Determinants of Maths Performance of FirstYear Business Administration and Economics Students
 1.2k Downloads
Abstract
This paper examines the determinants of mathematics performance of firstyear students enrolled in several business administration and economics study programmes at the beginning of the first semester (T1) and nine weeks later (T2). A simple model of educational production, which is developed in accordance with the model of Schrader and Helmke (2015), is used as the theoretical basis for our analysis. As predictors, we choose numerous variables which represent studyspecific, socioeconomic and biographical, motivational and cognitive aspects as well as variables that reflect the learning behaviour, working habits and the use of voluntary support (e.g., tutorials). Data from two skills tests and two surveys that were carried out at the University of Kassel in the winter semester 2011/12 and regression techniques are used to identify the determinants of mathematics performance. It turns out that the type of school graduation, final school grades, preknowledge and selfbelief are essential determinants.
Keywords
Determinants of maths performance Firstyear students Business administration and economics Education production function  Empirical analysisIntroduction
Motivation
Mathematics plays a growing role in business administration and economics (B&E). In particular, the increasing importance of quantitative methods in both science and practice has led to curricula in B&E study programmes at universities that include at least one or two modules in mathematics and statistics (Voßkamp 2016). Moreover, starting with Paul Samuelson’s textbook Economics (Samuelson, 1948/1997) – to acknowledge one milestone – the importance of mathematics has increased in all areas of B&E. A short look at widely used textbooks in economics illustrates this development (see, e.g., Blanchard and Johnson 2012, and Pindyck and Rubinfeld 2013). The mathematics used in B&E courses covers a wide range of topics from analysis (in particular calculus) to linear algebra. The textbooks on mathematics for economic analysis by Simon and Blume (2010), Sydsaeter et al. (2012) and Chiang and Wainwright (2005) give an idea of the mathematics topics taught in B&E study programmes at the university level.
In the light of this, it seems to be clear that students with stronger mathematical skills obtained at secondary schools will do better in B&E study programmes. For example, a student’s grade in mathematics at the end of the secondary school has a significant impact on the probability of passing a bachelor or master exam in B&E (see, e.g., Heublein 2014). Anderson et al. (1994) identified background knowledge of calculus as one of the most important determinants of the probability of dropping introductory economics courses in the first semester. It “is a wellestablished finding in the literature on economic education” that mathematical skills are a key determinant of study progress in economics (Arnold and Straten 2012, p. 33). Consequently, from the perspective of B&E students as well as the perspective of B&E faculties, a high level of secondary school mathematics is desirable.
However, many mathematics educators are confronted with inadequate firstyear students’ mathematical skills. “This is often referred to as the ‘Mathematics Problem’ and relates to students entering thirdlevel whose mathematics at school level is insufficient for the demands of their thirdlevel Service mathematics courses and careers” (Liston and O’Donoghou 2009, p. 77). In Bausch et al. (2014) and Hoppenbrock et al. (2016) numerous studies are published which report the deficits of firstyear students’ mathematical skills at German universities. In B&E study programmes in particular deficits are reported to be serious. Empirical research on B&E students at the University of Kassel supports these results (Voßkamp and Laging 2014; Laging and Voßkamp 2016; Sonntag 2016). Results from skills tests at the beginning of studies show dramatic deficits. Moreover, skills tests after several weeks show that in many cases firstyear students are not able to compensate deficits, despite the substantial voluntary support (e.g., remedial courses) offered to the students.
Among mathematics educators, these facts are often communicated with a dramatic tenor. Many mathematics educators like to point out that in earlier times things went better, adding conjectures on negative changes with respect to the students’ characteristics (e.g., decreasing preknowledge and motivation) and the negative impact of institutional changes (e.g., the implementation of new educational standards).
The transition from school to university is a stressful, demanding, lifechanging experience that requires many changes (Clark and Lovric 2008, p. 29). Especially the secondarytertiary transition in mathematics education has become a major issue with different focuses and different theoretical approaches (Gueudet 2008). Most of research on transition takes place in a few specialized areas (Thomas et al. 2012). Clark and Lovric (2008) postulate the absence of a theoretical model in research of transition. They provide an anthropological approach based on the ‘rite of passage’ concept and reflect the different kinds of changes students are confronted with. Gueudet (2008) structures mathematical educational literature about secondarytertiary transitions into three views of transition: Research belonging to ‘Advanced Mathematical Thinking’ (AMT), research focussing on proof and language and research focussing on didactical transpositions.
Therefore, most studies focus on the nature of mathematics and how it is taught at university level. However, there are only very few studies that analyse determinants of firstyear students’ achievement, especially determinants of mathematical skills of firstyear students (e.g. Hailikari et al. 2008; Pajares and Miller 1994). In contrast, there are many studies on the determinants of secondary school achievements. The metaanalysis ‘Visible Learning’ (Hattie 2009) gives a good overview of determinants of school achievements based on over 800 metaanalyses. However, it is questionable whether the results of these studies are applicable to understand the causes and consequences of the achievements in mathematics in higher education. In particular, the impact of studyspecific, socioeconomic and biographical, motivational and cognitive aspects as well as aspects that reflect the learning behaviour, working habits and the use of voluntary support (e.g., tutorials) is unclear.
Therefore, studies on higher education are needed in order to reveal the determinants of the mathematical competencies of firstyear students. Moreover, the interactions between all these variables and further variables have to be analysed. The knowledge of these relationships is necessary to identify effective and efficient interventions in order to increase the success of students.
Purpose of the Study
In this article we will contribute to the above mentioned issues. In a first step, we will develop a simple model of student achievement in higher education based on wellknown theories from various scientific disciplines. Our model which can be interpreted as a model of educational production provides a link between the mathematical skills (output variable) and several explanatory variables (input variables).
In a second step, we will test the model empirically. For this purpose wellknown methods from empirical educational research (especially regression methods) will be applied. We used two instruments to obtain data: a skills test on basic mathematics and a questionnaire with questions concerning students’ socioeconomic and biographical background, motivational aspects, learning behaviour, working habits and the use of voluntary support services. The sample consists of 447 B&E students enrolled at the University of Kassel; all students were enrolled in a firstyear course, ‘Mathematics for B&E’, in the winter semester 2011/12. Both instruments were used (in variants) at two points of time: at the beginning of the course (time T1) and in the ninth week of the course (time T2).
Outline
We start with a brief overview of relevant empirical literature. Then, we discuss theoretical contributions related to our purposes. On this basis, we develop a simple model of educational production. After elaborating on the quantitative methods applied, the data and variables used, we present and discuss the results. The paper ends with a conclusion including summary remarks, statements on limitations, implications as well as an outlook on further research.
Empirical Background
Previous research has highlighted several determinants of academic performance. An overview of important factors for school achievement is given in the synthesis of over 800 metaanalyses in ‘Visible Learning’ by Hattie (2009). Hattie structures over 130 potential determinants around six factors: the child, the home, the school, the curricula, the teacher and the approaches to teaching. Most of the metaanalyses included combine different kinds of academic performance. A focus on only mathematical performance and constructs related to mathematics can change the importance of factors. For instance, a more recent metaanalysis about the relationship of maths selfefficacy and maths performance in school and university (Laging 2016a) produced a considerable larger effect size than the synthesized metaanalyses by Hattie (2009).
Most of the studies and metaanalyses are focussed on single aspects, but there are very few studies that analyse these variables simultaneously (Schiefele et al. 2003), especially in the case of higher education.
Moreover, many studies related to secondary schooling show the relevance of many predictors in detail. PISA 2000 data confirmed socioeconomic background as one of the strongest predictors of performance (OECD 2003) with an exceedingly strong relation in Germany (Stanat and Lüdtke 2013). Also, effects of family characteristics, student motivation and country resources on mathematics achievement in 41 countries are analysed by Chiu and Xihua (2008) with the PISA 2000 database. The multilevel analyses revealed that 44 % of the variance in students’ mathematics scores occurred at student level, 25 % at school level and 31 % at country level. Multilevel regression with these variables explained 36 % of the variance in students’ mathematics scores. The results indicate the importance of family characteristics. Students living with two parents from families with greater socioeconomic status, more investment in educational resources and more family involvement obtain higher scores in mathematics performance.
Intrinsic motivation, selfefficacy and selfconcept were also found to be significant positive predictors of mathematics achievement. Analysing the PISA data of Turkish students, a factor analysis with 14 items from a student questionnaire revealed four factors accounting for approximately 34 % of the variance in mathematics scores (Demir et al. 2009): student background (e.g., economic, social and cultural status, HISEI), selfrelated cognitions in mathematics (selfconcept, interest, selfefficacy and anxiety in mathematics), learning strategies and school climate. Multivariate regression revealed that student achievement is predominantly influenced by student background and selfrelated cognitions in mathematics (Demir et al. 2009).
In higher education, “various predictors, including precollege characteristics, traditional assessments of one’s cognitive abilities, and a battery of psychological and noncognitive variables” (Strayhorn 2013, p. 17) have been investigated to explain academic achievement. As “academic preparation is the most significant predictor of academic achievement in higher education” (Strayhorn 2013, p. 18), past grades in school are important predictors of academic achievement in higher education. In multiple regression analyses Liston and O’Donoghou (2009) reveal mathematical competencies and past experiences in terms of results of final maths exams as the strongest predictors of mathematics marks in the first semester.
The amount of time and effort students devoted to their study influences their achievement in college (Kuh and Hu 1999; Strayhorn 2013). A metaanalysis to predict college performance shows that ACT/SAT scores and high school GPA were the strongest predictors, but academic selfefficacy and achievement motivation contribute meaningfully to the prediction of college performance (Robbins et al. 2004). Hailikari et al. (2008) revealed domainspecific prior knowledge as strongest predictor of students’ achievement on a university mathematics course using structural equation modelling. Prior knowledge test performance was strongly influenced by academic selfbeliefs (expectation of success, selfefficacy, selfperception of mathematics ability).
Anthony (2000) identified factors that students and lecturers perceived as most influential for students’ academic success or failure in mathematics courses in their first year at university. Factors that students and lecturers listed as most important in an openended survey were categorized in factors related to lectures, courses, students and other external factors. Students and lecturers rated student factors most often, but lecturers placed more responsibility for success and failure on student factors than students did. The importance of identified factors was analysed using a Likerttype scale questionnaire and “motivation was seen by both students and lecturers as the most influential factor related to levels of success” (Anthony 2000, p. 9).
Theoretical Background
General Remarks
A general theory to explain the maths skills of firstyear students does not exist. Therefore, we will start with theoretical approaches which were originally formulated to explain achievements of students at secondary schools.
Numerous theories from very different fields such as pedagogical psychology, sociology and economics can be used to explain student achievement at school. The subdiscipline classroom research alone has developed several different theories in the recent decades: personality paradigm, teaching style, processproduct paradigm, expert paradigm, systemic model (for a review see Klieme 2006). In many theories the focus is set on only a few predictive variables, e.g. the personality paradigm focusses on teachers’ personality while the processproduct paradigm focusses on teachers’ behaviour in the classroom.
We want to show a comprehensive picture of the determinants of mathematics performance. We, therefore, have to start with complex models which simultaneously take into account many variables. A model developed by Schrader and Helmke (2015) meets this criterion. This model, which is based on earlier works by A. Helmke, F. W. Schrader and F. E. Weinert (in particular Helmke and Weinert (1997)), will be the starting point for the development of our model.
The Model by Schrader and Helmke (2015)

learning activities (LA)

individual motivational determinants (IMD)

individual cognitive determinants (ICD)

cultural background, media, peers (CB)

home learning environment (HLE)

school organisation, climate and classroom context (SO)
The authors postulate several important interdependencies. Learning activities (LA) decisively determine a pupil’s school achievement (SA) and are influenced by motivation (IMD) and cognition (ICD). Moreover, both are influenced by cultural background (CB). The environment at home (HLE) and at school (SO) also influences all these variables. Furthermore, there are also impacts from student achievements. Schrader and Helmke (2015) assume that the pupil’s achievements influence motivational (IMD) and cognitional (ICD) variables as well as the environment at home (HLE) and at school (SO). This model integrates many theories that can be used to specify the variables of the blocks. For example, the variables that are part of IMD can be identified by expectancyvaluetheory (Wigfield and Eccles 2000). Constructs related to expectancies are for example selfconcept and selfefficacy, constructs related to values are for example interest and goal orientation.
The Model
The Applicability of the Model of Schrader and Helmke for Higher Education Issues
The model of Schrader and Helmke (SH model) was originally designed to explain school achievement, but it integrates many theories that are not restricted to secondary schooling. For that reason, the model is transferable from a secondary schooling context to a higher education context. Most of the interdependencies which work in the case of secondary schooling are also relevant in the case of higher education at universities, but not all of them. The influence of HLE is assumed to be weaker because university students are more independent than secondary school students. On the other hand, motivational factors play a bigger role because attendance, invested time and effectiveness of learning in higher education is less controlled. Chemers et al. (2001) argue that “confidence in one’s relevant abilities (i.e., selfefficacy) and optimism play a major role in an individual’s successful negotiation of challenging life transitions” (p. 55). However, in the case of higher education the list of variables, which belong to the different blocks, has to be adjusted. And, before starting with empirical issues we have to specify scales and variables which represent the mentioned blocks.
Need for a Simplification of the Model
The SH model is a suitable theoretical basis that can be used to explain the mathematical achievements of firstyear students. In principle, the SH model is also an adequate basis for empirical analysis. However, fundamental problems arise when the SH model has to be verified empirically because the quantitative methods which have to be applied require a large number of observations. In our case, the number of students represented in our data set is too small. The application of education production functions which implies a simplification of the SH model allows us to continue.
Education Production Functions (EPF)
As a rule, it is assumed that the relationship between output variable and input variables can be represented mathematically by using functions (e.g., CobbDouglas functions and linear functions).
The EPF approach assumes that an educational output (e.g., a certain grade or a level of students’ performance in mathematics) is a result of a production process where educational inputs (e.g., investment of time, motivation, quality of school) determine the output. The most crucial problem with EPF concerns the choice of variables. In standard production theory it is clear which variables have to be taken into account for both output variable and input variables. Moreover, the measurement of the variables is straightforward because widely accepted accounting principles are applied (in Europe: European systems of accounts (ESA); Eurostat 2013).
With EPFs, things are more difficult. First, for many input and output variables, there is no consensus for measurement of the variables. Second, there is no clearly defined set of input variables which have to be taken into account if we look for the determinants of a specific educational output variable. Therefore, the application of EPFs is associated with heuristic and empirical methods in order to make the approach valuable.
Operationalisation of Input and Outcome Variables
Because of this background, we are working with a simple model which covers the main features of the SH model. In particular, we use the EPF approach to link student achievement and important predictors.

studyspecific variables (S)

socioeconomic and biographic variables (B)

motivational factors (M)

learning strategies (L)

working habits (W)

use of support (U)
A Comparison of the Variables Represented in the SH Model and the Simplified Model
Comparison of blocks of variables represented in the SH model and the simplified model
SH model  Simplified model 

School organization (SO)  
Home learning environment (HLE)  Socioeconomic and biographic variables (B) (partly) 
Cultural background (CB)  Socioeconomic and biographic variables (B) (partly) 
Individual motivational determinants (IMD)  Motivational factors (M) 
Individual cognitive determinants (ICD)  Socioeconomic and biographic variables (B) (partly) and learning strategies (L) 
Learning activities (LA)  Working habits (W) and use of support (U) 
Studyspecific variables (S) 
Nevertheless, there are some differences. First, we do not take into account variables representing the school organisation, etc. (SO) because of our sample. All students represented in our sample are enrolled in the same study programme at the same university. Therefore, all students are confronted with the same organisational circumstances and the same mathematics educator. In general, we suppose that these aspects influence the development of mathematical achievement. In lecturers’ and students’ perception, course design and organization influence study success (Anthony 2000). Aspects that should be considered for other samples are the kind of mathematics and the way mathematics is taught. According to Tall (2008), pure mathematics in university requires a transition from school mathematics to formal thinking. However, we do not focus on advanced mathematical thinking because B&E students learn maths as a service subject. Mathematics for economics has – to note one aspect – a strong focus on the application of mathematical methods and not on proofs. Second, we do not take into account most of the variables representing the home learning environment (HLE). We suppose that parental personalities, expectations, theories, support, climate and sanctions do not play such a big role in tertiary education, because university students are more independent than secondary school students. Only socioeconomic variables that are measured by university degree of parents are supposed to influence achievement. Third, we do not take into account a separate block representing aspects of the cultural backgrounds of the students. Cultural background is represented by the variable migration background in block B. More aspects of cultural background would only be relevant within a broader sample with higher degree of diversity. Finally, we take several variables into account which cover studyspecific aspects.
How to Operationalize the Blocks? First Remarks
Method
In this section we present the method we apply to estimate the determinants of maths performance at T1 (first week of the semester) and T2 (nine weeks later). In both cases we will apply the same linear regressions method. However, we have to take into account that at T1 and T2 different blocks of variables are relevant. Working habits (W _{ t }) and the use of support services (U _{ t }) refer to learning activities during the semester and are therefore not surveyed at T1. These blocks are relevant at T2. Moreover, at T2 previous knowledge has to be taken into account.
The following remarks might be helpful to understand the approach with respect to the use of data. The variables of block S and block B do not change from T1 to T2. Variables concerning motivational factors (block M) and learning strategies (block L) will change from T1 to T2. Variables of block W and block U are only relevant at T2. For that reason, variables belonging to the blocks S, B, M and L are surveyed at T1. At T2, the survey includes questions related to the blocks M, L, W and U.
Mathematical Representation
 k

index for students
 i

index for variables within a block
 π _{ k } ^{ T1}

maths performance of student k in T1
 π _{ k } ^{ T2}

maths performance of student k in T2
 S _{ ik }

k’s value for variable S _{ i } in T1 (and T2)
 B _{ ik }

k’s value for variable B _{ i } in T1 (and T2)
 M _{ ik }

k’s value for variable M _{ i } in T1, respectively T2
 L _{ ik }

k’s value for variable L _{ i } in T1, respectively T2
 W _{ ik }

k’s value for variable W _{ i } inT2
 U _{ ik }

k’s value for variable U _{ i } in T2
 s, b, m, l, w, u

number of variables included in block S, B, M, L, W, U
Both models will be the theoretical basis for our empirical analysis.
Linear Regression Models
The starting points for our further analysis are Eqs. (1) and (2). In accordance with most studies in this research domain, we assume a linear relationship between student maths performance and the predictors motivated above. Scatterplots confirm the assumption of a linear relationship; see Figs. 5 and 6 (Appendix) as samples. This allows us to apply standard regression methods (e.g., Wooldridge 2015).
In both Eqs. (3) and (4), the variables β _{ i } ^{ j } represent canonically the regression coefficients which have to be estimated. The error terms are denoted by ε _{ k } . We use the standard ordinary least squares method to estimate the regression coefficients.
Nested Regression Methods
Nested regression methods are used where blocks of variables are taken into consideration. Every block of variables is successively integrated in the model if the determination coefficient R ^{2} increases significantly. For details see Acock (2014). Due to the structure of our model this regression method should be applied.
Data
General Setting
Data is drawn from two surveys which were carried out at University of Kassel in the winter semester 2011/12. At the beginning of the semester (T1, October 2011) students were asked to participate in a skills test and a questionnaire for the first time. The survey was completed in the first session of the lecture. The students were asked to participate in another skills test and questionnaire nine weeks later (T2, December 2011). In both cases, students had to complete the survey (skills test and questionnaire) in 75 min. All surveys were anonymous. Students were asked in both surveys to generate a unique password. This allows us to match data from T1 with data from T2.
The authors and one additional tutor managed the surveys. The surveys were voluntary. The skills tests were not part of the mark of the course, but they were used to give students feedback about their basic maths skills. There might be students who did not make much of an effort and could have performed better. However, we had the impression that most students took the survey seriously.
Participants
The sample consists of B&E students who were enrolled in the course ‘Mathematics for B&E students’ in the winter semester 2011/12. For most participants this course was compulsory. At time T1, 447 students participated in the survey; at time T2, 237students. Only 183 students participated in both surveys. Due to missing values for several variables the number of observations is smaller.
Skills Tests
Comparison of blocks of variables represented in the SH model and the simplified model
Topic  Description  Example 

terms  evaluation and simplification of terms  Simplify this term as far as possible: \( \frac{1{y}^2}{y+1} \) 
equations and inequalities  solving linear, quadratic and cubic equations and inequalities  Solve following quadratic equation: (x − 2)^{2} − 2 = − 1 
functions  determination of properties, drawing graphs  Draw the graph of the following function: \( y=\frac{1}{x} \) 
differential calculus  determination of derivatives  Determine the first derivation of f(x) = e ^{3x } 
For each task, a student could obtain not more than one credit. The skills tests include different types of tasks: single choice tasks, multiple choice tasks, calculations and drawing graphs. The topics included in the skills tests are only briefly considered in lectures. However, voluntary support is provided for these topics.
Originally, the main purpose of the skills tests was to identify students with problems in basic mathematics skills in order to encourage them to use the voluntary support services provided. Therefore, tasks of the maths skills tests are structured by topics.
Questionnaires
At both points in time the students were asked to answer a questionnaire. At time T1, the students were asked questions which are related to the blocks of variables S, B, M, and L. The questionnaire consists of 16 questions and 47 Likerttype items. At time T2, in the middle of the semester, we also asked questions related to blocks W and U that included additional 44 Likerttype items. The questionnaires start with questions about studyspecifics and biographic background. The Likerttype items are organized in blocks. The items are randomly scattered within these blocks.
Independent and Dependent Variables
Dependent variables
Code  Description  Comments  Type  Values  Items  Mean  CA  Mean  CA 

(T1)  (T1)  (T2)  (T2)  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
π ^{ T1}  maths performance at T1  number of correct answers to tasks in T1  scale  0 to 30  30  5.99  .873  x  x 
π ^{ T2}  maths performance at T2  number of correct answers to tasks in T2  scale  0 to 30  30  x  x  12.35  .863 
Variables block S
Code  Description  Comments  Type  Values  Items  Mean  CA  Mean  CA 

(T1)  (T1)  (T2)  (T2)  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
S1  participation preparation course  participation in preparation course: yes = 1; no = 0  binary  0 / 1  1  .436  x  .490  x 
S2  exam already taken  student has already taken exam without success: yes = 1; no = 0  binary  0 / 1  1  .116  x  .226  x 
S3  course of studies  business administration & economics = 1; other programmes = 0  binary  0 / 1  1  .712  x  .757  x 
Variables block B
Code  Description  Comments  Type  Values  Items  Mean  CA  Mean  CA 

(T1)  (T1)  (T2)  (T2)  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
B1  gender  male = 1; female = 0  binary  0 / 1  1  .479  x  .449  x 
B2  type of graduation  ‘FOS’ = 0; ‘Abitur’ = 1 (details: see text)  binary  0 / 1  1  .459  x  .551  x 
B3  years since graduation  years between end of secondary school and the start of studies at the university  metric  0 to n  1  2.17  x  3.32  x 
B4  university degree parents  mother or father obtained a tertiary degree =1; else = 0  binary  0 / 1  1  .384  x  x  x 
B5  apprenticeship  student has completed an apprenticeship: yes = 1; no = 0  binary  0 / 1  1  .385  x  x  x 
B6  grade of graduation  final grade at second school: excellent = 1; insufficient = 5  metric  1 to 5  1  2.48  x  x  x 
B7  maths grade in school  final grade in maths: excellent = 1; insufficient = 5  metric  1 to 5  1  2.60  x  2.43  x 
B8  migration background  student has a migration background: yes = 1; no = 0  binary  0 / 1  1  .267  x  .268  x 
Variables block M
Code  Description  Comments  Type  Values  Items  Mean  CA  Mean  CA 

(T1)  (T1)  (T2)  (T2)  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
M1  maths selfefficacy  based on common procedure (see e.g. Pajares and Miller 1994)  scale 30  1 to 8  30  4.79  .946  5.00  .962 
M2  maths selfconcept  see LIMA  scale 3  1 to 6  3  3.44  .891  3.41  .886 
M3  maths interest  see Co^{2}CA  scale 4  1 to 6  4  3.54  .932  3.55  .945 
M4  mastery goal orientation  based on LIMA  scale 3  1 to 6  3  3.42  .849  3.34  .836 
M5  maths anxiety  see LIMA  scale 3  1 to 6  3  4.09  .850  4.00  .847 
M6  perceived value of maths  based on Shell et al. (1989)  scale 9  1 to 6  9  4.67  .880  4.40  .876 
Variables block L
Code  Description  Comments  Type  Values  Items  Mean  CA  Mean  CA 

(T1)  (T1)  (T2)  (T2)  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
L1  memorising strategies  see LIMA (and PISA)  scale 3  1 to 6  3  3.99  .694  3.60  .751 
L2  elaboration strategies  see LIMA (and PISA)  scale 4  1 to 6  4  2.86  .705  2.76  .801 
L3  control strategies  based on LIMA (and PISA)  scale 5  1 to 6  5  4.15  .751  4.04  .764 
L4  planning strategies  based on Rakoczy et al. (2005)  scale 4  1 to 6  4  4.33  .678  4.30  .721 
L5  heuristic strategies  based on Rakoczy et al. (2005)  scale 5  1 to 6  5  4.63  .750  4,80  .759 
Variables block W
Code  Description  Comments  Type  Values  Items  Mean  CA  Mean  CA 

(T1)  (T1)  (T2)  (T2)  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
W1  effort  based on Rakoczy et al. (2005) and PISA  scale 7  1 to 6  7  x  x  4.45  .838 
W2  persistence  based on Rakoczy et al. (2005)  scale 4  1 to 6  4  x  x  4.20  .838 
W3  regularity  based on Rakoczy et al. (2005) and PISA  scale 3  1 to 6  3  x  x  4.96  .819 
W4  preparation of lecture  hours spend on the preparation of the lecture  metric  0 to n  1  x  x  2.70  x 
W5  preparation of assignments  hours spend on the preparation of the assignments  metric  0 to n  1  x  x  2.60  x 
Variables block U
Code  Description  Remarks  Type  Values  Items  Mean  CA  Mean  CA 

(T1)  (T1)  (T2)  (T2)  
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
U1  use of lecture  students were asked how often they make use of the lecture: at no time = 1; at all times = 6  metric  1 to 6  1  x  x  5.70  x 
U2  use of tutorial  students were asked how often they make use of the tutorial: at no time = 1; at all times = 6  metric  1 to 6  1  x  x  5.40  x 
U3  use of assignments  students were asked how often they make use of the assignments: at no time = 1; at all times = 6  metric  1 to 6  1  x  x  5.26  x 
U4  use of additional tasks  students were asked how often they make use of additional tasks: at no time = 1; at all times = 6  metric  1 to 6  1  x  x  3.72  x 
U5  use of an open learning environment  students were asked how often they make use of the open learning environment: at no time = 1; at all times = 6  metric  1 to 6  1  x  x  2.01  x 
U6  use of tests with feedback  students were asked how often they make use of the tests with feedback: at no time = 1; at all times = 6  metric  1 to 6  1  x  x  1.82  x 
U7  use of tests without feedback  students were asked how often they make use of the tests without feedback: at no time = 1; at all times = 6  metric  1 to 6  1  x  x  2.61  x 
Dependent Variables: Mathematical Achievement in T1 and T2
 π _{ k } ^{ T1}

maths performance of student k in T1
 π _{ k } ^{ T2}

maths performance of student k in T2
Table 3 presents information about the two dependent variables. In particular, the table shows that mathematics performance improves over time.
Block S: StudySpecific Variables
With block S, we gather data about the students’ institutional study characteristics. The block consists of three variables (Table 4).
First, it is important to know whether the student has taken the preparation course in mathematics, which is offered just a few weeks before the test in T1 (variable S1). Like the test, this course is built on the maths curricula of secondary schools. For that reason the following results seem to be natural: if a student participated in the preparation course than we can expect a positive impact on maths performance. And, if a student who failed the final exam takes the main course twice (variable S2) than we conjecture that there is also a positive impact on the mathematical achievement. Finally, variable S3 represents the study programme a student has chosen. Most students are enrolled in the study programme ‘Business Administration & Economics’ (in German: Wirtschaftswissenschaften).
Block B: SocioEconomic and Biographic Variables
Block B of independent variables (Table 5) consists of important socioeconomic and (educational) biographical variables that can also be found in wellknown school performance studies like PISA or TIMSS.
Most of the eight variables are related to the educational background of the students. B2 indicates the type of school graduation. With respect to the complex German educational system, we distinguish between only two types of graduation: graduation from a ‘Fachoberschule’ ends with ‘Fachhochschulreife’ (FOS), graduation from a ‘Gymnasium’ with ‘Allgemeine Hochschulreife’ (Abitur). The tracks are quite different with respect to the number of school years (FOS: 12; Abitur 12 or 13) and the quantitative and qualitative workload in mathematics. For a description of the differences see EACEA (2015). Furthermore, we assume that the length of the period between the secondary school graduation and the start of higher education (B3) will have an impact on mathematical achievements. Faulkner et al. (2011) found that mature students who have not engaged in mathematics for a number of years (‘nonstandard students’) are mathematically less prepared. Nonstandard students perform below average in the diagnostic test at the beginning of the semester, but they improve and are more likely to use the voluntary support services. B6 und B7 represent the students’ success at school. We take the overall final grade and the final grade in mathematics into consideration.
Besides this, we consider four variables which are also present in many studies on secondary school achievement (e. g., PISA, TIMSS): gender (B1), migration background (B8) and the academic background of the parents (B4). Finally, B5 indicates that a student has completed an apprenticeship.
Block M: Motivational Factors
Block M, as part of the individual motivational determinants (IMD), includes six approved motivational scales (Table 6) that have been used for several studies (e. g. LIMA: Fischer et al. 2012; CO^{2}CA: Bürgermeister et al. 2011; PISA: Kunter et al. 2002; PYTHAGORAS: Rakoczy et al. 2005). All scales are developed maths specific.
First, we include maths selfefficacy (M1) as a predictor for mathematical achievement. The concept of selfefficacy is based on a theoretical framework by Bandura (1977). “Perceived selfefficacy refers to beliefs in one’s capabilities to organize and execute the courses of action required to produce given attainments. […] Such beliefs influence the courses of action people choose to pursue, how much effort they put forth in given endeavors, how long they will persevere to adversity, whether their thought patterns are selfhindering or selfaiding, how much stress and depression they experience in coping with taxing environmental demands, and the level of accomplishments they realize” (Bandura 1997, p. 3). The scale is based on a common procedure applied e.g. by Pajares and Miller (1994). Just before the skills test, students rate for each task of the skills test how confident they are to solve the tasks.
Maths selfconcept (M2), master goal orientation (M4) and maths anxiety (M5) are further scales. These scales are adopted from the German project LIMA (Fischer et al. 2012) that is concerned with creating new standards for the maths teachers’ education and implementing teaching innovations. Shavelson et al. (1976) define selfconcept as a person’s perception of himself that can be described as “organized, multifaceted, hierarchical, stable, developmental, evaluative, differentiable” (p. 411). Based on the hierarchical model of selfconcept we measure maths selfconcept that is a subarea of the academic selfconcept. Students rate e.g., how well they understand mathematical subjects and how gifted they are in mathematics. The distinction between selfefficacy and selfconcept is difficult because the two constructs share a number of similarities. For a better distinction, we measure maths selfefficacy on a taskspecific scale and maths selfconcept on a domain specific scale. For a detailed review on the differences between the two constructs, see Bong and Skaalvik (2003).
The achievement goals construct is important in achievement motivation literature. Different conceptual models have been developed (Elliot et al. 2011, p. 632). The dichotomous model distinguishes between mastery goals and performance goals as purpose for engaging in achievement (Elliot et al. 2011, p. 632). Mastery goal orientations are “used to refer to goals that orient the individual to focus on the task in terms of mastering or learning how to do the task” (Pintrich 2000, p. 95). In contrast, performance goals are “goals that orient the individual to focus on the self, ability, or performance relative to others” (Pintrich 2000, p. 95). This model has been extended, for further information see Elliot et al. (2011). The metaanalysis of Huang (2011a) shows a weak positive correlation between mastery goal orientation and achievement and no correlation between performance goal orientation and achievement in the twofactormodel. Therefore, we only measure mastery goal orientation. A sample item is “I prefer challenging tasks, so I can learn a lot”.
Students with maths anxiety perceive mathematicsrelated situations as threatening to their selfesteem. The metaanalysis of Ma (1999) confirms the negative relation between maths anxiety and achievement in mathematics with an average correlation coefficient of .27.
Maths interest (M3) is modelled in accordance with Rakoczy et al. (2005). According to Schiefele (2009) we measure individual interest that “is defined as a relatively stable set of valence beliefs” (p. 201). Interest focuses on the content of learning and “is always related to a specific object, activity, or subject area” (Schiefele 2009, p. 197). Therefore, we measure interest domainspecific, e.g. with “I find mathematics fascinating”.
We received a scale for the perceived value of maths (M6) by adaptation of a scale developed by Shell et al. (1989). Students were asked to rate the importance of mathematics for their study and career aspirations. Sample items are “How important are skills in mathematics to get a job?” and “How important are skills in mathematics to get good grades at university?”
Block L: Learning Strategies
Block L (Table 7) focuses on the learning strategies of the students. Capturing learning strategies is one way to measure aspects of selfregulated learning. The different models of selfregulated learning (e.g. the threelayer model by Boekaerts 1999) have the major components in common: cognitive, metacognitive and motivational components. Motivational aspects are captured in Block M. Cognitive and metacognitive components are a part of the individual cognitive determinants (ICD) that are operationalized by learning strategies. We expect that the type of learning strategies the students apply will influence the mathematical performance. Research indicates “the positive effect of students’ use of selfregulated learning strategies on their academic performance” (Zimmerman 1990, p. 185).
As in the case of motivational factors, we adopt scales that are applied in several studies on school achievement. Specifically, we define five scales representing the following learning strategies: memorising strategies (L1), elaboration strategies (L2) and control strategies (L3), planning strategies (L4), heuristic strategies (L5). While the scales L1 to L3 are based on scales used in LIMA and PISA, L4 and L5 are based on scales used in Rakoczy et al. (2005). The scales are domainspecific. The skills tests capture basic mathematical skills, not advanced mathematical thinking. Therefore, we apply scales that are originally used in studies on secondary school mathematics.
L1, L2 and L5 represent cognitive learning strategies; L3 and L4 represent metacognitive learning strategies. Sample items are: “To learn mathematics, I try to remember every step of the solution process” (L1), “When I am learning maths, I try to link the content with things I have already learned in other subjects.” (L2), “I ask myself questions about the content to be sure that I have understood everything.” (L3), “When I am learning maths, at first I plan what exactly I have to practise” (L4), “When I am solving a difficult task, I bring to my mind what is the main issue of the task.” (L5).
Cognitive strategies can be classified into surface cognitive and deep cognitive strategies. “Surface strategies refer to rehearsal, involving the repetitive rehearsal and rote memorization of information […]. Deep cognitive strategies, pertaining to elaboration, organization and critical thinking involve challenging the veracity of information encountered and attempting to integrate new information with prior knowledge and experience.” (Vrugt and Oort 2008, p. 128). Accordingly, L1 are surface cognitive strategies, L2 and L5 are deep cognitive strategies.
Empirical research supports a positive relationship between deep approaches of learning and achievement, and a negative relationship between surface approaches of learning and achievement (e.g. Ainley 1993). Other studies showed that a mix of surface and deep cognitive strategies is most effective (e.g. Vrugt and Oort 2008).
Block W: Working Habits
Block W (Table 8) consists of five variables which represent the working habits of the students.
In the questionnaire, the students were asked about their effort (W1), persistence (W2) and regularity (W3) regarding maths assignments. The fundamental idea of these variables is based on Rakoczy et al. (2005) and PISA. We adjusted the structure of the scales and the wording of the items. Sample items are “In mathematics I try to do everything as well as possible.” (W1), “I always endeavour to solve the assignments in mathematics.” (W2), “I do not give up even if the mathematical tasks are very difficult.” (W2) and “I work on the assignments every week as good as I can.” (W3). Furthermore, we asked the students how many hours they spent on the preparation of the lecture (W4) and the assignments (W5). Thus, the variables reflecting the students’ working habits are part of learning activities (LA). Obviously, these variables are only used as predictors for the explanation of the mathematical performance in T2. There is empirical support for this assumption (e.g. Kuh and Hu 1999; Strayhorn 2013).
Block U: Use of Support
Block U comprises variables that capture the use of (voluntary) support services. Besides the lecture, students can use a variety of voluntary support services. All services are offered every week throughout the semester. The variables included in block U (Table 9) show how often the students have used the services.
U1 shows how often the students participated in lectures (4 h per week). The mean of the variable is quite high. However, it should be mentioned that the questionnaire was conducted in the lecture at time T2.

U2: Tutorials (2 h per week, about 20 students per group) are voluntary courses; the tutors are experienced students (mostly third and fourthyear students).

U3: Assignments are offered to the students every week. The solutions to the tasks of these assignments are discussed in the tutorials. The assignments are voluntary. In general, there is no individual feedback.

U4: Beside the assignments further tasks are offered. These tasks are not subject to the tutorials. There is no feedback.

U5: During the semester the students can visit the ‘MatheTreff’. This is an open learning environment. Four hours per week a seminar room is opened for students; a lecturer (PhD student or experienced student) is present who answers questions if required.

U6: Each week a voluntary test is offered. U6 shows the intensity of use of the tests with feedback. It means that the students submit solutions; after a week, the students receive written feedback. The reviewers are experienced students supervised by a lecturer (PhD student).

U7: This variable shows the intensity of use of the test without feedback. It means that students work on the test, but they do not submit solutions to the reviewers.
In general, we expect that the use of the support services will have positive impact on the mathematical performance. However, one aspect of the concept for the support services is crucial: While the skills tests take only tasks related to secondary school mathematics into account the support services cover the curriculum of the lecture for which secondary school mathematics plays only a limited role.
Results
Results for T1
First, we will present the results obtained for variables at T1 by determining correlation coefficients (see Table 14 in the Appendix). All motivational variables, except perceived value of maths (M6), correlate moderately with maths performance at T1. We find the strongest correlation with maths selfefficacy (M1) (r = .44). In addition, some studyspecific and biographical background variables correlate moderately with maths performance. Exam already taken (S2) and type of graduation (B2) correlate positively and maths grade in school (B7) negatively with maths performance at T1. Learning strategies show a diverse pattern of correlations. Only elaboration strategies (L2) correlate positively, but weakly with maths performance. Control, planning and heuristic strategies (L3, L4, and L5) show no correlation with maths performance and memorising strategies (L1) even a weak negative correlation. All motivational variables correlate from moderately to strongly among themselves and with maths grade in school, but not strongly. Learning strategies correlate from moderately to strongly among themselves, too.
Model comparison with π ^{ T1} as dependent variable at T1
Model  Independent variables  Adj. R^{2}  Comparison  Fstatistic 

M1a  S _{ i }  .117  
M1b  S _{ i }, B _{ i }  .395  M1a vs. M1b  F(8,289) = 18.06; p < .001 
M1c  S _{ i }, B _{ i }, M _{ i }  .448  M1b vs. M1c  F(6,183) = 5.61; p < .001 
M1d  S _{ i }, B _{ i }, M _{ i }, L _{ i }  .462  M1c vs. M1d  F(5,278) = 2.48; p = .032 
Results of regression analyses at T1
Code  Predictor  Regressions predicting π ^{ T1} T1  

M1a  M1b  M1c  M1d  
S _{1}  participation preparation course  .201***  .205***  .180***  .176*** 
S _{2}  exam already taken  .301***  .325***  .324***  .347*** 
S _{3}  course of studies  .105  .034  .041  .040 
B _{1}  gender  .098*  .072  .039  
B _{2}  type of graduation  .346***  .322***  .317***  
B _{3}  years since graduation  .015  .015  .005  
B _{4}  university degree parents  .104*  .099*  .099*  
B _{5}  apprenticeship  .094  .061  .046  
B _{6}  grade of graduation  .089  .116*  .118*  
B _{7}  maths grade in school  .321***  .147*  .155*  
B _{8}  migration background  .050  .063  .053  
M _{1}  maths selfefficacy  .141**  .132*  
M _{2}  maths selfconcept  .023  .005  
M _{3}  maths interest  .160*  .188**  
M _{4}  mastery goal orientation  .023  .021  
M _{5}  maths anxiety  .047  .040  
M _{6}  perceived value of maths  .034  .015  
L _{1}  memorising strategies  .087  
L _{2}  elaboration strategies  .054  
L _{3}  control strategies  .149*  
L _{4}  planning strategies  .081  
L _{5}  heuristic strategies  .038  
Adj. R^{2}  .117  .395  .448  .462 
Most important in all three models are the participation of the maths preparation course (S1), having already taken the exam unsuccessfully (S2), the type of graduation (B2) and maths grade in school (B7). Coefficients of these variables decrease slightly when motivational variables are integrated (M1b); the influence of maths grade in school, especially, is decreasing. Maths selfefficacy and maths interest are significant predictors of maths performance in both models. The only significant coefficient for learning strategies in M1d we find for control strategies, but with a negative influence on maths performance.
Results for T2
All correlation coefficients of variables at T2 among themselves and with maths performance at T1 and T2 are presented in Table 14 (Appendix). Maths performance at T1 correlates very strong with maths performance at T2 (r = .80). These results are similar to results of correlational analysis of variables at T1. All motivational variables, except perceived value of maths, correlate from moderately to strongly with maths performance at T2. Type of graduation, elaboration and control strategies correlate moderately with maths performance at T2. Effort, persistence and regularity of working habits correlate moderately and positively with maths performance and strongly among each other. Hours spent for preparation show no correlation with maths performance and only weak correlations with other working habits. Use of assignments is the only variable of use of support that correlates moderately with maths performance.
Model comparison with π ^{ T2} as dependent variable at T2
Model  Independent variables  Adj. R^{2}  Comparison  Fstatistic 

M2a  π ^{ T1}  .632  
M2b  π ^{ T1}, S _{ i }  .629  M2a vs. M2b  F(3,125) = .65, p = .582 
M2c  π ^{ T1}, B _{ i }  .664  M2a vs. M2c  F(8,120) = 2.54; p = .014 
M2d  π ^{ T1}, B _{ i }, M _{ i }  .710  M2c vs. M2d  F(6,114) = 4.167; p < .001 
M2e  π ^{ T1}, B _{ i }, M _{ i }, L _{ i }  .703  M2d vs. M2e  F(5,109) = .463; p = .803 
M2f  π ^{ T1}, B _{ i }, M _{ i }, W _{ i }  .741  M2d vs. M2f  F(5,109) = 3.757; p = .004 
M2g  π ^{ T1}, B _{ i }, M _{ i }, W _{ i }, U _{ i }  .742  M2f vs. M2g  F(7,102) = 1.076; p = .384 
Model M2c shows that the successive integration of socioeconomic and biographic variables as well as the motivational variables (model M2d) raises R ^{2} significantly. The same holds for variables concerning working habits (model M2f). However, variables concerning learning strategies (model M2e) as well as variables representing the use of voluntary support (model M2g) do not increase R ^{2} significantly. The final model M2f explains about 75 % of the variance.
Results of regression analyses at T2
Code  Predictor  Regressions predicting π _{ k } ^{ T2}  

M2a  M2c  M2d  M2f  
π ^{ T1}  maths performance at T1  .797***  .714***  .623***  .603*** 
B _{1}  gender  .124*  .064  .066  
B _{2}  type of graduation  .182**  .144**  .153**  
B _{3}  years since graduation  .042  .052  .062  
B _{4}  university degree parents  .034  .038  .049  
B _{5}  apprenticeship  .049  .094  .084  
B _{6}  grade of graduation  .125  .165**  .160**  
B _{7}  maths grade in school  .035  .070  .047  
B _{8}  migration background  .055  .036  .036  
M _{1}  maths selfefficacy  .067  .052  
M _{2}  maths selfconcept  .109  .153*  
M _{3}  maths interest  .006  .011  
M _{4}  mastery goal orientation  .033  .020  
M _{5}  maths anxiety  .164**  .119*  
M _{6}  perceived value of maths  .070  .079  
W _{1}  effort  .008  
W _{2}  persistence  .041  
W _{3}  regularity  .233***  
W _{4}  preparation of lecture  .031  
W _{5}  preparation of assignments  .054  
Adj. R^{2}  .632  .629  .710  .741 
Discussion
In this section we discuss the final estimates for model M1d for T1 and model M2f for T2 in more detail. It has to be stressed that the skills tests measure basic maths skills that are subject of secondary school mathematics. Tasks do not require advanced mathematical thinking. Therefore, applicability of the results of this study is limited. Aspects dealing with the nature and teaching of mathematics need to be added in the theoretical model in order to transfer it to achievement in pure mathematics.
Results for T1
At time T1, the maths skills of students are not influenced by the learning activities at the university. However, there are two exceptions. Some students participated in a preparation course before the semester started. Moreover, some participants did not pass the exam one semester earlier and had to repeat the module. In both cases we observe an expected positive influence. The regression coefficients for S1 and S2 are positive and significant at the 1 %level. These results indicate that the preparation course helps students to refresh their basic maths skills, exactly as intended.
Students start their studies at university with very different educational backgrounds. In particular, the success depends on the type of graduation. Two types have to be distinguished in the case of the German educational system. The estimates show that students who have obtained an ‘Abitur’ achieve significantly better test results at time T1. The regression coefficient for B2 (‘Abitur’ = 1; ‘FOS’ = 0) is positive and significant.
Furthermore, it appears that better school grades also imply better performance at time T1. This holds for the grade in mathematics B7 (1 = excellent; 5 = insufficient) and the final grade B6 (1 = excellent; 5 = insufficient). These results about the influence of type of graduation and school grades are in accordance with the metaanalysis by Robbins et al. (2004) who identified ACT/SAT scores and high school GPA as the strongest predictors of college performance.
An academic background of the parents (B4) has a negative influence on the maths skills of students. This is a surprising result. Here, further analysis is necessary to find explanations.
Regarding gender (B1; male = 1, female = 0), the number of years between the end of secondary school and the start of the studies at the university (B3), apprenticeship (B5; yes = 1; no = 0) and migration background (B8; yes = 1; no = 0) we found no significant influence. The results for B3 and B5 do not support the results about nonstandard students by Faulkner et al. (2011). Students who probably have not engaged in mathematics for some time (B3, B5) do not perform worse in the basic maths skills test at the beginning of the semester, but correlation analysis shows that on average they have worse final grades and maths grades, lower maths selfefficacy, lower maths selfconcept and higher maths anxiety. One possible explanation might be that the influence of B3 and B5 is mediated by grades and motivational variables. Further analysis (e.g. with structural equation modelling) is needed.
The nonsignificant influence of migration background could be interpreted in a positive way. However, selection before starting university might be the reason.
Block M contains six variables representing motivational factors. Two of them show a significant influence. Maths selfefficacy has a positive influence (M3; low = 1; high = 8) on maths performance at T1 as well as maths interest M6 (low = 1; high = 6). Both results are compatible with findings from empirical studies on secondary schooling and higher education (see Multon et al. 1991; Laging 2016a). The other variables of this block (maths selfconcept, mastery goal orientation, maths anxiety, perceived value of maths) have no significant influence in the multivariate regression analysis. The absolute values of these correlation coefficients of these variables with mathematics performance at T1 are between .20 and .39. But they also correlate moderately among each other and with maths selfefficacy and maths interest. This is consistent with the theoretical background. These constructs are distinct, but share several aspects. Especially selfefficacy and selfconcept are similar concepts. Consequently, it is not surprising that they do not all show significant influence on maths performance in a multivariate approach. From a theoretical and empirical perspective it is not clear which of these variables is most important for performance. Many researchers (e.g., Bandura 1997; Pajares and Miller 1994; Zimmerman 2000) argue in favour of selfefficacy, because it is a more specific construct than selfconcept. Our results for T1 support this conjecture.
Block L with a total of five variables reflects the learning strategies of students. Clearly, at time T1 the variables map the learning behaviour the students applied at school. Correlation analyses support the assumption of a positive relationship of deep cognitive strategies (elaboration) and basic maths skills, and a negative relationship of surface cognitive strategies (memorising) and basic maths skills. The correlation coefficients are small and multivariate analysis does not support these findings. Only the variable which represents control strategies has a statistically weak significant negative influence when applying multivariate regression techniques.
Comparing the estimated regression coefficients for the different models, it turns out that the inclusion of blocks does not have a major effect on magnitude and significance of the factors. But this is not unexpected, because the inclusion of blocks M and L increases R ^{2} only slightly.
In summary, it can be stated that in our study studyspecific, biographical and motivational variables determine the mathematical success at T1 with the type of graduation as one of the most important factors. Students who start with ‘Abitur’ have attended the preparation course more frequently, are stronger interested in mathematics and have higher maths selfefficacy beliefs. Therefore, we have good and bad news. On the one hand, we can show that the use of voluntary support has a positive effect. On the other hand, the type of graduation remains of great importance.
Results for T2
The estimates for the final model M2f for T2 show similar results. Discussing the results, we have to keep in mind that the skills test at T2 covers the topics which were already relevant at T1. In both cases, all 30 tasks are related to secondary school maths. However, at T2 we expect better maths skills for a number of reasons. First, in a period of nine weeks between T1 and T2, students had the opportunity to refresh secondary school maths in selfstudy. Second, we expect that working habits (block W) and the use of support (block U) will have a positive influence on maths skills at T2 although secondary school maths plays only a minor role in lectures and voluntary support.
First, the preknowledge measured by π ^{ T1} is the most important factor influencing maths skills at T2. This factor explains about 60 % of the variance if only π ^{ T1} is taken into account as a predictor (see model M2a). The strong influence is not surprising, because the period between T1 and T2 is quite short and experience has shown that despite clear instructions, students underestimate the importance of basic mathematical skills. Therefore, many students spend only a very limited time and other resources in order to reduce deficits. The results for T1 show that maths skills at T1 are partly explained by several variables. As a consequence, π ^{ T1} is influenced by these variables. Further analyses, e.g. with structural equation modelling, which require a larger sample, are needed to separate the influences.
In the estimates for π ^{ T2} we find no evidence for an influence of studyspecific effects. It is clear that the introduction of π ^{ T1} reduces the importance of the studyspecific variables. As before, however, the B variables play an important role. Again, the type of graduation (B2; ‘FOS’ vs. ‘Abitur’) is crucial. Moreover, the final grade (B6) has a significant influence, but not the grade in maths.
The inclusion of the motivational variables leads to a significant increase of R ^{2}. In general, it is the same as in the case of T1. But in detail, the results are different. Here, the factors maths concept (M2) and maths anxiety (M5) are significant at the 5 %level. This underlines the assumption that it is not clear which factor influences maths performance the most, but in general selfbelief is important.
The L block of variables representing learning strategies does not contribute to a significant increase of R ^{2}. Correlation analyses show moderately positive correlations between elaboration strategies (L2) and control strategies (L3) with maths skills at T2. There is a moderately to high correlation of these strategies with motivational variables. These results indicate that the learning strategies we measured do not contribute for the explanation of maths skills at T2. The interpretation that learning strategies are not important for developing maths skills would be prejudged. There might be other reasons for these results. For example, it might be crucial how learning strategies are measured and which specific learning strategies are chosen. Measuring learning strategies by students’ ratings is problematic. Knowing specific strategies and applying these strategies are two different things. In addition, transition from school to university requires several changes including adjusting learning strategies. The scales we use to measure learning strategies might be inapplicable for this research. An examination of strategies students really need and use to acquire basic maths skills at university would be useful to adjust measurements of learning strategies.
The block of Wvariables that represent the working behaviour of the students increases R ^{2} significantly. Only regularity (W3) shows a significant positive influence on maths skills in the multivariate approach. Correlation analyses show moderate correlations of effort, persistence and regularity with maths performance at T2, but no correlation of invested time and maths performance. Results of correlation analyses support research in the domain of homework. For example, results of multilevel modelling show that homework effort positively predicts maths achievement at the student level and time spent on homework has a negative influence on maths achievement (Dettmers et al. 2010). Spending much time on assignments can be a result of high effort, but also of lower cognitive abilities, missing knowledge, inappropriate learning strategies or missing concentration. Trautwein and Köller (2003) suggest distinctions between time spent on an assignment and time actively spent on it, and between time that is needed to complete an assignment and time actually spent on finishing it.
The use of various kinds of voluntary support does not contribute to a significant increase in R ^{2}. At first glance, this is surprising. However, a substantial explanation may be that the voluntary support is only a limited form of assistance for the development of basic mathematical knowledge. Most of the voluntary support is substantially aligned with higher education mathematics, which covers most of the relevant topics of the module. Furthermore, many students did not use voluntary support in the winter semester 2011/12. One reason might be that most of the voluntary support services were established for the first time in that semester. Research on the use of voluntary support revealed that a small number of the students (especially students who have already taken the exam) uses the various kinds of support extensively, but most students use them rarely and irregularly (Laging and Voßkamp 2016).
To sum up, it can be stated that the mathematical skills at T2 are significantly influenced by preknowledge, the type of graduation, the school grade, maths selfconcept and regular learning. Most of these variables represent students’ characteristics that are defined before they start their studies. This underpins the experience of students and maths educators that the negative effects of insufficient preknowledge and disadvantages due to educational background cannot be overcome in a short period. However, students whose working behaviour is characterised by regularity will show better results in maths performance.
Validity of the Models
It must be noted that the variables in M1d explain about 46 percent of variance in basic maths skills at T1, but still this model cannot explain a large proportion of variance. Consequently, factors have remained unconsidered or not sufficiently operationalised which are essential for the explanation of mathematical performance. Against this background, we cannot rule out that other variables mentioned in the model of SH may play an important role, e.g. home and school variables.
The variables in M2f explain about 74 % of variance in basic maths skills at T1. The inclusion of maths performance at T1 as an indicator for preknowledge increases R ^{2} substantially. A measure for preknowledge or cognitive ability to explain maths performance at T1 should be considered. At T1 preknowledge is measured by type of graduation and grades in school which are good predictors.
At both points of time, not all variables considered influence maths performance, which is not surprising by applying multivariate regression analyses. Individual cognitive and individual motivational determinants represented by a number of variables are confirmed as important predictors of maths performance. The weak support for learning activities as predictors might be due to the rare and irregular use of support services in this sample. Several variables do not show the assumed influence on maths performance as discussed above.
For our research, we adjusted the SH model to our sample. Therefore, several determinants need to be considered to transfer the model to other samples. Within a broad sample, variables measuring organization, climate and course context as well as cultural backgrounds should be added. In addition, the nature of mathematics and the way mathematics is instructed should be considered to capture specific issues of the transition of school mathematics to university mathematics.
Conclusion
Summary
In this paper, we have made an attempt to determine the predictors of mathematical skills of firstyear B&E students. Based on data from skills tests and questionnaires, regression results have shown that for B&E students the type of graduation, the final school grade and maths grades in school are very important predictors of maths performance when they start their studies. Also, motivational variables contribute to the explanation of maths performance. Other socioeconomic and biographical variables do not play a significant role. The same applies to the block of variables representing learning strategies. Moreover, the multivariate regression analysis shows that the maths skills in T1 representing preknowledge significantly influence the maths skills at T2.
Limitations
The results presented should be interpreted with reference to several limitations.
Sample: the sample includes only students of the University of Kassel. External validity is therefore limited. However, we have no indications that the results presented are driven by specific factors which would lead to biased results.
Measurement of mathematical achievements: the skills test covers different mathematical competencies (e. g., Blum et al. 2010), but not to the same extent (Laging 2016b). An analysis for specific domains of competencies is not possible due to the limited number of tasks. Moreover, we used classical test theory to construct the scales representing maths performance.
Measurement of the independent variables: the variables, respectively scales, related to the blocks M, L, W and U, have been created on the basis of proven and recognised concepts. However, it should be noted that the scales are based on selfassessments. Alternatives are not available in the framework of this approach.
Method: theoretical basis of the estimates is a simple model based on the concept of EPF. This simple model allows an empirical examination using regression techniques. Note, however, that the relevant theoretical contributions underpin strong arguments for the development of a path model or a structural equation model. In our case the number of observations is too small in order to test such models empirically.
Implications
Beside the limitations, our analyses show that a broad range of variables determines maths performance. For that reason, we can expect that numerous inventions exist which will have a positive impact on maths skills of firstyear students. However, we can expect that an implementation of a single intervention will not cause a fundamental improvement of mathematics performance. Only a mix of interventions will work. In addition, it should be noted that many interventions cannot be implemented by the maths educators or the students. Universities are faced to many factors they cannot control. In particular, this applies to factors which we take in the blocks S and B into account. For example, (German) universities have only limited options to define criteria for admission which affect the composition of the cohorts of the firstyearstudents.
Despite these general remarks, from the perspective of lecturers and students our analyses justify at least two concrete recommendations. First, our results show that voluntary support can have a significant positive impact on maths performance. In our case, this prevails especially for a preparation course. Second, our results show that regular learning is of great advantage. The positive influence of regularity brings up the question how incentives or command and control can force regularity. Further exams and weekly compulsory assignments might be appropriate inventions.
Outlook
The results presented are a first step towards the explanation of mathematical skills of firstyear students. However, the identified limitations leave scope for further studies that can be done only within the framework of a larger project.
An extension of the database by the inclusion of students at other universities is desirable. This could in particular contribute to the external validity of the results.
The mathematical skills of students were measured using classical test theory. In a further step the variables representing maths performance could be calculated in line with item response theory (IRT). However, we do not expect significant changes of the results.
The explanatory variables have been formulated on the basis of the questionnaires. The associated problems were mentioned. Using alternative methods would improve the analysis. For example, learning diaries offer better opportunities to capture the way students learn and work. However, this would go beyond the scope of this project.
Finally, from a theoretical point of view an investigation of indirect effects of the observed variables is desirable. This would drive the analyses towards structural equation modelling. However, such a research strategy is only meaningful if the analysis is not limited by the availability of data.
Notes
Acknowledgments
We would like to thank the participants of the Oberwolfach Workshop “Mathematics in Undergraduate Study Programs: Challenges for Research and for the Dialogue between Mathematics and Didactics of Mathematics” and the participants of khdm workshops for stimulating discussions. Moreover, we would like to thank Patrick Thompson and the anonymous reviewers for their very helpful comments. In particular, reviewer 2’s comments have greatly improved the article.
References
 Acock, A. C. (2014). A gentle introduction to stata (4th ed.). College Station: Stata Press.Google Scholar
 Ainley, M. D. (1993). Styles of engagement with learning: multidimensional assessment of their relationship with strategy use and school achievement. Journal of Educational Psychology, 85(3), 395–405.CrossRefGoogle Scholar
 Anderson, G., Benjamin, D., & Fuss, M. A. (1994). The determinants of success in university introductory economics courses. The Journal of Economic Education, 25(2), 99–119.CrossRefGoogle Scholar
 Anthony, G. (2000). Factors influencing firstyear students’ success in mathematics. International Journal of Mathematical Education in Science and Technology, 31(1), 3–14.CrossRefGoogle Scholar
 Arnold, I. J. M., & Straten, J. T. (2012). Motivation and math skills as determinants of firstyear performance in economics. The Journal of Economic Education, 43(1), 33–47.CrossRefGoogle Scholar
 Bandura, A. (1977). Selfefficacy: toward a unifying theory of behavioural change. Psychological Review, 84(2), 191–215.CrossRefGoogle Scholar
 Bandura, A. (1997). Selfefficacy. The exercise of control. New York: W. H. Freeman and Company.Google Scholar
 Bausch, I., Biehler, R., Bruder, R., Fischer, P. R., Hochmuth, R., Koepf, W., Schreiber, W., & Wassong, T. (Eds.). (2014). Mathematische Vor und Brückenkurse: Konzepte, Probleme und Perspektiven (Mathematical preparatory and bridging courses: concepts, problems and perspectives). Heidelberg: Springer.Google Scholar
 Blanchard, O., & Johnson, D. R. (2012). Macroeconomics. Boston: Pearson.Google Scholar
 Blum, W., DrükeNoe, C., Hartung, R., & Köller, O. (Eds.). (2010). Bildungsstandards Mathematik. (Educational standards). Berlin: Cornelsen.Google Scholar
 Boekaerts, M. (1999). Selfregulated learning: where we are today. International Journal of Educational Research, 31, 445–457.CrossRefGoogle Scholar
 Bong, M., & Skaalvik, E. M. (2003). Academic selfconcept and selfefficacy: how different are they really? Educational Psychology Review, 15(1), 1–40.CrossRefGoogle Scholar
 Brewer, D. J., Hentschke, G. C., & Eide, E. R. (2010). Theoretical concepts in the economics of education. In D. J. Brewer & P. J. McEwan (Eds.), Economics of education (pp. 3–8). Amsterdam: Elsevier.Google Scholar
 Bürgermeister, A., Kampa, M., Rakoczy, K., Harks, B., Besser, M., Klieme, E., Blum, W. & Leiß, D. (2011). Dokumentation der Befragungsinstrumente des Laborexperimentes im Projekt ‘Conditions and Consequences of Classroom Assessment (Co ^{2} CA)’ (Documentation of the survey instruments of the laboratory experiment in the project ‘Conditions and Consequences of Classroom Assessment (Co ^{2} CA)’). Frankfurt am Main: DIPF.Google Scholar
 Chemers, M. M., Hu, L., & Garcia, B. F. (2001). Academic selfefficacy and firstyear college student performance and adjustment. Journal of Educational Psychology, 93(1), 55–64.CrossRefGoogle Scholar
 Chiang, A. C., & Wainwright, K. (2005). Fundamental methods of mathematical economics (4th ed.). Boston: McGrawHill.Google Scholar
 Chiu, M. M., & Xihua, Z. (2008). Family and motivation effects on mathematics achievement: analyses of students in 41 countries. Learning and Instruction, 18, 321–336.CrossRefGoogle Scholar
 Clark, M., & Lovric, M. (2008). Suggestion for a theoretical model for secondarytertiary transition in mathematics. Mathematics Education Research Journal, 20(2), 25–37.CrossRefGoogle Scholar
 Demir, I., Kilic, S., & Depren, Ö. (2009). Factors affecting Turkish students’ achievement in mathematics. USChina Education Review, 6(6), 47–53.Google Scholar
 Dettmers, S., Trautwein, U., Lüdtke, O., Kunter, M., & Baumert, J. (2010). Homework works if homework quality is high: using multilevel modeling to predict the development of achievement in mathematics. Journal of Educational Psychology, 102(2), 467–482.CrossRefGoogle Scholar
 EACEA (Education, Audiovisual and Culture Executive Agency) (2015). Description of national education systems. Germany. Resource document. https://webgate.ec.europa.eu/fpfis/mwikis/eurydice/index.php/Germany:Redirect.
 Elliot, A. J., Murayama, K., & Pekrun, R. (2011). A 3 X 2 Achievement goal model. Journal of Educational Psychology, 103(3), 632–648.CrossRefGoogle Scholar
 Eurostat (2013). European systems of accounts. Luxembourg: Eurostat.Google Scholar
 Faulkner, F., Hannigan, A., & Gill, O. (2011). The changing profile of third level service mathematics in Ireland and its implications for the provision of mathematics education (1998–2010). In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of the 7th Conference of European Researchers in Mathematics Education (pp. 1992–2001). Rzeszow, Poland.Google Scholar
 Fischer, E., Bianchy, B., Biehler, R., Hänze, M. & Hochmuth, R. (2012). Lehrinnovationen in der Studieneingangsphase ‘Mathematik im Lehramtsstudium’. Hochschuldidaktische Grundlagen, Implementierung und Evaluation. Skalendokumentation (Teaching innovations in the first study phase ‘Mathematics in teacher training programmes’. Fundamentals, implementation and evaluation. Scales documentation). Unpublished.Google Scholar
 Gueudet, G. (2008). Investigating the secondarytertiary transition. Educational Studies in Mathematics, 67, 237–254.CrossRefGoogle Scholar
 Hailikari, T., Nevgi, A., & Komulainen, E. (2008). Academic selfbeliefs and prior knowledge of student achievement in mathematics: a structural model. Educational Psychology, 28(1), 59–71.CrossRefGoogle Scholar
 Hattie, J. (2009). Visible learning. A synthesis of over 800 metaanalyses relating to achievement. Abingdon: Routledge.Google Scholar
 Helmke, A., & Weinert, F. E. (1997). Bindungsfaktoren schulischer Leistungen (Factors of school achievement). In F. E. Weinert (Ed.), Enzyklopädie der Psychologie, Band 3 (Psychologie der Schule und des Unterrichts) (pp.71–176). Göttingen HogrefeVerlag.Google Scholar
 Heublein, U. (2014). Student dropout from German education institutions. European Journal of Education, 49(4), 497–513.CrossRefGoogle Scholar
 Hoppenbrock, A., Biehler, R., Hochmuth, R., & Rück, H.G. (Eds.). (2016). Lehren und Lernen von Mathematik in der Studieneingangsphase (Teaching and learning of mathematics in the first study phase). Wiesbaden: Springer.Google Scholar
 Huang, C. (2011). Achievement goals and achievement emotions: a metaanalysis. Educational Psychology Review, 23(3), 359–388.CrossRefGoogle Scholar
 Klieme, E. (2006). Empirische Unterrichtsforschung: Aktuelle Entwicklungen, theoretische Grundlagen und fachspezifische Befunde. Einleitung in den Thementeil (Empirical classroom research: current developments, theoretical background and specialist results). Zeitschrift für Pädagogik, 52(6), 765–773.Google Scholar
 Kuh, G. D., & Hu, S. (1999). Unraveling the complexity of the increase in college grades from the mid1980s to the mid1990s. Educational Evaluation and Policy Analysis, 21(3), 297–320.CrossRefGoogle Scholar
 Kunter, M., Schümer, G., Artelt, C., Baumert, J., Klieme, E., Neubrand, M., Prenzel, M., Schiefele, U., Schneider, W., Stanat, P., Tillmann, K.J., & Weiß, M. (2002). PISA 2000: Dokumentation der Erhebungsinstrumente (PISA 2000. Documentation of survey instruments). Materialien aus der Bildungsforschung Nr. 72.. Berlin: MaxPlanckInstitut für Bildungsforschung.Google Scholar
 Laging, A. (2016a). A metaanalysis about the relation of selfefficacy beliefs and achievement in mathematics. In preparation.Google Scholar
 Laging, A. (2016b). Stärke und Exaktheit der mathematischen Selbstwirksamkeitserwartungen bei Studienanfänger/innen (Strength and accuracy of firstyear students’ mathematical selfefficacy). In preparation.Google Scholar
 Laging, A., & Voßkamp, R. (2016). Identifizierung von Nutzertypen bei fakultativen Angeboten zur Mathematik in wirtschaftswissenschaftlichen Studiengängen (Identification of user types in the case of voluntary support services to mathematics in economics courses). In A. Hoppenbrock, R. Biehler, R. Hochmuth, & H.G. Rück (Eds.), Lehren und Lernen von Mathematik in der Studieneingangsphase (pp. 585–600). Wiesbaden: Springer.CrossRefGoogle Scholar
 Liston, M., & O’Donoghue, J. (2009). Factors influencing the transition to university service mathematics: part I a quantitative study. Teaching Mathematics and Its Applications, 28, 77–87.CrossRefGoogle Scholar
 Ma, X. (1999). A metaanalysis of the relationship between anxiety towards mathematics and achievement in mathematics. Journal for Research in Mathematics Education, 30(5), 520–540.CrossRefGoogle Scholar
 Multon, K. D., Brown, S. D., & Lent, R. W. (1991). Relation of selfefficacy beliefs to academic outcomes: a metaanalytic investigation. Journal of Counseling Psychology, 38(1), 30–38.Google Scholar
 OECD (Organisation for Economic Cooperation and Development) (2003). Literacy skills for the world of tomorrow: Further results from PISA 2000. Paris: OECDGoogle Scholar
 Pajares, F., & Miller, D. (1994). Role of selfefficacy and selfconcept beliefs in mathematical problem solving. A path analysis. Journal of Educational Psychology, 86(2), 193–203.CrossRefGoogle Scholar
 Pindyck, R., & Rubinfeld, D. (2013). Microeconomics. Global edition (8th ed.). Boston: Pearson.Google Scholar
 Pintrich, P. R. (2000). An achievement goal theory perspective on issues in motivation terminology, theory, and research. Contemporary Educational Psychology, 25(1), 92–104.CrossRefGoogle Scholar
 Rakoczy, K., Buff, A. & Lipowsky, F. (2005). Befragungsinstrumente. Teil 1. In Klieme, E., Pauli, C. & Reusser, K. (Eds.). Dokumentation der Erhebungs und Auswertungsinstrumente zur schweizerischdeutschen Videostudio ‘Unterrichtsqualität, Lernverhalten und mathematisches Verständnis’ (Documentation of the data collection and analysis tools for the SwissGerman video study ‘Teaching quality, learning and mathematical understanding’). Materialien zur Bildungsforschung. Band 13. Frankfurt am Main: DIPF.Google Scholar
 Robbins, S. B., Lauver, K., Le, H., Davis, D., & Carlstrom, A. (2004). Do psychological and study skill factors predict college outcomes? A metaanalysis. Psychological Bulletin, 130(2), 261–288.CrossRefGoogle Scholar
 Samuelson, P. (1948/1997). Economics: The original 1948 edition. McGrawHill Education.Google Scholar
 Schiefele, U. (2009). Situational and individual interest. In K. R. Wentzel & A. Wigfield (Eds.), Handbook of Motivation at School (pp. 197–222). New York: Routledge.Google Scholar
 Schiefele, U., Streblow, L., Ermgassen, U., & Moschner, B. (2003). The influence of learning motivation and learning strategies on college achievement: Results of a longitudinal analysis. German Journal of Educational Psychology, 17(3/4), 185–198.Google Scholar
 Schrader, F. W., & Helmke, A. (2015). School achievement: motivational determinants and processes. International Encyclopedia of the Social & Behavioral Science, 21, 48–54.CrossRefGoogle Scholar
 Shavelson, R. J., Hubner, J. J., & Stanton, G. C. (1976). Selfconcept: validation of construct interpretations. Review of Educational Research, 46(3), 407–441.CrossRefGoogle Scholar
 Shell, D. F., Murphy, C. C., & Bruning, R. H. (1989). Selfefficacy and outcome expectancy mechanisms in reading and writing achievement. Journal of Educational Psychology, 81(1), 91–100.CrossRefGoogle Scholar
 Simon, C. P., & Blume, L. (2010). Mathematics for economists. New York: Norton & Company.Google Scholar
 Sonntag, G. (2016) Studienerfolg ohne allgemeine Hochschulreife? Wie Herkunft, Bildungsverlauf und Wahlmotive den Studienerfolg beeinflussen (Academic success without higher education entrance qualification? The impact of origin, course of education, and motivation for the study programme choice on academic success). Marburg: Tectum Verlag.Google Scholar
 Stanat, P., & Lüdtke, O. (2013). International largescale assessment studies of student achievement. In J. Hattie & E. M. Anderman (Eds.), International guide to student achievement (pp. 481–483). New York: Routledge.Google Scholar
 Strayhorn, T. L. (2013). Academic achievement. A higher education perspective. In J. Hattie & E. M. Anderman (Eds.), International guide to student achievement (pp. 16–18). New York: Routledge.Google Scholar
 Sydsaeter, K., & Hammond, P. (2012). Essential mathematics for economic analysis (4th ed.). Harlow: Prentice Hall.Google Scholar
 Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24.CrossRefGoogle Scholar
 Thomas, M., de Freitas Druck, I., Huillet, D., Ju, M.K., Nardi, E., Rasmussen, C., & Xie, J. (2012). Survey team 4: key mathematical concepts in the transition from secondary to university. ICME12, Seoul, Korea.Google Scholar
 Trautwein, U., & Köller, O. (2003). The relationship between homework and achievement – still much of a mystery. Educational Psychology Review, 15(2), 115–145.CrossRefGoogle Scholar
 Voßkamp, R. (2016). Mathematics in economics study programmes in Germany: Structures and challenges. khdmreport No. 5. In press.Google Scholar
 Voßkamp, R., & Laging, A. (2014). Teilnahmeentscheidungen und Erfolg: Eine Fallstudie zu einem Vorkurs aus dem Bereich der Wirtschaftsmathematik (Participation decisions and success: a case study on a preparation course in the field of business mathematics). In I. Bausch, R. Biehler, R. Bruder, P. R. Fischer, R. Hochmuth, W. Koepf, S. Schreiber, & T. Wassong (Eds.), Mathematische Vor und Brückenkurse: Konzepte, Probleme und Perspektiven (pp. 67–83). Heidelberg: Springer.CrossRefGoogle Scholar
 Vrugt, A., & Oort, F. J. (2008). Metacognition, achievement goals, study strategies and academic achievement: pathways to achievement. Metacognition Learning, 30, 123–146.CrossRefGoogle Scholar
 Wigfield, A., & Eccles, J. S. (2000). Expectancyvalue theory of achievement motivation. Contemporary Educational Psychology, 25, 68–81.CrossRefGoogle Scholar
 Wooldridge, J. M. (2015). Introductory economics (6th ed.). Mason: Thomson.Google Scholar
 Zimmerman, B. J. (1990). Selfregulating academic learning and achievement: the emergence of a social cognitive perspective. Educational Psychology Review, 2(2), 173–201.CrossRefGoogle Scholar
 Zimmerman, B. J. (2000). Selfefficacy. An essential motive to learn. Contemporary Educational Psychology, 25, 82–91.CrossRefGoogle Scholar