it was another person who had shown for example failure or misbehavior, just because the persons identified themselves with the other person (Salice & Sánchez, 2016).
The experience of shame leads to various consequences. According to the control-value theory, unpleasant emotions show negative effects on achievement via mediating variables such as motivation (Pekrun & Perry, 2014). Hence, shame can be conceptualized as an achievement emotion that is closely related to requirements and knowledge in a domain (Thompson et al., 2004). Studies have also revealed that ashamed individuals motivationally tend to avoid shame-eliciting contexts (Schmader & Lickel, 2006).
To sum up the described findings, shame is understood as an unpleasant and activating emotion. It occurs predominantly in social situations due to specific beliefs concerning these situations, especially in relation to oneself. These appraisals show various effects and lead, for example, to poor performance and a high avoidance tendency.

Shame and mathematics
Research has shown that the development of emotions in mathematics depends on a variety of aspects such as one's beliefs or teacher's instruction (Schukajlow et al., 2017). Both aspects can be related to appraisals or attributions in terms of control-value theory (Pekrun & Perry, 2014) or attribution theory (Weiner & Kukla, 1970) in relation to the subject of mathematics. For example, subject specificity of shame in mathematics may be due to mathematics-specific beliefs (Panagi, 2013). Mathematical achievement is commonly understood as an innate ability or is seen as a proxy for general intelligence (Goldin, 2014). However, at the same time, there is a collective belief that mathematics is an important cultural asset and that the correct understanding and use of mathematical tools is an essential twenty-first-century skill (Goldin, 2014). Thus, there is the threat of attributing failures globally to one's own abilities in a subjectively significant domain. This observation can facilitate the experience of shame. The conceptualization of mathematics based on a static perspective as a collection of algorithms and formulas (as opposed to a dynamic and developing perspective) can increase the potential for subjective failure (Roesken et al., 2011). Also, correct solutions are attributed an enormous importance in mathematics (at least in rather traditional mathematics classrooms), in that solutions of given problems in class are constantly evaluated as either right or wrong. Such a permanent evaluation may lead to unpleasant emotions for students when they fail to find the right solution (Bibby, 2002;Goldin, 2014).
Moreover, studies showed that specific instructional methods can influence students' emotional experiences (Bieg et al., 2017). Finlayson (2014) has suggested that methods like direct instruction promote a higher right-or-wrong orientation than reform-based practices valuing a discourse orientation in the mathematics classroom. In addition, it may be also likely that methods with high social exposure (e.g., calculating at the blackboard, competition games) enhance a student's experience of shame over time. These methods are still more frequently used in teaching mathematics than in other subjects (Bieg et al., 2017).
To summarize, the theoretical assumptions and the empirical studies emphasize that the occurrence of shame in mathematics can be seen as specific because of an individual's characteristic belief system about mathematics, the predominant conceptualizations of mathematics, and mathematics-specific instructional methods applied in mathematics education.

Pre-service primary school teachers' shame
Pre-service primary school teachers report shame during their school years more often compared to other subjects, for example sports, but rather less often compared to other emotions (Jenßen et al., 2020). This can also be due to modes of assessments or because shame is a rather fleeting emotion (Frenzel, 2014), which rapidly expresses itself in other emotions like anger (Tangney et al., 1992). However, shame is usually experienced intensively. One shameful situation can be enough for teachers to doubt their own abilities as teachers (Basso, 2014). Shame can be found especially among teachers, because they are constantly socially exposed in the classroom (Frenzel, 2014), and they are constantly subject to social evaluations and comparisons (Miles, 2017). Moreover, shame can be understood as an identity-determining emotion of teachers (Zembylas, 2003). Shame is particularly relevant for prospective primary school teachers because they are often trained as generalists (Bibby, 2002;Cooke et al., 2019). They therefore have less specialized content knowledge (Bibby, 1999), and this is one of the most important facets of teacher knowledge (Loewenberg et al., 2008). This means that they have fewer resources to meet mathematical requirements (this leads in turn to a lower control-appraisal which can promote the experience of shame). This is problematic because mathematics is a core and compulsory subject matter of practicing primary school teachers.
Emotions play an essential role for choosing and working in the teacher profession (Skaalvik & Skaalvik, 2011). Unpleasant emotions in particular play a role in the avoidance of mathematics-related professions (Chipman et al., 1992;Huang et al., 2018). Because shame triggers a high avoidance tendency, one can hypothesize that pre-service primary school teachers with a high experience of shame neither choose mathematics as a subject in their studies nor as a specialization. This vicious circle could maintain the experience of shame.

Current study and research questions
The investigation underlying this paper took place in the context of teacher education at a university in a greater urban area in Germany. During primary teacher education in Germany, mathematics can be chosen as one of various subjects. However, it is possible to avoid mathematics as a core subject. During the course of their studies (after about 2 years), prospective primary teachers may decide to choose mathematics in advanced courses, when they already have chosen mathematics as a core subject (specialization subject). Students then have more courses in mathematics and mathematics education (Cooke et al., 2019). It has to be noted that in Germany education systems are different in different federal states. In some federal states, it is not possible to avoid mathematics as it is a mandatory subject. In some of these federal states, mathematics can already be chosen as a specialization at the beginning of the first semester.
The current study attempts to close major research gaps in relation to shame of preservice primary school teachers in mathematics. The paper reports on two studies, a qualitative and a quantitative one. The qualitative study examines the experiences of shame that prospective primary school teachers have had during their own primary school time in mathematics. The research question is: RQ1: "What are the characteristics of shame-eliciting situations experienced by preservice primary teachers during their mathematics education at school?" The second study is based on a quantitative approach and examines the effects of experienced shame during schooling on the choice of mathematics during pre-service primary school teachers' education at university. There are two stages of decisions (first decision, mathematics as a core subject, second decision, mathematics as a specialization), and the second decision depends on the first one, that is only those who have chosen mathematics as a core subject can choose mathematics as a specialization. Therefore, two decisions including two groups each can be distinguished. For the decision about mathematics as a core subject, non-choosers and choosers can be distinguished and also non-choosers and choosers regarding mathematics as a specialization. In total, four groups have been examined in the present study.
We examined two independent research questions: RQ2a: "Do pre-service primary teachers who have chosen mathematics as a core subject differ from those who have not in their experienced shame at school when they were students?" RQ2b: "Do pre-service primary teachers who have chosen mathematics as a specialization differ from those who have not in their experienced shame at school when they were students?" Because of theoretical assumptions about the avoidance tendency following the experience of shame and empirical evidence for lower mathematics achievement when highly experiencing shame, we hypothesize that the two groups of RQ2a differ in their experienced shame at school (H1). We also assume this difference for the two groups described in RQ2b (H2). However, as the choice of mathematics as a specialization only can apply to those pre-service primary teachers who have already chosen mathematics as a core subject, we assume that this effect is lower than the effect proposed in H1. Based on data, we potentially differentiated shame for each school level with respect to theory and empirical findings regarding developmental differences of shame. One might assume that shame at different school levels might affect pre-service teachers' decisions in a different way. Theories and studies suggest that shame develops with increasing achievement requirements in mathematics at school as the potential of failing rises. As achievement requirements at university are also high, one could conclude that the effect of pre-service primary teachers' shame in mathematics experienced in secondary school is more associated with their decisions regarding mathematics during teacher education than their shame in mathematics experienced in primary school.

Participants
The study was conducted with the participation of n = 311 pre-service primary school teachers of a university in a greater urban area in Germany. The majority of the participants were female (84%), reflecting the gender composition in the primary school teaching profession in Germany. The average age of the participants was M = 25.66 years (SD = 6.91) with a minimum of 16 years and a maximum of 56 years. Participants indicated M = 2.44 (SD = 1.04) as their last grade in mathematics at school. In Germany, grades range from 1 (= best grade) to 6 (= worst grade). The participants were recruited from the Bachelor program (76%) and also from the Master program (24%) for pre-service primary teachers. During their studies, all of the participants take courses in arithmetic, geometry, and stochastics, 1 including both a mathematical and mathematics education perspective. Although no specific courses are dedicated to exploring emotions in general or shame specifically, these aspects are discussed within several courses, for instance, when reflecting pre-service primary teachers' experiences during the practical term at school.
Participants worked voluntarily on the survey during regular instruction and did not receive any incentives. At any time, they were given the option to stop participating in the study.

Data collection
In the present study, a survey was used to collect qualitative as well as quantitative data (Bergin, 2018). At the beginning of the survey, socio-demographic data on the person (gender, age, semester/study program, last grade in mathematics at upper secondary school) were gathered. To ensure that participants understood what we meant by the term "shame," the phenomenology was described at the beginning of the survey: The feeling of shame is understood as an unpleasant emotion, which is often accompanied by negative thoughts (e.g., 'I'm not good enough') in achievement situations. Additionally, shame can lead to the wish that the ground would open and swallow oneself up. Feelings like embarrassment, humiliation and disgrace can also be related to the emotion of shame.
The first part of the survey consisted of two dichotomous items, one multiple-choice item and one open response item. In the first dichotomous item, participants were asked to answer whether they have experienced a shame-eliciting situation in mathematics as an observer and in the second dichotomous item, as a directly ashamed person when they were students at school. After the two dichotomous items, participants were asked to remember their subjectively worst shame-eliciting situation when they were students at school (cf. Bekdemir, 2010). The next multiple-choice item asked for the school level in which this situation was experienced. The open-response item then asked the participants to describe in detail their most unpleasant experience that initiated a shameful experience. Participants were encouraged to name persons involved in the situation, mathematical content domains, and instructional methods used, when possible. These items were part of the qualitative study while the next items were part of the quantitative study.
In the quantitative study, participants rated on a 5-point scale ranging from 0 (= never) to 4 (= always) three questions regarding frequency (one for each of the three school levels: "How often did you experience shame in mathematics at this school level?"), and a 5-point scale ranging from 0 (= not intense) to 4 (= very intense) to answer three questions regarding the intensity (one for each of the three school levels: "How intensely did you experience shame in mathematics at this school level?"). At the end of the survey, participants were asked for their subject choices ("Have you chosen mathematics as a core subject at university?" and, if applicable "Have you chosen mathematics as a specialization subject at university?").
Prior to administration, we tested the survey to ensure practical application. We assume high face validity and poor item subtlety of the items as we have directly asked for the experiences of shame at school. The combination of high face validity and poor item subtlety seems to be desirable in terms of item validity (Holden & Jackson, 1979). Reliabilities of scores are reported in the next section.

Data analysis
The qualitative data were analyzed using content analysis techniques (Bergin, 2018;Mayring, 2014). In the first deductive step, the main aim was to find patterns in the data and to examine which themes tended to appear frequently in the participants' responses to the open-response item mentioned above. All the passages in which participants directly or indirectly mentioned "persons involved," "mathematical content domain," and "instructional methods used" were coded for this purpose. These categories were derived from the research literature about shame with regard to its social relatedness (e.g., Lewis, 2003) and literature about specific instructional methods (e.g., Bieg et al., 2017) and mathematical content domains (e.g., Goldin, 2014) which can affect emotional experiences. In the second step, the category "persons involved" was divided into deductive categories (teacher, classmates, student themself). For the two other categories, responses were coded inductively from the material to get sub-categories, so that we were able to examine which specific instructional methods and which specific mathematical content domains are related to shame experiences at school. About 30% of the given responses were coded twice by two independent and trained raters to ensure inter-rater-reliability. Findings showed high interrater-reliabilities for sub-categories (Cohen's Kappa between 0.84 and 1.00, M = 0.89).
The quantitative analyses of the six shame items comprised testing for differences in shame experiences across school levels in dependence of two groups (first analysis: preservice primary teachers who have chosen mathematics as a core subject at university vs. those who have not; second analysis with a subsample of the first analysis: pre-service primary teachers who have chosen mathematics as a specialization vs. those who have not). To reduce the number of dependent variables of our survey data without losing empirical information, based on the correlational pattern of the shame items (see Appendix Table 4), we performed a principal component analysis (PCA) to extract the most important independent factors. The Kaiser-Meyer-Olkin measure of sampling adequacy was 0.71, representing a relatively good factor analysis fit (Kaiser & Rice, 1974), and Bartlett's test of Sphericity was significant (p < 0.001), indicating that correlations between items were sufficiently large for performing a PCA. The varimax-rotated solution revealed three factors with each item loading on one factor (see Appendix Table 5). Each factor represented shame experiences (frequency and intensity) for one of the three school levels. Shame scores were generated by adding the frequency item and the intensity item for each school level separately. Reliabilities (Cronbach's) of the shame scores were high ( primary = 0.89, lowersecondary = 0.90, and uppersecondary = 0.91).
These shame scores were used as dependent variables in the further analysis. In light of the aim of the analyses, we decided for a multivariate analysis of variance (MANOVA) (cf. Pituch & Stevens, 2019). As the sample of the second analysis represents a sub-sample of the first analysis and consequently, sample sizes differ, we decided for two one-way MANOVAs. As basic assumptions of a MANOVA were given in our application (interval level of dependent variables, two categorical groups as independent variable, independence of observations), we have checked empirically for further assumptions of a MANOVA (neither univariate nor multivariate outliers, multivariate normality, no multicollinearity, linear relationships, homogeneity of error variances and covariances). Multivariate normality was analyzed through normality of each of the residuals for each group of the independent variable, because all analyses were performed by using SPSS 25.0 (IBM Corp., 2017) and the possibility to test for multivariate normality is not implemented in SPSS. However, it has to be noted that violations of some assumptions still allow the application of MANOVA. According to Finch (2005), MANOVA delivers robust results if the assumption of normality is violated. If linear relationships between the variables do not exist, statistical power decreases, but the application of a MANOVA is still allowed (Pituch and Stevens, 2019). Partial 2 was estimated as effect size (Ellis, 2010). In case of significant results, post hoc univariate ANOVAs and discriminant analyses were performed to detect significant differences between the groups in detail. Level of significance was Bonferroni-corrected.

Qualitative study: characteristics of situations eliciting shame at school
In the qualitative part of the study, pre-service primary school teachers were asked for situations that they subjectively experienced as shameful. To describe typical situations eliciting shame at school, we first distinguish between situations experienced as an observer or as a directly ashamed person. We then report at which level (primary, upper, or lower secondary) the worst shame-eliciting situations were experienced. Finally, we categorize the reported situations with respect to "involved persons," "mathematical content," and "instructional methods." Sub-categories of these three main categories describe characteristics of situations eliciting shame at school besides the characteristics "observer vs. directly ashamed person" and the school level. Coded categories, sub-categories, and descriptions are given in Appendix Table 3.
The majority of the 311 participants stated that they had observed shameful situations during mathematics lessons at school (78%). Each participant reported one situation. All of these participants reported that they had an unpleasant feeling in this situation, except for two participants who experienced a pleasant feeling while observing the situation. Nearly 69% of all participants reported they have experienced shameful situations themselves during mathematics lessons in which they were the ashamed person. The worst shame-eliciting situations were more often reported for secondary school (lower secondary school, 48%; upper secondary school, 35%) than for primary school (only 17%).
In the following, we classify the situations that the pre-service primary teachers considered as shameful in mathematics during schooling with respect to the persons involved, the mathematical content at hand and the instructional methods used by the teacher. In 38% of the situations described, the teacher was significantly involved in the induction of shame. Hostile behavior patterns on the part of the teacher were cited as particularly shameful (37%), with physical aggression also being reported in two cases (objects thrown at students). A participant (P) reported: I had to calculate a homework task on the blackboard. I didn't understand the homework at home, which I also told the teacher before the lesson. But the teacher insisted that I work on the homework on the blackboard. None of my classmates were allowed to help me and the teacher did not support me in working on the homework, but rather embarrassed me in front of the class. The whole thing lasted about 10 minutes, then I was allowed to sit down again and another student was supposed to work on the task. (P79) In this case, the teacher's hostility is evident in that he or she forced the participant to work on the task on the blackboard even though the teacher knew that participant P79 was not able to do so. The teacher also deprived participant P79 of social support from classmates, making a strategy use impossible to overcome his or her perceived helplessness.
In 10% of cases, classmates were identified as the main actors in the situation. In most of the cases, the combination of teacher and classmates as inducers was reported, for example, in the situation of participant P1: I had to work on math tasks on the blackboard. The teacher forced me to do so. The teacher commented on my lack of knowledge at the blackboard and made fun of me so that the whole class laughed at me. (P1) In 14% of the situations, no other actors than the student themself were named. Only the participant's own failure without the reaction of others was named as the cause of the experience of shame. P256's response also emphasizes the avoidance tendency, which may exist in anticipation of shame (fear of failure).
In primary school I was one of the best pupils. Whenever I had problems with a task, I always went to the rest rooms. So, I could not get to the discussion of the solutions. Everyone always thought that I could do everything in math. (P256) The response shows that shame in mathematics can also occur in people who otherwise tend to see themselves as high performers in mathematics. P256's strategy seems to be to maintain this attribute also before others.
Regarding "mathematical content," arithmetic (50%) or stochastics (30%) were reported most often by the participants in the worst-shame-eliciting situation. Geometry, calculus, or elementary linear algebra were reported less often (7% each) by the participants. Other mathematical content domains were not mentioned.
Most frequently, "instructional methods used" by the teacher with high social exposure in mathematics teaching were reported: Calculating at the blackboard (86%), public feedback on performance (20%), competition games (19%), and the presentation of homework (11%). In about 2% of the situations, the shameful experience of mathematics occurred during group work in the sense that the person was excluded from normal group work. Regarding competition games, P36, for example, reported the following aspects: We had a competition with mental arithmetic in class. Very difficult tasks were set by the teacher, which I could not solve. But you were only allowed to sit down when you had solved a task correctly. If you did something wrong, the teacher laughed at you. I was the last one standing there and everyone could see that I was the worst in math. (P36) The situation shows that subjective task difficulty can also be a condition that can promote a shameful experience. It becomes clear that P36 subjectively perceives the tasks as difficult, although everyone else can solve them. The comparison with the group's achievement level is particularly evident in this public competition.
The reported situation of P197 also describes the public feedback on performance as a shame-eliciting feature: During independent silent work, the teacher checked my solutions and then said loudly in front of the class "I think you're just worse in math." (P197) The teacher provides P197 with an explanation of his or her achievement, attributing it to be stable, internal, and uncontrollable factors. As a result, negative self-concepts and dysfunctional beliefs can become established, which can continue to negatively influence the participant's mathematical achievement.

Descriptive results
Descriptive results for shame scores of each school level for all four groups are presented in Table 1. The possible range of each score was between minimum = 0 and maximum = 8 (as each score consisted of two items). It is obvious that the parameters indicate fewer shame experiences regarding frequency and intensity across school levels. Mean values seem to increase from primary school to secondary school. The highest mean value was found at lower secondary school level in both samples. It has to be noted that N 2 (mathematics as a specialization) is a subsample of N 1 (mathematics as a core subject), because pre-service primary school teachers decide on mathematics as their specialization after they have decided on mathematics as a core subject.

Shame and choice of mathematics as a core subject
The majority (94%) of the participants have chosen mathematics as a core subject. Only n = 16 participants did not choose mathematics as a core subject. Before we analyzed the data by the application of a MANOVA, we had checked empirically for the assumptions of such an analysis. No univariate or multivariate outliers were found, as assessed by the Mahalanobis distance (p > 0.001). Normality was given for the groups who have not chosen mathematics as a core subject, but not for the groups who did so (assessed by the Shapiro-Wilk test, = 0.05). Correlations between the shame scores of the school levels (dependent variables) were low (r between 0.40 and 0.59), indicating that multicollinearity was not a confounding factor in the analysis. Linearity between each pair of the shame scores within each group (chooser vs. non-chooser) was given. Homogeneity of the error variances for all the shame scores was given as assessed by Levene's test (p > 0.05). There was also homogeneity of the covariances, as assessed by Box's test (p > 0.001). The one-way MANOVA showed a statistically significant difference between the participants who had chosen mathematics as a core subject and the participants who did not on the combined dependent variables, F(3, 285) = 7.395, p < 0.001, partial η 2 = 0.072, Wilk's Λ = 0.928. According to Ellis (2010), the effect size can be regarded as medium.
In the next step, post hoc univariate ANOVAs were conducted for every dependent variable (see Table 2). Results showed a statistically significant difference between both groups (chooser vs. non-chooser of mathematics as a core subject) for shame experienced in primary school, for shame experienced in lower secondary school, and also for shame experienced in upper secondary school. Effect sizes were small to medium in accordance with Ellis (2010). The MANOVA was followed up with a discriminant analysis, which revealed one discriminant function (canonical R 2 = 0.07). The discriminant function significantly differentiated both groups (chooser vs. non-chooser of mathematics as a core subject), Λ = 0.93, 2 (3) = 21.40, p < 0.001. The correlations between outcomes and the discriminant function revealed that shame experienced in primary school, in lower secondary school, and in upper secondary school loaded highly onto the function (r = 0.86 for shame experienced in primary school, r = 0.48 for shame experienced in lower secondary school, and r = 0.76 for shame experienced in upper secondary school). The coefficient was lower for shame experienced in lower secondary school than for the other two school levels.

Shame and choice of mathematics as specialization subject
As a specialization is only possible in the second half of the bachelor study, this aspect was analyzed only for a subsample of N 2 = 168 pre-service primary school teachers. In this analysis also, no univariate outliers were found. However, two multivariate outliers were found, as assessed by the Mahalanobis distance (p > 0.001). Nevertheless, both cases remained in the sample, because the scores of these two participants were theoretically possible and justifiable. 2 The correlations between the dependent variables were low (r between 0.37 and 0.57), indicating that multicollinearity was not a confounding factor in the analysis. Therefore, linearity was assured. There was homogeneity of the error variances for all shame scores (p > 0.05) as assessed by Levene's test, and also there was homogeneity of covariances, as assessed by Box's test (p > 0.001).
The one-way MANOVA found no statistically significant differences between the participants who had chosen mathematics as specialization and those participants who did not on the combined dependent variables, F(3, 164) = 1.458, p = 0.23, partial η 2 = 0.026, Wilk's Λ = 0.974.

Discussion
Our results show that all the pre-service primary teachers in our sample reported shameful experiences in mathematics during their schooling time, either as observers or as directly involved persons and that these shame-eliciting situations cover many different characteristics. That is, different characteristics in terms of the persons involved (teacher, classmates, and the student themself), the mathematical content at hand, and the instructional methods used by the teacher are relevant. It might therefore not be enough to consider only one single element or characteristic for the explanation (and also for the reduction) of shame experiences in mathematics at school. Our qualitative data show that the role of the teachers is particularly important, both in terms of their own behavior and the choice of instructional methods. Categories that covered these aspects were the most frequently reported ones (e.g., in comparison to classmates). Competitive games have also been reported as aspects of shame-eliciting situations. Reasons for that might be that social exposure and social comparisons are assumed as shame-eliciting factors in general (Smith et al., 2002) and competitive games highly expose individuals and encourage them to compare themselves with others with regard to mathematical achievement. Our results also show that shame might be experienced when doing group work at school, when students are, for instance, abandoned from the group by its members as reaction to the student's poor performance in mathematics. There is also some social exposure in group work situations which can cause shame.
Regarding the role of the mathematics content, our findings reveal that shame can be experienced across different mathematical domains. We suppose that arithmetic was so often mentioned as a context because it represents the majority of mathematics in school across all school levels. In addition, the handling of numbers dominates in this area, and traditional teaching styles focus on executing procedures and not on inquiry-based learning (Finlayson, 2014). Thus, such a mathematics classroom can quickly trigger students' negative associations or unpleasant emotions. Shame is experienced in the domain of stochastics too. One reason could be that the potential for cognitive confusion is high (Goldin, 2014) due to intuitive approaches that often conflict with the mathematically required ones.
From a theoretical perspective, our research extends the findings from other studies on pre-service primary teachers' unpleasant experiences in mathematics. Our results differ from those in the study of Bekdemir (2010) who has investigated negative experiences of math-anxious pre-service teachers at school. Bekdemir (2010) reported more frequent associations between situations such as exams and mathematics anxiety. In contrast, we found an association between situations with high social exposure and shame in mathematics. Thus, different characteristics of situations in mathematics classes might be related to different unpleasant emotions (in this case anxiety vs. shame). Additionally, we did not find any hints for negative attitudes towards mathematics per se. In addition to identifying the teacher and classmates as sources of shame, our participants also identified themselves, as devaluating their self-view. One could assume that self-related beliefs in mathematics may be more relevant for shame, while beliefs concerning the nature of mathematics are more related to the experience of mathematics anxiety. It has to be noted that participants in Bekdemir's study (2010) had already completed mathematics courses during their teacher education. In our study, participants who did not choose mathematics as a core subject and consequently did not take mathematics courses during teacher education were also included in our qualitative examination. Therefore, we can assume that their shame in mathematics is mainly based on previous experiences, prior to their teacher education. The participants' statements in Bekdemir (2010) may be confounded by their emotional experiences during teacher education, given that the study excluded pre-service primary teachers who did not choose mathematics. It is precisely the group avoiding mathematics that is of interest when it comes, for example, to increasing the number of pre-service primary teachers in mathematics. Our findings suggest that these pre-service primary teachers show higher levels of shame and therefore present a vulnerable group. However, the number of these participants was small in our study (n = 16).
In the quantitative study, shame was assessed as an emotional disposition. The results of our quantitative analyses indicate that experiences of shame in terms of frequency and intensity (Bailen et al., 2019;Diener et al., 1985) are associated with the decision to study mathematics at university, at least with a medium effect. The association between shame experiences and the decision to study mathematics at university was found for each school level. Unexpectedly, these experiences are not related to the choice of mathematics as a specialization. This may show that selection processes already take place in advance.
The results of our studies have to be seen against the background of some limitations. Design of the present study and the survey methodology do not allow causal conclusions. Because of the retrospective assessment, a limited memory performance, especially for primary school, must be assumed. The results may be affected by this memory bias. Moreover, our study is based on an occasional sample from only one university. Sample sizes appear small in the cases of non-choosers of mathematics as a core subject and choosers of mathematics as a specialization. Nevertheless, it has to be noted that homogeneity of error variances and covariances was given, what seems to be more important for performing ANOVAs than sample sizes. Furthermore, Monte Carlo simulation studies revealed the ANOVAs to be robust when analyzing data with unequal sample sizes (Blanca et al., 2017).
The results of both studies have implications for pre-service primary teachers' education at university. In light of our quantitative results, one might argue that mathematics should be a mandatory subject at university for becoming a primary teacher, so that avoidance of mathematics is not possible. Nevertheless, pre-service teachers' shameful experiences should be considered during teacher education. However, our results do not deliver concrete hints for interventions during pre-service primary teachers' education at university. Based on theories and empirical findings on shame in general, some points can be discussed. In regard to the motivational facet of shame in general (e.g., Tangney et al., 1992), pre-service teachers might endure mathematics courses at university or overcompensate during teacher education (e.g., getting angry when working on mathematical tasks without a deeper understanding of mathematical ideas behind these tasks). Furthermore, pre-service teachers' shame in mathematics might affect their shame experiences when teaching mathematics later at school (Bibby, 2002) and following the teacher emotion model by Frenzel (2014), also their instructional quality. With regard to the phenomenology of shame, interventions on reducing primary teachers' shame in mathematics should pay attention to one's own identity, and, in particular, the importance of the social context must be made clear. In this sense, mathematics must also be understood as a social process. An essential aim of teacher education should also be becoming aware of emotions related to mathematics (Goldin, 2014;Liljedahl et al., 2007). A systematic and guided sharing of experiences of unpleasant emotional situations during schooling can be a helpful building block in teacher education. Here, the self-reference of good mathematical performance should also be positively connoted, for example "I can acquire mathematical knowledge" or "I am able to solve mathematical problems." With respect to our results, from a point of view with regard to teacher education, it seems to be not enough to reflect only on experiences from primary school, but also those from secondary school. Additionally, pre-service primary teachers might be supported in strengthening their shame resiliency through fostering their mathematics self-efficacy and to develop goals related to teaching mathematics at school (e.g., "I will need pronounced content knowledge for teaching mathematics effectively at school."). According to Turner and Schallert (2001), pre-service primary teachers could learn that current experiences of shame are emotional signals that they have to work on reaching their goals. During teacher education, pre-service teachers should also be encouraged to build positive role models to hinder shame experiences for students at school. However, we have not examined the effectiveness of such interventions in the present study. Our results only reveal knowledge about characteristics and effects of pre-service primary teachers' shameful experiences in mathematics at school and do not provide causal explanations.

Conclusions
The present study is the first ever to systematically investigate pre-service primary teachers' shame experiences in mathematics. We have used both qualitative and quantitative methods to explore this under-researched area. From a theoretical point of view, our findings reveal the importance of educational characteristics of situations at school for shame in mathematics. By so doing, it extends biopsychosocial models for causes of shame by an educational dimension. Likewise, our analyses validate theories on the complexity of emotions' development in achievement situations, for example, control-value theory. They also highlight the meaning of emotions, shame in particular, for subject choices of preservice primary teachers at university. Shame experienced in mathematics at school affects the beginning step in a process of various decisions concerning subject choice at university.
Shame thus turns out to be an emotion among prospective primary school teachers that merits attention in teacher education. One could assume that studies on unpleasant experiences in mathematics education, which explicitly include feelings of shame, might lead to an increased knowledge of negative teaching practice in mathematics classes. In particular, future research is needed to better understand a variety of factors: shame experience at university, effects on pre-service teachers' knowledge in mathematics, or the prognostic effects on later teaching. Calculating at the blackboard Participant describes how he or she or another student had to calculate at the blackboard in front of the classmates (e.g., solving equations) and doing a mistake while working on the task Public feedback on performance Someone (e.g., the teacher) gives feedback on the participant's or a student's performance (e.g., test score) in mathematics in front of others (e.g., classmates) and the performance is subjectively perceived as poor by the participant Competition games Participant describes a game for the whole class with students being in a competitive position to each other (e.g., "Home run!": the student who gives the correct answer is allowed to go to the next corner of the classroom, students who give four correct answers have a "home run" and win the game) where he or she or another student showed subjectively perceived low performance in mathematics Presentation of homework Participant describes that he or she or another student had to present their findings from homework in mathematics in front of their classmates and the presenting person made a mistake in homework Exclusion from group work Participants describes that he or she or another student had to work with other classmates who have excluded the person from the group work, for example, because of his or her low performance in mathematics Table 4 Correlations between shame items All coefficients were significant (p < 0.001).

Frequency in primary school
Intensity in primary school Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.


Social tie formation of COVID-19 students: evidence from a two-cohort longitudinal study

The COVID-19 pandemic and the consequent move of higher education to online courses has disrupted the learning paths of many students. Social network data were collected from two cohorts of students, those starting their higher education in normal conditions in 2017 and those starting in 2020 during the pandemic. The findings showed that students in the 2020 cohort reported making fewer connections at the beginning of the first semester and developed significantly fewer connections during the first semester. Female students lost the relative advantage they had compared with male students in developing new connections. Based on our findings, and because of the importance of social connections made during the first year of study, the 2020 cohort will need considerable support in catching up with previous cohorts. The findings provide strong support for the assumption that online studies offer limited possibilities in building social connections compared with on-campus education.

Introduction
Social networks, in particular friendship networks, matter for students in higher education (Felten & Lambert, 2020;McCabe, 2016). Having more connections with fellow students has been shown to lead to better performance (Baldwin et al., 1997) and for firstyear students, connecting with similar students is important for their retention (Wilcox et al., 2005). Recent research on Swiss students during the COVID-19 lockdown has, however, shown that students reported fewer interactions with students, and their mental health deteriorated compared with reports by the same students before the lockdown . This could potentially have serious implications for higher education, as students who started their education during the pandemic do not have the same social capital as those starting their studies on campus.
This social network analysis compares the social networks of students starting their education during the COVID-19 pandemic to social networks of students who started the same undergraduate courses 3 years earlier in a normal on-campus environment. In both cohorts, data on social ties along with some background information (such as gender, grade point average, and age) was gathered 1 week after the beginning of the school year and then again in the middle of the second semester of studies. The number of ties each cohort has at these two points in time are compared together with formation of new ties. The findings indicate that the students starting their undergraduate degree program during the pandemic have fewer connections and their social network did not develop over the first few months of studies the way that the social network of students starting the same degree programs 3 years earlier did. Because of this, university administration staff and teachers need to take into account that students who started their education during the pandemic might not have the same peer support as previous cohorts as outlined in the discussion.

Social networks in higher education
The use of social network analysis is useful in a wide variety of social settings. The social network perspective encompasses "theories, models and applications that are expressed in terms of relational concepts or processes" (Wasserman & Faust, 1994, p. 4). This approach has been applied to online social networks and social media sites, but it is not limited to such platforms. Rather, it is characterized by the assumptions that individuals, who are sometimes referred to as "actors" or "nodes" in the network, are interdependent, and that this interdependence can be understood by examining their relationships with other individuals, which are sometimes referred to as "connections," "ties," or "edges" in the network. Social networks have been shown to have important implications for information transfer (Baer, 2010;Granovetter, 1973) and research in higher education show that the number of connections students have are important for their performance (Baldwin et al., 1997). There are also indications that informal connections are the most important in facilitating students' advancement (Hommes et al., 2012). Interviews with first-year students further indicated that connecting with similar students is important in preventing "drop out" (Wilcox et al., 2005). The results of a long-term study of social networks among higher education students indicated that early connections are important throughout the time of study (Kossinets & Watts, 2009). Building social connections in the first year of study thus seems important.
Previous findings showed that in developing their social networks, students who entered with larger networks tended to create more connections than those who began with a smaller network (Zander et al., 2018). Students with larger networks at the beginning of their studies were further found to be more likely to keep their connections (Cho et al., 2007) and increase their network (Zander et al., 2018). Conversely, research has also shown that students at the margins of social networks, despite having fewer connections, were found to be more likely to create more connections (Cho et al., 2007). Although these findings could be seen as being in conflict, the inconsistency could also reflect the fact that the focus of the above research was on the two ends of the spectrum with regards the size of social networks. In other words, it could be that both well-connected students with large networks and marginalized students with small networks are more likely to create new connections than students with average-sized networks.
Social networks in higher education have been found to have a dual connection, as they are both connected to study relationships (instrumental) and friendship (support) (Fjelkner-Pihl, 2022b;McCabe, 2016). The distinction between these is however not always clear as research has shown that friendship ties that evolve into study relationships have been found to be particularly important for students' success (Stadtfeld et al., 2019). Research on complex social ties that cover both friendship as well as connections related to studies further indicate that they often overlap (Fjelkner-Pihl, 2022a, 2022bMcCabe, 2016). McCabe (2016) finds that students' social networks differ in that some students have a tight network, where the connections are both related to study and friendship, whereas other students compartmentalize these connections or even "sampled" them in individual connections. McCabe findings do however indicate that the support is greater in the tight networks and those who sampled their connections did not seek support from their network.
The size and growth of social networks are however not the only thing that matters. Social networks do need to change and adapt as students progress in their studies. Wang et al. (2018) showed that connections with other students with negative attitudes toward their studies could have a negative impact on a student's own attitudes. It is therefore important for students to build supportive connections and shed those that do not support their education (Rios-Aguilar & Deil-Amen, 2012). Losing ties that are not beneficial might, however, not be as easy if the student is in a tight network (McCabe, 2016).
There is limited amount of research on social networks in online courses in higher education. Research has shown that students centrally located with higher number of connections did better than students with fewer connections (Cho et al., 2007). However, there are indications that the networks created in online courses are limited to the course and do not extend to longer term relationships when teachers stop initiating activities (de Lima & Zorrilla, 2017). Thus, even if students have managed to build some connections in their online courses during the COVID-19 pandemic, these connections might not be strong enough to carry over into subsequent academic years.
Given the lack of connections in online education the question remains whether higher education institutes can support connections between students. Research has shown that making large classes seem smaller fosters connection between students (Hommes et al., 2014) and two experimental studies indicate that groups created by the university can influence social networks (Boda et al., 2020;Rientes & Nolan, 2014) although the effect might diminish over the academic year (Boda et al., 2020). The limited long-term effect might, however, be explained by students acknowledging that interventions help but that they only keep those connections that "work" for them (Fjelkner-Pihl, 2022a).

Higher education during the COVID-19 pandemic
Most previous research has been on emergency online teaching during the COVID-19 pandemic. Students felt that studying online increased workload and required more selfdiscipline than a traditional setting (Gelles et al., 2020). The attitude of students might further be important in how well prepared they were to deal with the move to online courses (Alqahtani & Rajkhan, 2020). Student groups were also affected differently by the transfer to online teaching during the pandemic. Female students were increasingly burdened by domestic duties, whereas male students reported having more free time (Gelles et al., 2020). When it came to online teaching male students were, however, less satisfied than female students (Flores et al., 2021).
Research on the effects of the COVID-19 pandemic on social networks has shown that students reported fewer connections during COVID-19 compared with reports by the same students before the pandemic. Under the assumption that social interactions should typically be relatively constant over the course of study, this finding suggests that the pandemic affected social networks, even of students who had opportunities to build face-to-face connections before all classes were moved online . This is in alignment with research on work during the pandemic as comparison between network activities before and during the pandemic when people were working from home, showed that connections became more siloed and static during the pandemic (Yang et al., 2021).
The purpose of the present study is to examine the development of students' social networks during the pandemic compared with that of students who started the same studies 3 years earlier, during what we now refer to as a normal situation in higher education. Comparing the social networks of students that were forced into online education to those of traditional students at the same university could yield important insights into how online education might lead to different patterns in making connections. We further believe that understanding the development of social networks of students in an online program is also important, as it is likely that the pandemic will result in increased offerings of online courses, as both academic staff and students have gained experience in and understanding of the possibilities of online education.

Research context and participants
The research was conducted at the University of Iceland, a comprehensive public university with around 15,000 students. Compared with the rest of the world, Iceland was not as severely affected by COVID lockdown. With the exception of a complete lockdown during the first wave of the COVID-19 pandemic in spring 2020, the campus remained at least partially open for most of the school year from 2020 to 2021. Although subject to strict regulations, teaching in small groups was possible, and most departments prioritized splitting first-year classes into smaller groups and offering first-year students the possibility of coming to the campus at least once a week. However, many students were reluctant to come to campus, and there was a strong demand by students that exams should be held online to limit the risk of contracting COVID-19. Hence, few students used this opportunity to connect with their fellow students.
The participants in the study were students in the School of Social Science and the School of Engineering and Natural Sciences, which are the two largest schools at the university. These two schools offer a broad range of study programs to a large student population. The students' initial participation in the study was at the beginning of their first semester. We then followed the same participants as they progressed through their studies. The first cohort of participants began in the fall of 2017, well before the onset of the COVID-19 pandemic. Because the 2020 cohort began their studies in the middle of the pandemic, we decided to replicate the study in order to collect data for a comparative analysis.

Data collection
The initial data collection was conducted during the first semester in September 2017 and 2020, 1 week after the beginning of the school year. Another wave of the data collection was conducted in February of the following year during the second semester. In the 2017 cohort, the data collection was then followed up by additional data collections in October during the third semester. The final wave of the data collection occurred in May during the sixth semester, near the end of the study program. Because corresponding milestones have not been passed in the 2020 cohort, comparison will only be made between the first two data points.
Data on student social networks were collected using a purpose-built survey tool, which was integrated into the student management system. This tool allowed the students to start typing a name and then select a student from a drop-down menu of all students registered in the respective schools. The collected data were divided by school, as the students could only name students in their own school, which led to two data sets based on the same survey. In each wave, students were prompted to name up to seven students, but they were allowed to list as many as they wanted. In the waves that were conducted after students had begun their university study activities, the students were asked to name the fellow students that they spent time with during their studies or in social activities. The first wave, however, was conducted at the very start of their studies, during their orientation programs or the very first course sessions that students attended, so time spent during studies or social activities was not applicable. In this wave, the students were therefore asked to name the students they knew best at the beginning of their studies. The procedure was identical for both cohorts.
Through the survey tool, we also asked students whether they had translocated before starting their university studies. Other covariates, including grade point average (GPA) on a 0-10 scale (range in data: 5 to 9.99), age (range in data: 18 to 59), and gender, were retrieved from the student management system. For two participants, gender was missing or non-binary. Because of the very low fraction (0.4% of responses), we omitted these observations because a statistical analysis of this group was not feasible. For the 2020 cohort, we also included, in the second wave, open questions on students' experience of the online learning during the pandemic.
To obtain consistent data on ties in the first and second semesters, as well as on new ties between the semesters, we utilized only responses by participants who responded in both waves of the data collection. After the exclusion of missing or incomplete responses in either or both waves, there were 203 observations in the 2017 cohort of 1252 enrolled students. In the 2020 cohort, there were 248 observations of 1887 enrolled students. When comparing the participating student groups to the ones that did not participate, we found that students with higher GPA and younger students were somewhat more likely to participate but the difference was similar in the two cohorts. In both cohorts, female students were more likely to participate but even more so in the 2017 cohort.
For certain types of network analysis, a low response rate can be of concern, since analyses of social networks are known to be sensitive to missing data. In the following analysis, however, the focus is on comparing the two cohorts. With similar response rates and composition of students in two cohorts, any potential bias due to low response rate should be at place in both groups resulting in relatively unbiased comparison between them. In this case the response rates are similar, but female students were more likely to answer in the first cohort as stated above. It was therefore decided to investigate how robust our findings are to the difference in gender proportions in the cohorts.
To accomplish this, we used an adapted bootstrapping approach, where we produced 500 datasets where we had randomly removed data from female students from the 2017 group to make the gender proportions equal in the two cohorts. We then calculated the average number of ties in these produced datasets and compared to our results and found that the largest difference was less than 4%. We therefore concluded that the effect of the difference in gender proportions in the two cohorts was minimal and used all 451 observations in the analysis.

Data analysis
In addition to examining networks at single points in time, we used information about students' social networks at two time points to identify variables regarding the formation of social ties. To identify variables related to tie formation, we calculated new ties as ties between a pair of students in the second wave, but not in the first. The number of new ties formed by a student was therefore not counted as simply the difference in the total number of ties between the two waves, but whether a new tie to a particular student was reported in the second wave. For example, a student with two ties in the first wave and three in the second wave could nevertheless have formed two new ties between them: one of the three ties might be an old tie (reported in both waves), one of the old ties might have been lost, and two new ties might have been formed.
In reporting descriptive statistics that compared the two cohorts and showed whether the populations differed, we applied Pearson's Chi-squared test to the categorical variables and the Wilcoxon rank sum test to the continuous variables. Several multiple linear regression models were estimated, in which new ties were the dependent variable. The coefficients of such models are readily interpretable as estimating the effect of one variable while holding other variables constant. In the main analysis, we used ordinary least squares (OLS) for the estimation. This allows interpretation of coefficients as indicating that an increase in the independent variable by 1 unit increases the expected value of the dependent variable by the numeric value of the coefficient. In the present study, because the number of new ties implied counted data, the multiple Poisson regression would also have been appropriate, but it has the drawback that the interpretation of Poisson coefficients is much less intuitive. To confirm robustness, we performed all regressions using both OLS and Poisson. The results were equivalent in the two models regarding the statistical significance of coefficients, sign of coefficients, and direction of change in coefficients between cohorts. The results of the Poisson regression are available from the authors upon request.
All visualizations and statistical analyses were performed using the R software environment for statistical computing (R Core Team, 2021). Network manipulation and visualization were performed using the igraph package for R (Csardi & Nepusz, 2006). In other aspects of the data manipulation and analysis, we relied heavily on the tidyverse set of packages for R (Wickham et al., 2019).

Descriptive statistics
The initial descriptive statistics are shown in Table 1. They suggested that the observable characteristics of the cohorts were similar. The distribution of students in the two schools was also similar, as were their GPAs. The percentage of male students was slightly higher in the 2020 cohort, although the difference was not significant (p = 0.07). The percentage of students who were translocated prior to starting their university studies was also similar. This indicates that overall, the number of people students knew at the start of their higher education would be expected to be similar.
The most significant difference was in student age. The average age of students in the 2020 cohort was slightly more than 24 years, whereas the average age in the 2017 cohort was slightly under 22 years. Specifically, there was a statistically significant difference of 2.5 years (p < 0.001) between the two groups. This finding was not surprising, as some students were older individuals who found themselves out of work because of the pandemic and chose to study rather than remain unemployed. This age difference needs to be taken into consideration in interpreting the results.
The network summary statistics are presented in Table 2, where the waves of data collected from both cohorts are shown side by side. The number of ties was much lower in the COVID-19 cohort in both the first and the second waves. In the first wave the difference in average number of ties was found to be 1.47 while in the second wave the difference was found to be 2.82. In both cases the difference was found to be significant (p < 0.001). However, the difference in the number of ties was not as pronounced as the difference in the number of new ties formed between the waves. The 2020 cohort reported only 1.40 new ties (sd = 1.85) on average, compared with 3.54 (sd = 3.04) reported by the 2017 cohort, a difference of 2.14, which was both statistically (p < 0.001) and substantively significant. The ratio of tie formation between the two cohorts was striking. The new tie formation reported by the 2020 cohort was less than 40% of that reported in 2018. In other words, over 60% of the relationships that first-year students could have expected to form with their fellow students in a regular year did not materialize for students who started their studies during the pandemic.
The reduction in new tie formation reported by the 2020 cohort was partially counteracted by a reduction in lost ties, as the 2020 cohort reported only 0.65 lost ties (sd = 1.21) on average, compared with 1.47 (sd = 1.81) reported by the 2017 cohort, which was a statistically significant difference (p < 0.001). This could be interpreted as an indication that students, finding fewer opportunities for forming new ties, compensate by better maintaining their existing ties. It is worth noting, however, that because the 2020 cohort reported fewer overall ties in the first semester, the percentage of such ties lost over the course of the semester is almost the same. In 2017 students lost 55% (that is, 1.47/2.69), of their initial ties, but in 2020, students lost 53% (that is, 0.65/1.22) of their initial ties. Overall, the findings imply that the networks in the COVID-19 cohort evolved at a significantly slower pace compared with the regular cohort and that a much greater proportion of the few ties present in the second semester were connections carried over from prior to the onset of the students' university studies.
The findings also showed that in the 2017 cohort, the time between the first and second semesters was the period of greatest renewal of ties. Between the start of the first and second semesters, these students reported an average of 3.54 new ties (sd = 3.04), whereas between the second and third semesters, they reported an average of 1.85 new ties (sd = 1.91). Between the third and sixth semesters, an average of only 1.44 new ties (sd = 1.68) were formed. Therefore, in each subsequent wave of the survey, fewer new ties on average tended to be reported. Figures 1 and 2 confirm the pattern demonstrated in Table 2. The difference in the number of ties was evident at the start of the first semester, but it became more pronounced in the second semester. The greatest difference in the number of new ties is shown in Fig. 3. In 2017, less than one of every ten (9.4%) students reported forming no new ties between the first and second semesters. In contrast, in 2020, 47.4% reported no new ties. In other words, almost half of the students did not report forming any new relationship at the start of the second semester. Some students formed new relationships during the pandemic, but the findings showed that the reduction occurred over the entire range. It is informative to look at the findings for a student in the 25th percentile, as shown in Fig. 3. In 2017, this somewhat socially active student would have been expected to report five new relationships; in 2020, the same student would have been expected to report only two.
Additional information about the structure of social networks and tie formation is provided in graphs of the actual networks. Figure 4 compares the networks of the two cohorts using a network graph. The figure shows four separate networks, each of which is formed by a different set of individuals. It shows the 2017 and 2020 cohorts of both engineering and natural science students and social science students. To allow the visualization of tie formation during the year, the ties that were reported to have formed at As shown in Fig. 4, the network of students in engineering and natural sciences is more clustered in both cohorts. However, the figure shows that the 2020 network was decreased compared with the 2017 cohort. Several findings should be noted. First, there was a much greater number of "social isolates" in the second cohort (i.e., people with no connections, which is illustrated as gray dots with no lines to other dots in Fig. 4). Second, the dark purple lines show new ties, which are scarce. Third, there was less evidence that existing relationships were complemented by new relationships or that existing groups were expanded. Instead, the findings indicated a bifurcation. On one hand, there were groups with existing ties that did not add new ties between the waves of data collection. On the other hand, there were groups that consisted of only new ties. It is possible that these groups were formed online. If so, they could be evidence of a split between new online relationships that were formed as a result of coursework and older in-person relationships without overlap.
As is evident from the above, in particular from Table 2 and Fig. 3, there was a large difference between the social networks of the two cohorts, especially in terms of new ties formed during the semester. However, it is important to rule out as far as possible that this difference was not caused by a third variable. For example, because the 2020 cohort was slightly older, it is possible that these older students were less inclined to form and maintain ties. Therefore, in Tables 3 and 4, we report the results of multiple linear regression models in which new ties were the dependent variable.
In Table 3, we show the difference between the cohorts regarding the maintenance of new ties when other key variables were considered. The coefficient estimate was negative at − 1.89 (p < 0.001). The negative and highly statistically significant coefficient estimate indicated that after controlling for all covariates, students in the 2020 cohort formed 1.89

Fig. 3 Number of new connections between the first and second semesters by year
fewer new relationships between the two waves of data collection, which was close to the raw difference of 2.14 shown in Table 2. The variance inflation factor (VIF) for the 2020 cohort variable was only 1.05, and the highest VIF (for the Engineering and Natural Sciences variable) was 1.14, indicating that multicollinearity is not an issue in this model.
The estimates for the other covariates are also worth discussing. Regarding gender, female respondents formed on average 1.23 more ties than the male respondents did (p < 0.001) across these two cohorts. Tie formation declined with age, but the decline was slight; only 0.04 fewer ties were formed per additional year of age (p = 0.017). Those who translocated before starting their studies tended to form more new ties, as did students who had high GPAs. Students in the engineering and natural sciences also formed more ties, which was consistent with the results shown in the network graph in Fig. 4.
The coefficient estimates shown in Table 3 were pooled across cohorts, which was relevant in determining whether the cohort difference persisted when control variables were included. However, they did not provide any insight into whether the predictors of new tie Fig. 4 Social networks of engineering and social science students in the regular cohort and in the COVID-19 cohort. Ties formed during the school year (between the start of the first semester and the second semester) are shown dark purple, and ties that existed at the start of the year are shown in light blue formation differed between the cohorts; that is, whether particular variables influenced tie formation either more or less during the pandemic compared with a normal situation. We therefore conducted a multiple linear regression model on each cohort, the results of which are shown in Table 4. As before, we calculated the variance inflation factors, and found them to be low (all variance inflation factors across both models and all variables presented in Table 4 were below 1.3) indicating that multicollinearity is not an issue in these models.
Although all coefficients were of the same sign for the two cohorts, there were some important differences. The estimate for gender was pronounced. In line with the pooled regression, the estimate for the 2017 cohort showed that female students reported significantly more new ties than the male students did. The adjusted difference was 2.13 ties (p < 0.001), indicating that after controlling for other variables, the female students reported 2.13 more new ties on average than the male students did. In the 2020 cohort, however, the gender difference was only 0.43 ties, which was not statistically significant (p = 0.073). These findings showed that tie formation was much more equal between the genders in the online setting. The effects of age and GPA were similar across the cohorts. Although the coefficient for GPA was not significant in the 2017 cohort, the point estimate was almost the same. The difference between engineering and natural science students and social science students was smaller in the 2020 cohort.
In the 2017 cohort, students who translocated before starting their studies formed on average 1.28 more ties (p = 0.004) than students who already resided in the city in which the university is located. In the 2020 cohort, this difference decreased to 0.17, which was not statistically significant. When all interactions were online, the increase in networks experienced by potentially marginalized students when they moved to a new town was not evident, which could have potentially increased their marginalization.
The final set of analyses was conducted to determine the way in which gender patterns in tie formation were affected by the pandemic. Figure 5 shows the distribution of connections at the start of the first semester according to gender and cohort. Figure 6 shows connections at the start of the second semester according to the same categories. Both figures show that in 2017, male and female students began with a similar number of connections, but during the year, the female students increased the size of their networks more quickly than the male students did. The comparison of the 2017 and 2020 cohorts did not show striking differences in how the networks of both genders evolved, with the exception that for both genders, the reported number of ties at each point was much lower in 2020 than in 2017. Figure 7 shows the number of new connections formed between the first and second semesters according to the same categories. The pattern indicated by the regression models presented in Table 4 is evident. Although the female students formed more new ties than the male students did, the difference was much smaller in the 2020 cohort than in the 2017 cohort.
Two open questions were included in the second wave of the 2020 survey where students could state what they appreciated and what they disliked about their online learning

Discussion
Our findings showed that students who started their undergraduate degrees in 2020 reported fewer connections than students who started their studies in 2017. This finding was consistent when we corrected for small differences in the ages of the two cohorts. This finding is in line with Elmer et al. (2020), who found that students reported fewer connections during COVID-19 than before the pandemic. We found that the differences in reported connections between the two cohorts increased after the first semester. Most pronounced was the difference in newly formed connections over the semester. Students in the 2020 cohort formed on average 60% fewer new connections than students in the 2017 cohort did in the same period. This decrease occurred despite the university's efforts to offer first-year students the opportunity to attend classes in smaller groups on campus. This indicates that even if diving students into smaller groups has shown an effect on social connection between students (Hommes et al., 2014) this was not enough to make up for the lack of connection in online classes. Given the importance of social connection for higher education students this is concerning (Felten & Lambert, 2020).
The lack of new connections during the pandemic clearly disadvantages the 2020 cohort, as more connections to fellow students have been shown to lead to better performance (Baldwin et al., 1997), and connections made in the first-year lead to better retention (Wilcox et al., 2005). Overall, connections made in the first year of study are important for students throughout their education (Kossinets & Watts, 2009). As the development of social networks reported by the 2017 cohort showed, networking activity was the highest in the first year, and it decreased as the students' progressed in their studies. Thus, there is considerable value in comparing two separate cohorts of students who are at the same point in their studies, one before and the other during online teaching during the pandemic.
It is noteworthy that regarding new connections, the female students lost some of the advantage they had compared with the male students before the pandemic. There were no direct indications in the data regarding why this happened, but it is possible this was an extension of findings that women in higher education faced greater stress than their male counterparts during the COVID-19 pandemic (Gelles et al., 2020). This finding is in particular interesting in light of the findings of Flores et al. (2021) that male students were less satisfied with the online teaching than female students. It is also important to note that students who translocated in 2020 to begin their studies did not build connections to the same extent as similar students in 2017, which could potentially lead to their marginalization.
Somewhat counteracting the drastic reduction in the formation of new ties for the 2020 cohort is a reduction in lost ties. This could indicate that due to fewer opportunities these students were more attentive to maintaining their existing ties. Given that these students reported fewer ties at the beginning of the year, however, this could also be due to these students only reporting their closest connections in the survey at the start of the year, unlike the 2017 cohort, who reported everyone they had known and met on campus. It is also worth noting that in the case of lost ties, a reasonable to focus on the percentage of one's existing ties that are lost over a certain period, and that because of the fewer initial ties in 2020, the percentage of ties lost was almost identical in the two cohorts.
It is to some extent positive that the findings indicate that students are actually able to maintain existing ties during these difficult times. This might indicate that they benefit from the support offered by tight social connections (McCabe, 2016). It is worth pointing out, however, that in some cases it can be important to shed connections that no longer support the progress and development of the student (Rios-Aguilar & Deil-Amen, 2012). As connections with students with negative attitudes toward school can affect a student's own views (Wang et al., 2018), it can be important for students to adjust their social network to support their education. In light of McCabes (2016) findings that losing connections that no longer serve the student is harder in a tight knit network this might mean that students did hang on to connections that do not serve them as they did not have opportunity to make new connections. Students might have entered the university with friendship connections but might not be building up the instrumental relationships they need in connections to their studies. Relying on older ties and a more static network structure could also impact the learning environment as a whole. Yang et al. (2021) reported similar patterns for information workers, as remote work led to more static and siloed networks that can make it more difficult to acquire and share new information. Thus, the potential silver lining of fewer lost ties in the pandemic cohort is considerably less bright than it might seem at first glance.
The question remains whether the 2020 cohort will be able to catch up with the 2017 cohort as they progress in their studies or whether the lack of connections in the first year set the students a difficult path in the long term. It is important for faculty who teach these students in later years to consider that students from these cohorts might not have the same peer support we are used to students having. And as the pandemic continued to impact teaching across the world in the years following 2020, other cohorts are likely to be similarly affected. Our view is that because the situation is so dire, it is also important that higher education institutions respond effectively to the lack of connections when they welcome students back on campus, offering students plenty of opportunities to collaborate so that they can pick up at least a few connections that work for them (Fjelkner-Pihl, 2022a).
The main conclusion derived from the comparison of the formation of new social ties between the 2017 cohort and students who started their education online in the fall 2020 during the COVID-19 pandemic indicates that compared to the 2017 cohort, the 2020 cohort did not adjust their social networks. Most significantly, they formed much fewer new connections. The implication is that students in the 2020 cohort may be at a considerable disadvantage throughout their higher education. It is our belief that universities need to play an active role in helping affected students make new connections and support the evolution of their social networks, as social networks play a significant role in students' futures.

Limitations and future research
The research reported here is limited to students in two academic fields: the social sciences and engineering and the natural sciences. Those two fields were separated into two different networks, limiting the ability of students to report cross-field connections which might have offered insights into university-wide social relationships. Further research could systematically investigate differences in the development of social networks in different fields of study, concurrently comparing online education and on-campus education. Our study was limited to one public university in Northern Europe, where the effects of the COVID-19 pandemic were not typical. It is possible that the difference between the 2017 cohort and the 2020 cohort would have been even greater in universities where all courses were taught entirely online.
Another limitation is that the university is a public university that accepts all students if they have passed grammar school tests. The findings might differ in universities with a more competitive student population. The fact that the students represented in this study were forced to learn online during the pandemic must also be considered. These students might differ from students who self-selected into an online course. Students who self-select online education might therefore approach social networks differently. However, our study aimed to compare the development of social networks between on-campus and online learning in similar student groups. By comparing the two cohorts at the same university, we were in a better position to show the effects of online learning versus on-campus learning, independent of differences in students who actively chose between the two modes of learning. As with any survey research, there is also the possibility of non-response bias if the people who participated in the survey differ systematically from those who do not participate. Because our focus is on the comparison of two cohorts, this is less of a threat because any non-response bias would also need to differ systematically across cohorts, but it cannot be completely discounted. The research is further limited in that it only relies on quantitative data where qualitative date would have added to the richness of the data (Froehlich et al., 2020).
We hope that future research will address the limitations of the present study by adding to the understanding of the development of the social networks of undergraduate students, particularly by comparing the development of social networks in online settings in diverse contexts. The findings of such research could benefit universities in developing and implementing programs and activities to help online students develop social networks to the same extent as traditional on-campus students. And as the most immediate effects of COVID-19 recede, they shed light on one of the many ways in which the pandemic has the potential to affect whole generations over the longer term.