As university dropout is not only a recent concern but has been an important issue for decades, the first theoretical models to explain this phenomenon have been developed long ago and maintained their relevance until today. One of the most famous pioneering models is the Student Integration Model by Tinto (1975) which states that a student’s level of social as well as academic integration into the college environment impacts college persistence. Another well-known model for explaining dropout is the Student Attrition Model by Bean (1980, 1983). It is based on the theory of organizational turnover and the theory of planned behavior. According to the model, dropout can be viewed as similar to employee turnover and is the result of an interplay of students’ background characteristics (e.g., performance) and organizational determinants (e.g., institutional quality). However, both models mainly refer to colleges or universities in the US. A well-known model for the German university context is the Model for Student Dropout Processes (Heublein et al., 2010, 2017). It frames the decision to drop out as a multidimensional, longer-term process which is influenced by a variety of impact factors concerning both the pre-study phase and the current study phase. The final decision to quit studies is understood as the result of an accumulation of factors that promote dropout.

In line with the model by Heublein et al. (2010, 2017) we take a process perspective on dropout and therefore focus on dropout intentions, which can be measured longitudinally, rather than actual dropout. Given the theory of planned behavior (Ajzen, 1985, 1987), which implies that intentions predict behavior, and given that the strength of an intention is positively related to the likelihood that the behavior occurs (Ajzen, 1991), it can be assumed that dropout intentions are indicators for actual dropout (see also Bean, 1982). Kelava et al. (2022) showed that among the students who actually dropped out, 86% had dropout intentions before which were detected by the authors’ statistical algorithm in advance.

In view of this process perspective, longitudinal studies seem best suited to study dropout intentions. Longitudinal designs make it possible to take into account the intra-individual as well as the inter-individual level. Both dimensions can be assumed to be relevant for the development of dropout intentions (e.g., Glaesser et al., 2021; Kelava et al., 2022). While the intra-individual level refers to changes within an individual over time, the inter-individual level refers to time-invariant differences between persons (e.g., with respect to cognitive attainment). On the intra-individual level, especially motivational states have been shown to be relevant for the development of dropout intentions (see, e.g., Dresel and Grassinger, 2013; Schnettler et al., 2020; Kelava et al., 2022). Such within-person changes in motivational states can for instance result from performance-related issues or external events.

The models for dropout which have been explained above address different levels of influence (e.g., individual, institutional, environmental). Even though institutional (e.g., academic selectivity; Pascarella and Terenzini, 2005) and environmental factors (e.g., working alongside studies; Sevilla et al., 2010) have been shown to play a role for university dropout, we concentrate on individual determinants in what follows.

The reason is that motivation and performance issues are believed to make an important contribution to the development of dropout intentions in the early phase of math studies (Andrà et al., 2011; Geisler, 2020; Glaesser et al., 2021).

Models from the field of career choice provide a theoretical foundation for the selection of individual-level variables that can help in explaining the development of an intention to drop out.

A theory which is frequently applied to the prediction of educational outcomes (e.g., Schneider and Preckel, 2017; Schnettler et al., 2020) is the so-called expectancy-value theory (EVT; Eccles and Wigfield, 1995; Wigfield and Cambria, 2010). The EVT postulates that individuals’ motivation to accomplish a specific task is determined by the interplay of their expectancies for success and the value placed on that task. Task value can be divided into several components, i.e., attainment value (personal importance of doing well on the task), intrinsic value (personal enjoyment of the task), and utility value (how well does the task relate to current and future goals). Costs is the fourth component of task value and represents the negative aspects which result from engaging in the task. These are for instance fear of failure or success, performance anxiety, the effort needed to complete the task or opportunity costs (Eccles et al., 1983; Eccles and Wigfield, 2002). Expectancies for success in the context of the EVT refer to individuals’ beliefs about their success in an upcoming task (Eccles and Wigfield, 2002).

It can be derived from the EVT that motivation is highest when individuals believe they can succeed (high expectancy) and when they perceive the outcomes as valuable (high task value). Therefore, we assume that students who have low expectations for success and devalue the outcomes of their university education are more likely to develop dropout intentions.

Based on the EVT we included items which relate to the four facets of task value in our study. To take into account expectancy beliefs, we included mathematical self-concept. Self-concept is commonly used as a proxy for expectancy (e.g., Nagengast et al., 2011) since these constructs are indistinguishable in empirical reality (Eccles and Wigfield, 2002). Understanding of the lecture content and perceived ability are concepts which are also closely linked to individual expectations for success and therefore included in the study as well.

The theory of vocational personalities and work environments (Holland, 1997), also known as the person-environment fit theory, is another theory that has been used in the context of academic decision-making (e.g., Wille et al., 2020). The theory focuses on the interaction between individuals’ personality traits and the characteristics of their work environment and therefore primarily addresses vocational choices and job satisfaction. However, it can also provide some insights into the factors that contribute to university dropout.

According to Holland (1997), individuals possess different vocational personalities which are characterized by their preferences, interests, and values. These are realistic (R), investigative (I), artistic (A), social (S), enterprising (E) and conventional (C). Vocational environments can also be classified based on these six different types. Thus, vocational personalities can either be matched or mismatched with the characteristics of the work environment, which influences for instance job selection, job changes and vocational satisfaction.

In the context of university dropout, a mismatch between the vocational personalities of students and their chosen academic path can lead to low study satisfaction or motivation and ultimately the decision to drop out.

While the theories pointed out above provide a helpful basis for the explanation of dropout intentions, previous empirical findings should be taken into account as well. Since it has been shown that dropout intentions depend on the same set of predictors like actual dropout (Fleischer et al., 2019) we also take into account studies that focus on dropout.

There exists a variety of individual characteristics that have been identified empirically as predictive factors for student dropout. Firstly, pre-university performance in the form of high school GPAs plays an important role for quitting studies. This relationship is mediated through the effect of high school GPAs on study performance at university, which is itself a predictor of dropout (Allen et al., 2008; Voelkle and Sander, 2008). Moreover, study motivation matters as revealed in a meta-analysis by Robbins et al. (2004). This holds true not only for motivational states that occur during the university course, for instance disappointed study expectations (Heublein et al., 2017) but also for motivational factors related to the study decision itself (e.g., intrinsic motivation; Heublein et al., 2017; Rump et al., 2017). Another possible determinant of leaving university early is gender. For instance, Leppel (2002) shows that women in general drop out less frequently than men on average. However, some studies point in the direction that a distinction between male- or female-dominated fields must be made. Nevertheless, results concerning the direction of the effect of gender proportions are found to be controversial (see Johnes and McNabb, 2004; Mastekaasa and Smeby, 2008). For the German context, it has been shown that among students who enrolled for the first time in the years 2009 to 2013, the success rates of women in the subject group mathematics and natural sciences were slightly higher than those of men, except for the year 2011 (Statistisches Bundesamt, 2023). As university entrance qualifications in Germany can be obtained not only at general-education Gymnasiums (similar to British grammar schools) which have an academic focus, but also at different types of vocational upper secondary schools, the type of school attended also has an impact (Müller and Schneider, 2013). Students who received their university entrance qualification at a general-education Gymnasium are less likely to drop out than students from other school types (Heublein et al., 2017). One possible reason is that there is a better fit between the imparted skills and the requirements of academic studies. Furthermore, the school type attended is associated with parental educational background (Lörz et al., 2011) which is itself related to dropout (e.g., Aina, 2013).

As previously pointed out, difficulties at the transition to university, particularly the gap between school and university mathematics, are believed to constitute a substantial contribution to high dropout rates in math studies. While in many subjects university studies directly build up on school contents (see, e.g., Nagy, 2006; Busker, 2010 for chemistry), tertiary mathematics has a different character compared to school mathematics (Hoyles et al., 2001; Rach and Heinze, 2013). At the university, formal-deductive proofs drastically gain in relevance compared to secondary school mathematics (see, e.g., Rach, 2014) which marks one of the most substantial differences between both disciplines (e.g., Gueudet, 2008; Engelbrecht, 2010). Given the complexity of mathematical proof and the lack of appropriate prior knowledge, the acquisition of skills for mathematical proof is a major challenge for new students that comes along with a variety of difficulties (Rach, 2014).

Difficulties at the transition to university mathematics are not limited to the German university context that is specifically considered in the present paper. In fact, they seem to be part of a general issue that is frequently addressed in the international literature (see, e.g., Moore, 1994; De Guzman et al., 1998; Daskalogianni and Simpson, 2002; Di Martino and Gregorio, 2019; Sönmez and Güven, 2022). Accordingly, all math students are exposed to such a discontinuity between school and university mathematics. Still, the way in which this challenge is dealt with and, in particular, to what extent it influences early dropout decisions differs across students with different characteristics. However, only little research has so far been carried out on individual risk factors for and mechanisms behind early dropouts in math studies.

Andrà et al. (2011), using a sample of undergraduate math students at an Italian university, provide evidence that not only a lack of knowledge but also students’ beliefs and motivations are relevant for dropout decisions. The importance of motivational factors for early dropouts is also supported by Geisler’s (2020) findings. Drawing on a sample of first-year math students at a German university, the author shows that students who dropped out were less interested in university mathematics and had a lower mathematical self-concept compared to those who did not drop out. Furthermore, he finds that only in the group of dropouts there was a decline in mathematical self-concept between the beginning and the middle of the first semester.

However, the studies mentioned have a retrospective design that might suffer from distortions and/or consider only a few points in time which are not sufficient to capture the changeability of motivational states. The work by Glaesser et al. (2021), which is based on the same dataset as the one used in the present study, resolves these issues. Using multilevel models with prospective data, the authors show that motivation related factors play an important role for the development of dropout intentions across first-semester math B.Ed. as well as math B.Sc. and physics B.Sc. students. This is shown to hold true for stable traits as well as changeable states. Regarding the changeable states, fear of failure appears to be a particularly strong predictor for dropout intentions. Kelava et al. (2022) drawing on the same dataset as well show that the emergence of an intention to quit can be forecasted by both stable student characteristics (cognitive abilities, i.e., IQ) and intra-individual changes in affective states 8 weeks before a student actually drops out, making it possible to identify students at risk in a timely manner.

Overall, the existing literature stresses the importance of performance related and motivational factors and suggests considering changeable states alongside stable characteristics for explaining dropout (intentions) in university math studies.

A special feature of the German tertiary educational system is the fact that in most cases, students have to decide right at the start of their studies whether they wish to become teachers. If that is the case, they embark on a teacher training course from the first semester. While math B.Ed. students get prepared for becoming math teachers and therefore must attend math specific as well as teaching specific modules, the B.Sc. students receive a more general mathematical education that allows them to decide on a particular future profession at a later point in time. Even though switching between B.Ed. and B.Sc. studies within a certain study subject is easily possible, the division between the courses is not purely formal since students enrolled in teacher training courses are quite certain about becoming teachers already at the beginning of their studies (Bauer et al., 2011). These circumstances make it possible for us to investigate whether math B.Ed. students preparing for teaching at the Gymnasium and math/physics B.Sc. students, who attend the same compulsory first-year math lectures, differ from each other regarding early dropouts.

Given the distinction regarding the professional fields that B.Ed. and B.Sc. prepare for, it can be assumed that expectations, motivations, abilities, and other personal characteristics differ right from the start between the two types of students.²

Probably the most intuitive difference between teacher candidates and other types of students can be found with respect to occupational interests. Within the STEM area, teacher candidates were shown to have lower investigative and realistic interests but higher social interests compared to the other students (Roloff Henoch et al., 2015).

Moreover, there exist demographic differences. In contrast to STEM studies in general, which are dominated by males, more than half of the students enrolled in studies which prepare for teaching mathematics at the Gymnasium are female (Hefendehl-Hebeker, 2013).

Furthermore, Kilian et al. (2020) provide evidence that math B.Ed. students have lower prior mathematical knowledge compared to B.Sc. students at the start of their studies and had worse math grades in school, but they did not have lower high school GPAs. Klusmann et al. (2009), by contrast, do not find differences in mathematical knowledge, GPAs or general cognitive abilities between student teachers who qualify for teaching at secondary schools and other types of students while controlling for gender. However, even though the authors differentiate between STEM and non-STEM majors, it has to be taken into account that these results do not specifically refer to math students.

Despite the findings being contradictory, negative stereotypes regarding the characteristics of trainee teachers versus other students for academic success persist (see Blömeke, 2005), a view which is also supported by the respective lecturers (see Günther and Koeszegi, 2015 for the subject of mathematics). As already shown in other contexts, such negative stereotypes can lower the academic self-concept (Ertl et al., 2017) and as a consequence the actual achievement (Marsh and Scalas, 2011).

Moreover, there are reasons to believe that the relative impact of extrinsic motives for study choice is higher among B.Ed. students than B.Sc. students. Firstly, the teaching profession in Germany offers high financial and job security as well as compatibility between family and work and is therefore favored by people that attach great importance to these factors (Weishaupt and Huth, 2012; Neugebauer, 2013). Secondly, some students choose math only as a strategically favorable supplementary subject for civil service positions instead of basing their choice on mathematical interest (Hefendehl-Hebeker, 2013). Thus, it comes as no surprise that the motivation for math contents is found to be lower for math trainee teachers than for other math students (Blömeke, 2009).

In addition, B.Ed. students in particular are likely to suffer from a perceived mismatch between the course contents and their chosen profession. The reason is that they are often not able to see the relevance of university mathematics for their future work as a math teacher (Höttecke et al., 2018). This is also linked to the self-image of student teachers as educators rather than academics (Bauer and Hefendehl-Hebeker, 2018).

Finally, one should bear in mind that B.Ed. students are enrolled in two subjects and furthermore have to take lectures in subject didactics, education, and psychology which imposes additional pressure and reduces the time that students are able to invest in math lectures.

Given that some of the factors mentioned above, such as mismatch between course content and profession (Radisch et al., 2018) or self-concept (Geisler, 2020), have already been shown to matter for dropout in math studies, we expect to find differences in the overall level of dropout intentions between math B.Sc. and B.Ed. students. In addition, we believe that the differences between the two groups of students outlined in this section may lead to differences in the strength of the associations between certain changeable motivational states and the intention to drop out, as noted above.

The data used in the present paper comes from a prospective, intensive longitudinal study on students enrolled in math courses at a large German university. The initial survey of that study took place in the second week of the first semester (t = 0) and included questions on students’ stable socio-demographic characteristics, interests, motivations as well as personality traits, whereby the latter are not part of the present study. Furthermore, a test of mathematical content knowledge was conducted. This initial survey was followed by online questionnaires that the students received each Monday, Wednesday, and Friday throughout their first semester. In these online questionnaires, students were asked about changeable affective, performance related and motivational states. In addition, at each measurement occasion, data on their current intention to drop out was collected (Glaesser et al., 2021; Kelava et al., 2022). In return for their participation, the students received vouchers for an e-commerce company.

Data collection took place during the winter semester of 2017/2018.We obtained survey data on 116 first-year students of math B.Ed., math B.Sc. and physics B.Sc. Over the course of the semester, each of them was invited to respond to a voluntary online survey at 50 points in time. As the participants did not answer all questionnaires, about 20 (19.88) data points per person were available on average.

In math studies, in order to be admitted to the final exams, a minimum number of points had to be reached in the exercise sheets handed out during the semester. Students who started studies earlier and received an exam admission in a previous semester might already have failed exams in the past. For those who have failed several times, dropping out might no longer be voluntary. Thus, their comparability to the first-semester students was limited. This held true for six students which we excluded from our sample.

Moreover, we excluded five students with fewer than two observations since they show no within-person variability, which is, however, needed for the usage of RDSEM as will become clearer in what follows.Thus, we ended up with 105 students, more precisely, 55 students in the math B.Ed. and 50 students in physics or math B.Sc.³

Of the students in the overall sample, about 48% were enrolled in the B.Ed. program, 43% were female and 73% had attended a general education Gymnasium. The average GPA, where lower numbers indicated better performance, was 1.86, which is considerably better than the German average (in the different federal states, the average GPA ranged between 2.16 and 2.67 in the school year 2020; KMK, 2021) and implies that the math students in our sample generally possessed favorable cognitive performance capacities. On the entrance test of mathematical knowledge, they scored on average 10.4 out of 20 possible points. For more detailed summary statistics of the between-level variables see Table A1 in the Supplementary material.

We used Stata (Version 17; StataCorp, 2021) and JASP (JASP Team, 2022) for descriptive statistics and group comparisons. To study the impact of inter-individual differences and intra-individual changes on dropout intentions, RDSEM were applied in Mplus Version 8.8 (Muthén and Muthén, 1998-2017). They can be seen as a special case of ordinary DSEM models and provide high flexibility since they encompass multilevel modeling, structural equation modeling (SEM), time-varying effects modeling (TVEM) as well as time-series modeling (Asparouhov et al., 2018). Thus, they are suitable for the complex structure of the ILD used in the present study.

Given the multilevel perspective, within-person associations are modeled on Level 1, whereas between-person differences enter the model on Level 2 (McNeish and Hamaker, 2020). In contrast to standard DSEM that put an emphasis on lagged or cross-lagged relationships, RDSEM focus on contemporaneous associations between variables. Therefore, the within-person model is further divided into a structural part and an autoregressive part. The structural part contains only concurrent relationships and can be interpreted analogously to within-person models in standard two-level SEM (Asparouhov and Muthén, 2020). To account for the autocorrelation in ILD as well, which can typically be approximated by an AR(1) process (McNeish and Hamaker, 2020), the dynamic part of the model is modeled on the residuals (Asparouhov and Muthén, 2020).Taking into account the content-related considerations from above, the following random intercept model was specified:⁶

Level 1:

Level 2:

Random effect

Fixed effects

where X i corresponds to the row vector of stable a priori predictors (except for the moderator variable t e a c h i ) which have been grand-mean centered in the case of those which were non-dichotomous. It is assumed that δ t i ~N (0, σ 2 ), u 0 i ~ N ( 0 , σ 2 ) and e t i ~ N ( 0 , σ 2 ) (McNeish and Hamaker, 2020).

On Level 1, we examined the relationship between changeable motivational states and dropout intentions. More precisely, we wished to assess whether deviations in the within-level predictors relative to the person’s own mean, rather than the grand mean over all cases, predict an increase or decrease in the outcome variable. This can be achieved by applying latent person-mean centering to all covariates used on the within level, as indicated by the superscript “c” (McNeish and Hamaker, 2020). Latent person-mean centering is the default option in Mplus and offers methodological advantages over observed centering which are discussed in Asparouhov and Muthén (2019) and Lüdtke et al. (2008). We considered the temporal stability of the intention to drop out by allowing for autocorrelation of the residuals, as explained above. To control for a possible linear time trend in the outcome variable, time was included as an additional within-level covariate (see Wang and Maxwell, 2015; Hamaker et al., 2018; McNeish and Hamaker, 2020).

On Level 2, we modeled between-person differences in the intercept and the slopes from our Level 1 equations. The intercept α i , which corresponds to an individual i’s mean level of dropout intentions averaged across all measurement occasions, was regressed on the set of stable variables from the initial survey. Furthermore, all person-specific slopes were regressed on the teacher candidate indicator t e a c h i to study differences between B.Sc. and B.Ed. students regarding the strength of the influence of the within-level predictors. Thus, the intercepts of the person-specific slopes ( γ 10 to γ 80 ) can be interpreted as the effects for B.Sc. students, whereas the coefficients γ 11 to γ 81 correspond to the additional slope increments for teacher candidates.

We assumed constant residual variances across individuals. In addition to what is displayed here, the between-parts of the changeable covariates entered the model as well as latent variables with estimated means and variances (see McNeish and Hamaker, 2020). Furthermore, all covariates used in the within-level equations (except for time) were brought into the model by allowing their residuals to be serially correlated. In this way, we were able to apply missing data theory and listwise deletion was avoided (see also Muthén and Muthén, 1998-2018; Wang and Wang, 2019).

For model estimation, we applied a Bayesian estimator with non-informative priors and 6,000 Markov Chain Monte Carlo (MCMC) iterations with the Gibbs sampler. We used the default option with two chains of which the first half was discarded as burn-in phase. Missing data treatment was likelihood based and yields consistent estimates under MAR (missing at random). To account for unequally spaced observations, Mplus’ “tinterval” option was used (for further details on RDSEM estimation see Asparouhov et al., 2018; McNeish and Hamaker, 2020). Convergence was checked graphically via trace plots and posterior parameter distribution as well as analytically via the potential scale reduction (PSR) statistic, Rhat, by Gelman and Rubin (1992), which corresponds to a weighted average of within- and between-chain variances. A value close to 1 suggests convergence. For all parameters, a Rhat <1.011 was reached.

The research questions for the present paper are based on the expectation that there exist a priori differences between math B.Ed. and physics or math B.Sc. students that might also be related to dropout decisions. Table 1 contains comparisons of both groups regarding initial characteristics.

As expected, there was a large and statistically significant difference in the share of women. While 68% of the B.Ed. students were female, this was the case for only about 20% of the B.Sc. students (p < 0.001, BF 10 = 96650.401 , log ( OR ) = 2.064 ) . Furthermore, the average mathematical knowledge of B.Ed. students was almost two points lower compared to B.Sc. students, which is remarkable given that the maximum possible score was 20 (p = 0.007, BF 10 = 5.971 , δ = 0.494 ) . However, it is evident that the B.Sc. students also may have deficits in their mathematical knowledge since they score only just over 11 points out of 20 on the mathematical knowledge test. In spite of the knowledge differences, high school GPAs did not differ significantly between the groups (p = 0.910, BF 10 = 0.208 , δ = 0.019 ) . Both findings are in line with previous results discussed above. Regarding the motivational variables measured at the start of the studies, we did, against our expectations, not find significant differences in IV (p = 0.380, BF 10 = 0.290, δ  = 0.151) or AV (p = 0.797, BF 10  = 0.213, δ  = 0.044). The same held true for UV (p = 0.587, BF 10  = 0.236, δ  = 0.094). Furthermore, teacher candidates did not have a worse mathematical self-concept (p = 0.655, BF 10 = 0.226, δ  = 0.077). However, B.Ed. students started with a more negative attitude towards the non-monetary costs the studies will cause (p = 0.017, BF 10 = 2.966 , δ = − 0.436 ) . As expected, the investigative interest of B.Sc. students was much higher (p < 0.001, BF 10 = 1.781 e 10 , δ = 1.583 ) and their social interest much lower (p < 0.001, BF 10 = 1.268 e 13 , δ = − 1.847 ) compared to teacher candidates which is in line with Roloff Henoch et al. (2015).

Overall, these findings confirm the existence of many of the suggested a priori differences between B.Ed. and B.Sc. students for the sample used in this analysis. Thus, we controlled for them in our RDSEM.

In this section, the results obtained from RDSEM are presented. All results are unstandardized so that, in contrast to average within-level standardized results, inferences can be drawn for other individuals from the population (Hamaker et al., 2018).

Reported point estimates for each parameter refer to the median of the respective posterior parameter distribution. If the corresponding 95% credible interval (central posterior interval; Bayesian analog of frequentist confidence interval) did not contain 0, parameters were considered statistically different from 0 (see, e.g., McNeish and Hamaker, 2020).