Christer Bergsten
email: christer.bergsten@liu.se

Eva Jablonka
email: eva.jablonka@fu-berlin.de

Understanding the secondary-tertiary transition in mathematics education: contribution of theories to interpreting empirical data

Keywords: university mathematics, sector transition, theories, socio-cultural studies

The paper reports a review of studies on the secondary-tertiary transition in mathematics. With an interest in the contribution of theories, it intends to explore what insights empirical studies using new theoretical lenses have added. While there is a move towards socio-cultural theorizing, some dimensions pertaining to issues of social (re)production still appear under-researched.

Introduction

In mathematics education, the expression 'transition problem' commonly refers to difficulties when students enter a new school sector, or begin a new more 'advanced' course. While the notion has been used also for passages between other contexts, this contribution focuses on the transition from school to university, which has long been described as particularly problematic (e.g., De [START_REF] De Guzman | Difficulties in the passage from secondary to tertiary education[END_REF], inter alia alerting to high dropout rates. Nevertheless, a difficult transition may not only be seen as a problem for the students, but also "as something potentially positive for their development and future opportunities" (Jablonka, Ashjari, & Bergsten, 2017, p. 76). 'Smoothing' the transition may not necessarily be desirable. Conceived as problematic, the transition has attracted much attention in mathematics education over the years, and the analytical lenses employed for its study have been diverse. However, a comparison of the overviews in De [START_REF] De Guzman | Difficulties in the passage from secondary to tertiary education[END_REF] and [START_REF] Gueudet | Investigating the secondarytertiary transition[END_REF] shows that new theories might contribute to deepening the understanding of the secondary-tertiary transition. While De Guzman et al. categorized 'types' of problems in 'theoretical' categories (epistemological/cognitive; sociological/cultural; didactical), their discussion also included under-theorized notions (e.g., type of mathematics, contact to teachers, ways of studying). [START_REF] Gueudet | Investigating the secondarytertiary transition[END_REF] used similar categories (thinking modes, mathematical language and communication, and didactical transposition and contract) but in contrast to De [START_REF] De Guzman | Difficulties in the passage from secondary to tertiary education[END_REF] explicitly contained the discussion within a range of theoretical frameworks with reference to empirical research drawing on these frameworks. In a later "survey of the state of the art" regarding transitions in mathematics education, [START_REF] Gueudet | Transitions in mathematics education[END_REF] pointed to the diversity of theoretical orientations, but also to the importance of complementarity.

In this paper studies from the last ten years on the secondary-tertiary transition in mathematics education will be reviewed with the aim to trace the development in terms of the theoretical approaches taken and how these have contributed to new insights regarding the transition.

Method

For an initial literature search, articles published during the last ten years in major (mathematics) education research journals and conference proceedings as well as articles listed in the MathEduc database that explicitly addressed the secondary-tertiary transition were identified. From these, only articles reporting empirical studies with an explicit theoretical approach were included in a first step of the analysis. From these around 100 articles, a final sample of around 50 articles drawing on theories not already discussed in [START_REF] Gueudet | Investigating the secondarytertiary transition[END_REF] was selected for a second step; weakly theorized quantitative studies were not included. The main stated results of the studies were used to identify how theoretical approaches contribute to insights regarding the transition. For the notion of theory, the selection was guided by the discussions in [START_REF] Radford | Connecting theories in mathematics education: challenges and possibilities[END_REF] and [START_REF] Jablonka | Theorising in mathematics education research: differences in modes and quality[END_REF]. Certainly, some of the theoretical orientations discussed in [START_REF] Gueudet | Investigating the secondarytertiary transition[END_REF] continued to be used and developed during the period studied; for example, in the first step of the analysis around ten articles drawing on ATD and/or TDS and five using a cognitive approach were found. These represent an important part of the theoretical development of the area but will not be included here due to space limitations. As the literature review was not based on a fully comprehensive search, some recent theoretical developments might have been omitted. The categorization made from the analysis of the sample, however, has a strong base in the articles found.

In the next section, the dominating theoretical approaches identified will be presented. These were found in 37 (of the 50) papers; single studies employing Horizon content knowledge (MKT) and Three worlds (Tall), use-and-exchange-value, communities of practice, a framework for textbook comparison, as well as a few studies using theoretically less specified notions such as competences, interest, expectations, transmissionist pedagogies, and perspective, will not be discussed here.

Contributions of different theoretical approaches

In the selected articles, five theoretical approaches not discussed in the review by [START_REF] Gueudet | Investigating the secondarytertiary transition[END_REF] were identified, which unavoidably only adumbrate distinct categories: Bourdieusian sociology, Bernsteinian sociology of education, Cultural anthropology, Cultural Historical Activity Theory and Commognition. How these contribute to the understanding of the transition will be illustrated by way of examples of specific studies.

Bourdieusian sociology

A group of studies drew on Bourdieu's notions of habitus and capital to analyse students' struggles during the transition. [START_REF] Kleanthous | Perceived parental influence and students' dispositions to study mathematically-demanding courses in Higher Education[END_REF] employed the notion of habitus to develop a framework for measuring parental influence on students' inclination to pursue mathematics studies in higher education. They interpreted their findings to be consistent with a Bourdieusian view in that "family influences might inculcate a mathematical habitus that encourages choosing mathematically-advanced courses at school, which thus enhance mathematical confidence and inclination, and shapes positive dispositions to study mathematically-demanding courses in the future." (p. 64).

That also those students who find the transition hard may hang on to successfully complete the transition was illustrated in [START_REF] Hernandez-Martinez | Against the odds: resilience in mathematics students in transition[END_REF], where these students were labelled as resilient, defined as a socio-cultural term. For the analysis the authors employed Bourdieu's notions of cultural and social capital, adding that "students can develop capital through reflection" (p. 49), a capital allowing agency for example during transition to align "their habituses with the conditions of the new field" (p. 49). Based on a narrative analysis of data from two student interviews, the authors identified biographical "narratives of resilience" (p. 50) that showed how reflexivity led to the development of a capital during schooling that eventually became valuable during the transition, for example when experiencing risks to progression. For education practice the authors therefore suggested that "processes that encourage reflexivity in students should be incorporated in pedagogical practices at all levels" (p. 57). [START_REF] Bergsten | Mathematics as "Meta-Technology" and "Mind-Power": Views of engineering students[END_REF] interviewed students at the beginning, middle and end of their first year of study in a five years engineering programme at two Swedish universities, and interpreted their data in terms of Bourdieu's notions of field as well as cultural, economic and symbolic capital. Regarding the choice of programme at university, the outcomes corroborated those of [START_REF] Kleanthous | Perceived parental influence and students' dispositions to study mathematically-demanding courses in Higher Education[END_REF], showing how "the cultural capital possessed by the families influence the choice to enrol at a university" [START_REF] Bergsten | Mathematics as "Meta-Technology" and "Mind-Power": Views of engineering students[END_REF], p. 2293). Students also appeared to have acquired a cultural capital in their science-and technology programmes at secondary school that has developed into a disposition towards academic studies (many talking about "a natural continuation"). While expectations about future professions differentiated between students from different study programmes in terms of the importance of cultural and economic capital as well as the symbolic capital that multiplies their impact, the views of the role of mathematics in this context were more coherent, with a focus "less on particular mathematical skills as a form of cultural capital", but more on "a general habituation to solve problems in a rational way" (p. 2294).

Bernsteinian sociology of education

To identify issues and operationalize notions seen as critical for the transition, some studies employed key concepts from Bernstein's theory of pedagogic discourse, in some instances combined with other theoretical constructs. [START_REF] Kouvela | This is what you need to be learning": an analysis of messages received by first-year mathematics students during their transition to university[END_REF] focused on the discourses that frame the changes in teaching and learning practices during the transition from school to university, and how these discourses transmit messages to the students that affect how they experience and handle the transition. Drawing on Bernstein the authors distinguished strongly classified and strongly framed messages. The empirical study involved first-year mathematics students and two lecturers, and included lecture observations, student questionnaires and interviews; a grounded theory approach amounted to categories of messages (what students should do during and after lectures, assessment, general study habits). These messages, transmitted by the lecturers to their students, were analysed in terms of classification and framing helping to decompose each message and "identify, through the underlying structures of power and control, in what ways the message can be influential on students' transition" (p. 173). The general conclusion was that messages common between school and university ease the transition, while differences can "prompt action to be taken by students and can change their thinking" (p. 181).

Jablonka, Ashjari and Bergsten (2017) investigated how first year students interpreted the type of mathematics they encountered at university. To explicate the expression 'type of mathematics' and create a language of description for the reading of the empirical data, an analytical framework drawing on Bernstein's notions knowledge classification and recognition rules, elements of social semiotics by Halliday and Hasan, Eco's conception of a model reader, and Sierpinska's idea of mathematical and didactical discourses as mathematical and didactical layers in textbooks was developed. In individual interviews, textbook excerpts were shown to 60 first year engineering students at two Swedish universities, asking them to rank the texts according to which they saw as "more mathematical". The framework allowed a description of students' articulation of their recognition rules, and the authors concluded that "recognition of differences in strength of classification of different mathematical discourses emulated in a range of pedagogic practices relates to success in the examinations" (p. 87). The study thus illustrates how academic success relates to a recognition of the university mathematical discourse.

Cultural anthropology

A couple of studies used theories that can be located in the rather broad field of cultural anthropology. Four research groups referred to the idea of a phased rite of passage, coined by Arnold van Gennep at the beginning of the 1900s. [START_REF] Clark | Suggestion for a theoretical model for secondary-tertiary transition in mathematics[END_REF] first suggested that this is a useful model for the study of the transition to university (mathematics). The transition period is characterised through a separation phase (end of school, anticipating university life), liminal phase (from finishing school to beginning university), and incorporation phase (first year at university). It thus constitutes first of all a formal grid for framing the transition but does not specify what or how to approach the particular issues related to the phases.

One attempt to apply the model is found in a pilot study by [START_REF] Bampili | The transition from high school to university mathematics: A multidimensional process[END_REF], which presents data from interviews involving 12 students during their transition to mathematics studies at a Greek university. At the three phases of Clark and Lovric's model questions were asked regarding the dynamics of the social-personal life, academic life and mathematical content dimensions to investigate how "the academic and social dimensions interact to shape the passage from the liminal to the incorporation phase regarding mathematical content" (p. 1987). In the paper, data from one student were discussed "because her responses during data collection strongly indicated that she was undergoing a rite of passage regarding mathematics" (p. 1987). Based on their findings the authors considered the study as "a good starting point for exploring specific issues considering transition more extensively" (p. 1991). [START_REF] Wade | A comparison of mathematics teachers' and professors' views on secondary preparation for tertiary calculus[END_REF] conducted an online qualitative survey involving secondary and tertiary level mathematics teachers in the USA about what they conceived as good high school preparation for students for the college calculus courses. As a framework for interpreting their data they used the rite of passage. In particular, the separation and incorporation phases were investigated through thematic analyses that revealed five categories in which secondary and tertiary teachers "differed significantly in the relative frequency of addressing them" (p. 7). While teachers at university pointed at preparation of specific areas within algebra and precalculus as vital, teachers at secondary level highlighted "classroom environment realities of teaching in the separation phase." (p. 7). The authors concluded that a better understanding of the "gap between the teachers' and the professors' outlooks" is needed and that these two groups need to "realize more clearly that they are part of one and the same rite of passage process" (p. 13). [START_REF] Corriveau | The secondary-tertiary transition viewed as a change in mathematical cultures: an exploration concerning symbolism and its use[END_REF] drew on cultural theory (referring to the work of Edward T. Hall), in particular the distinction between formal, informal, and technical modes of culture and on ethnomethodology. They sought to understand how differences in secondary and post-secondary teachers' "ways of doing mathematics" rather than differences in formal or technical aspects of mathematics present difficulties for students. More specifically, the secondary teachers aimed at "giving form" to symbols that "speaks" to the student, representing particular generalisations, while the post-secondary teachers worked to "give voice" to the pre-existing general symbols. The authors concluded that "the 'informal' mode of mathematical culture specific to each teaching level plays a key role in attempts to better grasp transition issues" (Corriveau & Bednarz, 2017, p. 1).

Cultural Historical Activity Theory

Several transition studies used Cultural Historical Activity Theory (CHAT) to frame their research. In a project in the UK on student participation in post-compulsory mathematics education, [START_REF] Black | Developing a 'Leading Identity': The relationship between students' mathematical identities and their career and Higher education aspirations[END_REF] interviewed 40 students at three occasions during the transition period. In doing so, they used a notion of identity, which in their reading of CHAT is "historical in origin and emerges from the subjectivities (how one views oneself) we experience in the process of doing activities" (Black, Williams, Hernandez-Martinez, Davis, & Wake, 2010, p. 58). The authors argued that what they termed leading identity (based on Leont'ev's concept leading activity) defines a student's motive for study and also influences the relation to mathematics. A narrative analysis of interviews with two students with different experiences and showing "contrasting aspirations in their interviews" (Black et al., 2010, p. 59) was conducted. For the analysis an operationalization of the constructs leading activity, leading identity, cultural models (such as 'math is hard') and troubles/obstacles used to read the data was developed. The two students were both drawing on "cultural models regarding learning mathematics and studying in general" (p. 68) which framed their leading identities; in one case to become an engineer, in the other to obtain a university degree. The authors suggested that a leading identity implying the use value of mathematics as a motive for its study has a stronger potential to "bring about a commitment to persisting with mathematics, particularly for at-risk students who may face more difficulties or challenges than others" (p. 71).

Within the same project, investigating the transition from school to college mathematics in the UK through student pre-and post-transition interviews, Hernandez-Martinez et al. ( 2011) categorised the transition 'problems' in social dimension, continuity of curriculum/pedagogy, and individualprogression/differences. Rather than only a discourse focusing on problems, the authors also identified "a more positive discourse on challenge, growth and achievement" (p. 119). They focused on the notion of identity, relying mainly on CHAT, in particular the ideas of self in practice and consequential transitions. As a result, the authors suggested it to be viable to "re-think transition as a question of identity in which persons see themselves developing due to the distinct social and academic demands that the new institution poses" (Hernandez-Martinez et al., 2011, p. 119).

Commognition

A number of studies drew on the theory of commognition [START_REF] Sfard | Thinking as communicating: Human development, development of discourses, and mathematizing[END_REF] to analyse the transition to university mathematics education as a discursive shift that may involve a commognitive conflict, that is "the encounter[s] between interlocutors who use the same mathematical signifiers (words or written symbols) in different ways or perform the same mathematical tasks according to differing rules" (Sfard, 2008, p. 161). This approach provides a language for describing mathematical work as discursive practices, characterized by their shared word use, visual mediators, endorsed narratives and routines. In a UK context, [START_REF] Thoma | Transition from school to university mathematics: Manifestations of unresolved commognitive conflict in first year students' examination scripts[END_REF] investigated how a group of first year mathematics students solved closed book exam tasks, and also interviewed the two lecturers involved in the courses. In the students' scripts four manifestations of (unresolved) commognitive conflict were observed, two of which (the numerical domains involved, and visual mediators and rules of school algebra and set theory) seen as "stemming from the use of visual mediators and rules from the school mathematics discourse" (Thoma & Nardi, 2018, p. 165). A challenge for beginning students at university is then to recognise the switch and adapt to different discursive contexts, which was not required in the arithmetic and algebra dominated school mathematics. Some further examples are found in [START_REF] Nardi | Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of Calculus[END_REF] who discuss empirical studies highlighting "commognitive conflicts that characterise the transition to literate mathematics" (p. 194).

Discussion

The studies reviewed above illustrate how the framing of the secondary-tertiary transition in mathematics education has widened in scope during the last decade. With the exception of commognition, the theoretical approaches identified as "new" have been developed outside mathematics education; however, most of them have been imported first into other research areas in mathematics education. Many of the studies recruit a combination of different theoretical resources in order to achieve their research aims. Overall, the transition as a process has come more into focus, which is also reflected in the methodologies (e.g., interviewing students at different points in time). Referring to recent work, [START_REF] Gueudet | Transitions in mathematics education[END_REF] made a similar observation in noticing a trend towards approaches that intend to grasp transition as a process in its complexity.

Notably, all approaches discussed might be subsumed under "socio-cultural" theorizing. This is a move away from views that essentialize students' mathematical and cognitive dispositions that may (or may not) make them successful in crossing the boundary between school and university mathematics education. The new approaches illustrated above certainly have enriched the field, as transition is not only pictured as problematic but as necessarily involving discontinuities. By drawing on theoretical tools developed within coherent frameworks in the analyses of mathematicsspecific practices, critical experiences of the transition as well as social and cultural (dis)continuities have been identified that otherwise would have remained hidden; some questions would not have been asked without turning to theories developed outside mathematics education. This applies not only to such issues as implicit rules for the use of mathematical symbols or modes of argumentation, but also to the organization of the content of mathematical courses, their modes of study and resources used. This development is also in alignment with the general trend in mathematics education research, including university level (cf. [START_REF] Inglis | Five decades of mathematics education research[END_REF]. However, the dominant focus here is on the "cultural" to the detriment of the "social" in "socio-cultural" theorizing. How social orders or hierarchies are (re)produced in the transition remains underresearched, despite numerous statistical reports on underrepresentation or dropout in higher (STEM) education of particular social groups. While, for example, ATD offers a precise language of description for conceptualising differences between mathematical cultures, sociological approaches, in particular if they allow a description of mathematical practices, discourse and knowledge structures in relation to mechanisms of inclusion and exclusion, are important to understand how particular social hierarchies are (re-)produced within particular mathematical cultures (see e.g., [START_REF] Bergsten | Reproduction and distribution of mathematical knowledge in higher education: constructing insiders and outsiders[END_REF][START_REF] Bergsten | Mathematics as "Meta-Technology" and "Mind-Power": Views of engineering students[END_REF]; and the review of sociological theories in mathematics education in [START_REF] Jablonka | Theories for studying social, political and cultural dimensions of mathematics education[END_REF].

As to the cultures of doing mathematics, similarly as noted by [START_REF] Gueudet | Investigating the secondarytertiary transition[END_REF], technology as one area relevant for the transition appears under-researched in our sample of papers. While as long as 30 years ago technology (in particular CAS) was seen as a key resource for 'reshaping calculus' in the USA, "the potential of technology in helping students in the 'rite of passage' to tertiary education has not yet been researched" (Bardelle & Di Martino, 2012, p. 787). There are however recent attempts in theorizing technology use in the transition [START_REF] Gueudet | Didactic contract at the beginning of university: a focus on resources and their use[END_REF]. Still, even after suspending traditional ways of doing university mathematics (e.g., through technology), going through a status of liminality will be necessary in the process of enculturation.


Ralf Nieszporek
email: ralf.nieszporek@math.upb.de

Rolf Biehler
email: biehler@math.upb.de

Retrospective competence assessment in a PD course on teaching statistics with digital tools in upper secondary schools

Keywords: Professional Development, retrospective competence self-assessment, probability and statistics, digital tools

Recent changes in German national standards and curricula have made statistics including digital tools an inherent part of the final examination at upper secondary schools. In previous years, many teachers focused only on mandatory content like algebra or analysis and almost completely avoided optional stochastic content. Thus they are now facing the challenge to create and implement lessons for a topic in which they are inexperienced and feel insecure. The professional development course presented in this study has the aim to increase teachers' knowledge and competence in statistics, including the use of digital tools like graphic calculators and GeoGebra. A newly developed questionnaire was used to assess teachers' development during the course and led to preliminary findings concerning the knowledge and competence for advanced topics of inference statistics and the expertise of using digital tools in this field.

Design of the professional development course

There is a high demand for professional development courses on teaching statistics and the use of digital tools (like GeoGebra and graphic calculators) in Germany. Due to a recent change in the national standards [START_REF] Kmk | Bildungsstandards im Fach Mathematik für die Allgemeine Hochschulreife[END_REF], a new state curriculum was implemented in the German federal state of North Rhine-Westphalia (NRW) which made statistics including digital tools mandatory for the final exams (Abitur) in upper secondary schools. This obligation forced teachers to negotiate new and difficult challenges in their teaching practice. Many of them had not come into contact with statistics during their university education and they managed to also completely avoid this topic in their previous teaching practice. Therefore, content knowledge for teaching statistics or using digital tools in class is often missing (Batanero, Burrill & Reading, 2011). Teachers consequently feel insecure when teaching this topic. To meet the steadily increasing need for support, a professional development (PD) course was developed at Paderborn University. A team consisting of experienced facilitators, teacher educators, and practitioners revised an already existing course concept [START_REF] Biehler | Professional Development for teaching probability and inference statistics with digital tools at upper secondary level[END_REF] in the sense of Design Research [START_REF] Mckenney | Conducting Educational Design Research[END_REF]. The improvements for the course were based on two foundations: on the one hand research on teacher education and proven ideas for designing teaching lessons [START_REF] Biehler | Developing students' computer-supported simulation and modelling competencies by means of carefully designed working environments[END_REF][START_REF] Prömmel | Das GESIM-Konzept -Rekonstruktion von Schülerwissen beim Einstieg in die Stochastik mit Simulationen[END_REF], and on the other hand concepts for PD course construction and research like the design principles of the German Center for Mathematics Teacher Education (DZLM) [START_REF] Barzel | Design principles and domains of knowledge for the professionalization of teachers and facilitators -Two examples from the DZLM for upper secondary teachers[END_REF] or the three-tetrahedron model for content-related PD research (Prediger, Leuders & Rösken-Winter, submitted, 2018).

The new state curriculum of NRW demands a wide variety of statistical topics to be covered in school lessons. It was possible to concentrate the essence of the curriculum in five PD attendance days (see Table 1 for the content of each day) distributed over half a year with opportunities for teachers to try out the newly learned concepts in their lessons. In addition to that, the PD course also offers self-learning material so that teachers can deepen their knowledge on various topics like simulations with graphic calculators (GC), combinatorics, or the normal distribution. The aim of our study is to implement and analyze a PD course with an emphasis on increasing participants' knowledge and competence in teaching statistics with digital tools. This excerpt of our larger study focuses day five which covers the field of teaching more advanced topics of inference statistics testing like:  Interpretation of hypothesis testing results (with focus on didactics and language)  Type I and type II errors  Choice of null hypothesis  Operation characteristics and power function More information on the PD design, PD aims and the course content can be found in Biehler, Griese & Nieszporek (to be submitted).

Theoretical framework

The TPACK framework created by [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF] expands the mathematical knowledge for teaching framework (MKT) by Hill, Ball, and Schilling (2008). In addition to content knowledge (CK) and pedagogical knowledge (PK), Mishra and Koehler enriched the model with elements of technology knowledge (TK). However, those knowledge facets and their intersections like pedagogical content knowledge (PCK) or technology pedagogical content knowledge (TPACK) are not sufficient as the only input for a PD course. It is essential for teachers and for their daily practice to acquire additional skills beside pure knowledge on the topic, as they are confronted with different classes or situations and have to constantly undergo planning activities. Planning activities can be performed with fictional classes where problems and difficulties have to be anticipated and adequate reactions must be found. Therefore, they can be implemented in PD courses rather easily to strengthen the skills of the participants. In contrast to knowledge dimensions these skills were grouped as competences in regard to [START_REF] Weinert | Concept of competence: A conceptual clarification[END_REF]. Table 2 illustrates some of the knowledge and competence dimension addressed in our PD course.

Knowledge

Content

I have content knowledge relating to the topic.

Background of the content I know the didactical and curricular background and the learning goals of the topic Ideas for implementation I know ideas for the implementation of the topic in lessons.

Approaches for digital tools I know approaches for using digital tools / GCs in the context of the topic.

Competences

Handling misconceptions I am capable of recognizing and reacting appropriately towards misunderstandings (…). Planning I can create a lesson plan incorporating the goals of the topic. Teaching the content I can teach the topic in a goal-oriented way.

Integrating digital tools I am capable of implementing the approaches for the use of digital tools / GCs (…) in school lessons in a didactically advantageous way.

Table 2: Knowledge and competences addressed in the PD course, via statements to be rated

Teachers' heterogeneous knowledge and teaching experiences

For a long time, statistics and especially inference statistics have not been compulsory for upper secondary level, and were often neglected in university courses for teachers. This resulted in deficits regarding statistics knowledge of current in-service teachers, and some teachers might have mixed or mistaken content and horizon knowledge in some fields [START_REF] Wassong | A model for teacher knowledge as a basis for online courses for professional development of statistics teachers[END_REF][START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. They are hardly able to simplify the content matter for class, which should also be taught in a PD course on statistics [START_REF] Godino | The onto-semiotic approach to research in mathematics education[END_REF].

In addition to this deficiency, knowledge and competences towards digital tools are also demanded by new curriculum in NRW. GCs (or computer algebra systems, CAS) have also become mandatory in the final examination. Therefore, it is necessary to have at least a basic knowledge in programming, simulating or visualizing and interpreting distributions to fulfil the new demands. Due to the mismatch of teacher training and curriculum, this lack of experience must be addressed by a PD course, too.

Research questions

We will address the following research questions:

1. General assessment of course success from participants' perspective. To what extent (from before until after the PD course) do participants report to have acquired knowledge and competence in reference to conducting lessons in advanced topics of inference statistics testing? 2. Suitability of PD course in respect to the heterogeneity of teachers' previous experiences.

What differences concerning the self-reported development of knowledge and competence facets (from before until after the PD course) can be observed in different groups of teachers with comparable previous experience in teaching statistics and/or using graphic calculators?

Results for PD day 4 including a detailed description of the underlying theory, competence models and methodology can be found in Nieszporek, Griese & Biehler (to be submitted). The procedures used in this article form the foundation for this paper and the analysis of the course success. Due to space limitations it is only possible to give a brief overview of the results for PD day 5 in this article.

The ReCoS questionnaire

With reconsideration of the course concept and the corresponding theoretical background, the ReCoS questionnaire (see Figure 1) was developed to fulfill our needs. ReCoS is an abbreviation for retrospective competence self-reports which describes the questionnaire quite well. It combines the knowledge and competence dimensions presented earlier (Table 2) with the specific content facets of each day. The PD course participants rate their before and after level of the facets subsequent to each PD day. The self-reports were made via German school grades (1=excellent, 2=good, 3=satisfactory, 4=pass, 5=poor, 6=fail). 2 for the content of the columns Pre/post or knowledge test designs were not feasible measurement tools for our project. Reasons range from organizational difficulties (lack of time for a detailed knowledge test due to a tight PD course schedule) to ethical concerns or motivation on behalf of the teachers. There are also good reasons for a retrospective test instrument like ReCoS. On the one hand different effects like response shift or practice effects were prevented. On the other hand, there was no sensitization towards a specific aspect or topic of the PD (Willson & Putman 1982).

Data collection and sample size

The ReCoS questionnaire matrix was adjusted to the topic of each PD day and distributed at the end of the respective day. A different questionnaire measuring individual experience levels via a sixpoint Likert scale (from inexperienced and highly experienced) was filled in by the participants at the beginning of the whole course. Individual anonymous codes were used to match the two questionnaires. 32 out of a total of 60 participants filled in both questionnaire types completely and thus only their data was used for further analysis.

Either SPSS 25 or R were used for statistical analysis like test for normal distribution (Kolmogorov-Smirnov), significance of competence increase (Wilcoxon signed-rank test) or building of scales of items and participants' groups.

Results

The general success of the PD course is illustrated in Figure 2. The ReCoS matrix shows the mean increase for each item. First of all, every change between the before and after competence selfassessment is positive. The changes were significant (p<0.001, Wilcoxon signed-rank test) and were around 1 which is equal to an increase of one school grade. Overall, the differences for each item do not vary much. Item II (I know the didactical and curricular backgrounds and the learning goals of the topic) is an exception. For almost every content facet the self-assessment increases less than 1 grade, which stands in contrast to the confidence in creating lesson plans (item VI) or teaching the topic (item VII). This might be a hint that teachers have insecure knowledge of the curriculum (a discussion of the curriculum was not part of the PD) or have difficulties understanding this specific item. Nevertheless, some scales within the questionnaire can be identified. The theoretical framework of the course suggests a separation of the items into two scales knowledge (Item I -III) and competences (Item V -VII) for each content facet. Item IV and VIII form a separate scale called Technology, because technology plays a big role in the PD course and the abilities for using digital tools in school differ from the content components combined within knowledge and competence. The theoretical considerations are supported by empirical results. As is presented in Table 3, the identified scales within ReCoS show good to excellent Cronbach's α. In the following passages we will focus on the content facet type I and type II errors.

Table 4 shows the heterogeneity of experience among the teachers. The expertise reported in the field of teaching statistics and using the GC is not very high overall. It is striking that the participants were more unexperienced in handling digital tools (n=21), even when tools like GeoGebra or GC are commonly used for topics like algebra or analysis. This lack of experience for statistics appears to be a singularity. For obvious reasons, we will disregard the groups consisting of only one person here, and instead focus on the participants who had low experience in teaching statistics and using GCs (group L), those who had varied experienced in both fields (group V) and those who reported medium experience (group M). This selection reduced the sample size to 29 for further analysis. The increase of the mean differences (Figure 2) can be rediscovered in the boxplots in Figure 3. This figure visualizes the development of the three groups (L, V and M) for the content facet type I and type II errors. The before scores of each group fit their previous experience on teaching statistics. This means that the lowest experienced group (L) has the lowest before scores compared to the other two more experienced groups. Only group V, who also rated their experience in using digital tools low, shows equally low before scores in the technology scale. The after scores illustrate the huge increase of knowledge, competence and capability for technology reported earlier for the mean (Table 2). Even the most experienced group M rated themselves higher in all three scales after the PD course.

It is striking that there is a larger spread in the before scores of each group, which is remarkably reduced in the after scores of all three scales. Especially the participants of groups V and M rate their skills uniformly high. Only the lowest experienced group L shows a reduced but still high spread in the knowledge scale. There might be a ceiling effect at grade 2 (good) because only very few participants rated their skills higher with grade 1 (excellent). The rating system based on school grades might be the reason for that.

Discussion and remarks

Overall, the PD course increased the knowledge and competences of the participants by a significantly high amount (Figure 2) and was a success. But at the same time there are some hints as to where the PD course can be improved in a future design research cycle. We expected that the participants would rate their knowledge higher than their competence level after the PD course, because they supposedly would need to test the new material and ideas for teaching in their classes.

There is a small difference between the score levels, but the competence after scores were almost as high as the corresponding knowledge scores. The different opportunities in the PD course to test and discuss the new input (for more details see Biehler et al., 2018) seem sufficient to encourage the participants to cover type I and II errors appropriately in class. Participants also feel more capable of using digital tools not only for calculating purposes but also for its didactical opportunities in the learning process.

The distribution of the before scores of each group (Figure 3) supports our division of the participants via their pre experience level. For example, the group of varied experience (V) shows medium high pre scores in knowledge and competences but has low scores in technology. The three different groups vary significantly from each other in the pre scores. This difference is still apparent in the after scores but relatively smaller since the after scores of groups L and V are almost on the same high level as group M. Also, the heterogeneity within each group is reduced on all scales.

It is important to have in mind that ReCoS only provides self-reported data and also that the sample size of 29 is not a large one. Therefore, we plan to contrast our results with data from other PD days and to use data from different other sources (questionnaires, audio transcript) to validate our findings. The comparison of the results from day 4 and 5 seems to be satisfying.

Figure 1 :

 1 Figure 1: ReCoS questionnaire matrix, see Table2for the content of the columns

Figure 2 :

 2 Figure 2: Mean differences between the before and after scores of each item, a difference of one positive point describes an increase by one school grade, sample sizes between 29 and 32

Figure 3 :

 3 Figure 3: Jitterplots and boxplots of the three scale pairs knowledge (K), competences (C) and technology (T) for the content type I and type II errors (Type I/II error), group L: blue (n=15), group V: orange (n=5), group M: green (n=9), scale range from 6=fail to 1=excellent, made with R

Table 1 : Topics covered in the five-day PD course

 1

	Day	Topic	Content (examples)
	1	Getting started in statistics and probability at upper secondary level, using GCs and simulation -in heterogeneous classrooms	Far-reaching and apposite examples for the law of large numbers and the -law, introduction to digital tools (GeoGebra, GC) and simulations
		Conditional probability, statistics	Natural (absolute) frequencies in tree diagrams,
	2	independence, and expectation as basic	complementing contingency tables with double
		concepts in stochastic modeling	tree diagrams
		Modeling with probability distributions,	The binomial distribution dynamically
	3	particularly with the binomial distribution: suggestions for teaching in a content-related	visualized with digital tools, interconnecting insights from previous day by using the same
		and process-oriented way	examples, strengthen modelling critique
	4/5	Teaching inference statistics (hypothesis testing) with the goal of understanding, with authentic examples for the binomial distribution	Hypothesis testing via p-values to predefined significance level, errors of the first and second kind, power function, choice of the null hypothesis

Table 3 : Cronbach's α for the different scales, separately for the before and after items, sample sizes between 29 and 32

 3

	Content-specific facet	Knowledge Competences α α	Technology α
	Before scores
	Interpretation of hypothesis testing results	0.910	0.982	0.897
	Type I and type II errors	0.924	0.948	0.909
	Choice of null hypothesis	0.933	0.979	0.927
	Operation characteristics and power function	0.942	0.969	0.910
	After scores
	Interpretation of hypothesis testing results	0.913	0.965	0.848
	Type I and type II errors	0.906	0.947	0.930
	Choice of null hypothesis	0.901	0.968	0.951
	Operation characteristics and power function	0.925	0.965	0.898

Table 4 : Distribution of participants in terms of self-reported experience in the use of GCs in statistics and in teaching inference statistics

 4

Acknowledgment

We would like to thank the Deutsche Telekom Stiftung who funds the DZLM and also the regional government of Arnsberg for cooperation and the opportunity to implement the course concept just as Birgit Griese for the help in the development process of the methodology and evaluation of day 4.


Mihir Sahasrabudhe
email: mihir.sahasrabudhe@ecp.fr

Centralesupélec

Zhixin Shu
email: zhshu@cs.stonybrook.edu

Edward Bartrum
email: ebartrum@turing.ac.uk

Rıza Alp Güler
email: r.guler@imperial.ac.uk

Ariel Ai

Dimitris Samaras
email: samaras@cs.stonybrook.edu

Iasonas Kokkinos

Lifting AutoEncoders: Unsupervised Learning of a Fully-Disentangled 3D Morphable Model Using Deep Non-Rigid Structure From Motion

: We introduce Lifting AutoEncoders, a deep generative model of 3D shape variability that is learned from an unstructured photo collection without supervision. Having access to 3D allows us to disentangle the effects of viewpoint, non-rigid shape (due to identity/expression), illumination and albedo and perform entirely controllable image synthesis.

Introduction

Computer vision can be understood as the task of inverse graphics, namely the recovery of the scene that underlies an observed image. The scene factors that govern image formation primarily include surface geometry, camera position, material properties, and illumination. These are independent of each other but jointly determine the observed image intensities.

In this work we incorporate these factors as disentangled variables in a deep generative model of an object category and tackle the problem of recovering all of them in an entirely unsupervised manner. We integrate in our network design ideas from classical computer vision, including structure-from-motion, spherical harmonic models of illumination and deformable models, and recover the threedimensional geometry of a deformable object category in an entirely unsupervised manner from an unstructured collection of RGB images. We focus in particular on human faces and show that we can learn a three-dimensional morphable model of face geometry and appearance without access to any 3D training data, or manual labels. We further show that by using weak supervision we can further disentangle identity and expression, leading to even more controllable 3D generative models.

We first introduce Lifting AutoEncoders (LAEs) to recover, and then exploit the underlying 3D geometry of an object category by interpreting the outputs of a Deforming AutoEncoder (DAE) [START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF] in terms of a 3D representation. For this, we train a network to minimize a non-rigid SfM objective, which results is a low-dimensional morphable model of 3D shape, coupled with an estimate of the camera parameters. The resulting 3D reconstruction is coupled with a differentiable renderer [21] that propagates information from a 3D mesh to a 2D image, yielding a generative model for images that can be used for both image reconstruction and manipulation.

Our second contribution consists in exploiting the 3D nature of our novel generative model to further disentangle the image formation process. This is done in two complementary ways. For illumination modeling we use the 3D model to render normal maps and then shading images, which are combined with albedo maps to synthesize appearance. The resulting generative model incorporates spherical-harmonics-based [START_REF] Zhang | Face synthesis and recognition under arbitrary unknown lighting using a spherical harmonic basis morphable model[END_REF]47,[START_REF] Wang | Face relighting from a single image under arbitrary unknown lighting conditions. Pattern Analysis and Machine Intelligence[END_REF] modeling of image formation, while still being end-to-end differentiable and controllable. For shape modeling we use sources of weak supervision to factor the shape variability into 3D pose, and non-rigid identity and expression, allowing us to control the expression or identity of a face by working with the appropriate latent variable code. Finally, we combine our reconstruction-driven architecture with an adversarially trained refinement network which allows us to generate photo-realistic images as its output.

As a result of these advances, we have a deep generative model that uses 3D geometry to model shape variability and provides us with a clearly disentangled representation of 3D shape in terms of identity, expression and camera pose and appearance in terms of albedo and illumination/shading. We report quantitative results on a 3D landmark localization task and show multiple qualitative results of controllable photorealistic image generation.

Previous work

The task of disentangling deep models can be understood as splitting the latent space of a network into independent sources of variation. In the case of learning generative models for computer vision, this amounts to uncovering the independent factors that contribute to image formation. This can both simplify learning, by injecting inductive biases about the data generation process, and can also lead to interpretable models that can be controlled by humans in terms of a limited number of degrees of freedom. This would for instance allow computer graphics to benefit from the advances in the learning of generative models.

Over the past few years rapid progress has been made in the direction of disentangling the latent space of deep models into dimensions that account for generic factors of variation, such as identity and low-dimensional transformations [8, [START_REF] Worrall | Interpretable transformations with encoder-decoder networks[END_REF]28,[START_REF] Worrall | Harmonic networks: Deep translation and rotation equivariance[END_REF][START_REF] Sundermeyer | Implicit 3d orientation learning for 6d object detection from rgb images[END_REF], or even non-rigid, dense deformations from appearance [57,[START_REF] Gaur | Weakly supervised manifold learning for dense semantic object correspondence[END_REF][START_REF] Thewlis | Unsupervised learning of object frames by dense equivariant image labelling[END_REF][START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF][START_REF] Wiles | X2face: A network for controlling face generation using images, audio, and pose codes[END_REF]. Several of these techniques have made it into some of the most compelling photorealistic, controllable generative models of object categories [START_REF] Pumarola | Ganimation: Anatomicallyaware facial animation from a single image[END_REF][START_REF] Karras | A style-based generator architecture for generative adversarial networks[END_REF].

Despite these advances, the disentanglement of the threedimensional world geometry from the remaining aspects of image formation still remains very recent in deep learning. Effectively all works addressing aspects related to 3D geometry rely on paired data for training, e.g. multiple views of the same object [START_REF] Tulsiani | Multi-view supervision for single-view reconstruction via differentiable ray consistency[END_REF], videos [START_REF] Novotny | Learning 3d object categories by looking around them[END_REF] or some pre-existing 3D mesh representation that is the starting point for further disentanglement [START_REF] Genova | Unsupervised training for 3d morphable model regression[END_REF]35,53,40] or self-supervision [START_REF] Zhou | Unsupervised learning of depth and ego-motion from video[END_REF]. This, however, leaves open the question of how one can learn about the three-dimensional world simply by observing a set of unstructured images. Very recently, a few works have started tackling the problem of recovering the three-dimensional geometry of objects from more limited information- [START_REF] Kanazawa | Learning category-specific mesh reconstruction from image collections[END_REF] use keypoints and masks to learn a 3D deformable model of birds, [START_REF] Tran | Nonlinear 3d face morphable model[END_REF] use keypoints and semisupervised pretraining using a 3DMM, while [50] use a 3D geometry-based reprojection loss and correspondences of object instances during training.

Even though these works present exciting progress in the direction of deep 3D reconstruction, they fall short of providing us with a model that operates like a full-blown rendering pipeline. By contrast in our work, we propose for the first time a deep learning-based method that recovers a three-dimensional, surface-based, deformable template of an object category from an unorganized set of images, leading to controllable photorealistic image synthesis.

We do so by relying on Non-Rigid Structure from Motion (NRSfM). NRSfM, developed originally to establish a 3D model of a deformable object by observing its motion [START_REF] Bregler | Recovering non-rigid 3d shape from image streams[END_REF], was developed to solve increasingly accurately the underlying mathematical optimization problems [START_REF] Torresani | Nonrigid structure-from-motion: Estimating shape and motion with hierarchical priors[END_REF]30,[START_REF] Akhter | Nonrigid structure from motion in trajectory space[END_REF][START_REF] Dai | A simple prior-free method for non-rigid structure-from-motion factorization[END_REF], extending to dense reconstruction [START_REF] Garg | Dense variational reconstruction of non-rigid surfaces from monocular video[END_REF], lifting object cat- egories from keypoints and masks [START_REF] Carreira | Lifting object detection datasets into 3d[END_REF][START_REF] Kanazawa | Learning category-specific mesh reconstruction from image collections[END_REF], incorporating spatio-temporal priors [38] and illumination models [START_REF] Liu-Yin | Better together: Joint reasoning for nonrigid 3d reconstruction with specularities and shading[END_REF], while leading to impressively high-resolution 3D Reconstruction results [START_REF] Gotardo | Photogeometric scene flow for high-detail dynamic 3d reconstruction[END_REF][START_REF] Liu-Yin | Better together: Joint reasoning for nonrigid 3d reconstruction with specularities and shading[END_REF][START_REF] Hernandez | Accurate 3d face reconstruction via prior constrained structure from motion[END_REF][START_REF] Kemelmacher-Shlizerman | Internet based morphable model[END_REF][START_REF] Kemelmacher-Shlizerman | Collection flow[END_REF][START_REF] Kemelmacher-Shlizerman | Face reconstruction in the wild[END_REF]. In [START_REF] Kong | Deep interpretable non-rigid structure from motion[END_REF] it has recently been proposed to represent non-rigid variability in terms of a deep architecture -but still the work relies on given point correspondences between instances of the same category. We now show this is no longer necessary -we delegate the task of establishing correspondences across image pixels of multiple images to a Deforming AutoEncoder [START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF] and proceed to lifting images through an end-to-end trainable deep network as we now describe.

Lifting AutoEncoders

We start by briefly describing Deforming AutoEncoders, as these are the starting point of our work. We then turn to our novel contributions of 3D lifting in Sec. 3.2 and shape disentanglement in Sec. 4.2.

DAEs: from image collections to deformations

Deforming Autoencoders, introduced in [START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF], and shown in Fig. 2, follow the deformable template paradigm and model image generation through a combination of appearance synthesis in a canonical coordinate system and a spatial deformation that warps the appearance (or, 'texture') to the observed image coordinates. The resulting model disentangles shape and appearance in an entirely unsupervised manner and also provides dense correspondes between images and the learnt template, while using solely an image reconstruction loss for training. Further details for training DAEs are provided in [START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF].

LAEs: 3D structure-from-deformations

We now turn to the problem of recovering the 3D geometry of an object category from an unstructured set of images. For this, we rely on DAEs to identify corresponding points across this image set and address our problem by training a network to minimize an objective function that is inspired from Non-Rigid Structure from Motion (NRSfM). Our central observation is that DAEs provide us with an image representation on which NRSfM optimization objectives can be easily applied. In particular, disentangling appearance and deformation labels all image positions that correspond to a single template point with a common, discovered U V value. LAEs take this a step further and interpret the DAE's UV decoding outputs as indicating the positions where an underlying 3D object surface position projects to the image plane. The task of an LAE is then to infer a 3D structure that can successfully project to all of the observed 2D points.

Given that we want to handle a deformable, non-rigid object category, we introduce a loss function that is inspired from Non-Rigid Structure from Motion, and optimized with respect to it. The variables involved in the optimization include (a) the statistical 3D shape representation, represented in terms of a linear basis (b) the per-instance expansion coefficients on this basis and (c) the per-instance 3D camera parameters. We note that in standard NRSfM all of the observations come from a common instance that is observed in time -by constrast in our case every training sample stems from a different instance of the same category, and it is only thanks to the DAE-based preprocessing that these distinct instances become commensurate.

3D Lifting Objective

Our 3D structure inference task amounts to the recovery of a surface model that maps an intrinsic coordinate space (u, v) to 3D coordinates: S(u, v) → R 3 . Even though the underlying model is continuous, our implementation is discrete: we consider a set of 2D points sampled uniformly on a cartesian grid in intrinsic coordinates,

S i = S(u i , v i ), (u i , v i ) ∈ D × D , (1)
with D = 0, 1 n , 2 n , . . . , 1 , i = 1, . . . , N = (n + 1) 2 , (2)
where n determines the spatial resolution at which we discretize the surface. We parameterize the three-dimensional position of these vertices in terms of a low-dimensional linear model, that captures the dominant modes of variation around a mean shape B 0 ,

S i = B 0 + S s=1 s s B s i . (3)
In morphable models [START_REF] Vetter | A bootstrapping algorithm for learning linear models of object classes[END_REF]3] the mean shape and deformation basis elements are learned by PCA on a set of aligned 3D shapes, but in our case, we discover them from 2D by solving an NRSfM minimization problem that involves the projection to an unknown camera viewpoint.

In particular, we consider scaled orthographic projection Π through a camera described by a rotation matrix R and translation vector t. Under this assumption, the 3D surface points project to the points x i , given by

x i = Π [RS i ] + t , Π = σ 0 0 0 σ 0 , (4)
where σ defines a global scaling.

We measure the quality of a 3D reconstruction in terms of the Euclidean distance of the predicted projection of a 3D point and its actual position in the image. In our case a 3D point S i is associated with surface coordinate (u i , v i ), we therefore penalize its distance from the image position xi that the DAE's deformation decoder labels as

u i , v i - xi = x : argmin x DAE(x) -(u i , v i ) 2 .
(

) 5
In practice, we find x by warping (u i , v i ) under the DAE's deformation grid, W , and locating the point in the image coordinates that it warps to. If we can find no such point in the image coordinates, we deem the point invisible, setting a visibility variable ν i to zero. We treat x and ν as data terms, which specify the constraints that our learned 3D model must meet: the 3D points S i must project to points x i that lie close to their visible 2D counterparts, xi . We express this reprojection objective in terms of the remaining variables,

L(R, t, σ, s, B) = N i=1 ν i xi -x i (R, t, σ, S, s) 2 , (6)
where we have expressed x i as a differentiable function of R, t, S, s through Eq. 4 and Eq. 3.

For a set of K images we have different camera and shape parameters (R k , t k ), s k , k = 1, . . . , K since we consider a non-rigid object seen from different viewpoints. The basis elements B are however considered to be common across all images, since they describe the inherent shape variability of the whole category. Our 3D non-rigid reconstruction problem thus becomes

L 3D = K k=1 L(R k , t k , σ k , s k , B) . (7)

LAE learning via Deep NRSfM

Minimizing the objective of Eq. 7 amounts to the common Non-Rigid Structure-from-Motion objective [START_REF] Bregler | Recovering non-rigid 3d shape from image streams[END_REF][START_REF] Torresani | Nonrigid structure-from-motion: Estimating shape and motion with hierarchical priors[END_REF]30,[START_REF] Akhter | Nonrigid structure from motion in trajectory space[END_REF][START_REF] Dai | A simple prior-free method for non-rigid structure-from-motion factorization[END_REF]. Even though highly efficient and scalable algorithms have been proposed for its minimization, we would only consider them for initialization, since we want 3D Lifting to be a component of a larger deep generative model of images. We do not use any such technique, in order to simplify our model's training, implementing it as a single deep network training process.

The approach we take is to handle the shape basis B as the parameters of a linear 'morphable' layer, tasked with learning the shape model for our object category. We train this layer in tandem with complementary, multi-layer network branches that regress from the image to (a) the expansion coefficients s k , (b) the Euler angles/rotation matrix R k , and (c) the displacement vector t k describing the camera position. These are components of a larger deep network that can learn to reconstruct an image in 3D -a task we refer to as Deep NRSfM.

If we only train a network to optimize this objective, we obtain a network that can interpret a given image in terms of its 3D geometry, as expressed by the 3D camera position (rigid pose) and the instance-specific expansion coefficients (non-rigid shape). Having established this, we can conclude the task of image synthesis by projecting the 3D surface back to 2D. For this, we combine the 3D lifting network with a differentiable renderer [21], and bring the synthesized texture image in correspondence with the image coordinates. The resulting network is an end-to-end trainable pipeline for image generation that passes through a full-blown, 3D reconstruction process.

Having established a controllable, 3D-based rendering pipeline, we turn to photorealistic synthesis. For this, we further refine the rendered image by a U-Net [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] architecture that takes as input the reconstructed image and augments the visual plausibility. This refinement module is trained using two losses, firstly an L 2 loss to reconstruct the input image and secondly an adversarial loss to provide photorealism. The results of this module are demonstrated in Figure 5 -we see that while keeping intact the image generation process, we achieve a substantially more realistic synthesis.

Geometry-Based Disentanglement

In this section we show that having access to the underlying 3D scene behind an image allows to further decompose the image generation into distinct, controllable sub-models, in the same way that one would do within a graphics engine. We first describe in Sec. 4.1 how surface-based normal estimation allows us to disentangle appearance into albedo and shading using a physics-based model of illumination. In Sec. 4.2 we then turn to learning a more fine-grained model of 3D shape and use weak supervision to disentangle perinstance non-rigid shape into expression and identity.

LAE-lux: Disentangling Shading and Albedo

As in several recent works [START_REF] Shu | Neural face editing with intrinsic image disentangling[END_REF]35] we consider a Lambertian reflectance model for human faces and adopt the Spherical Harmonic model to model the effects of illumination on appearance [START_REF] Zhang | Face synthesis and recognition under arbitrary unknown lighting using a spherical harmonic basis morphable model[END_REF]47,[START_REF] Wang | Face relighting from a single image under arbitrary unknown lighting conditions. Pattern Analysis and Machine Intelligence[END_REF]. We pursue the intrinsic where denotes Hadamard product, by constraining the shading image to be connected to the normals delivered by the LAE surface. In particular, denoting by L the representation of the scene-specific spherical harmonic illumination vector, and by H(N (x)) the representation of the local normal field N (x) on the first 9 spherical harmonic coefficients, we consider that the local shading, S(x) is expressed as an inner product, S(x) = L, H(N (x)) . As such the shading field can be obtained by a linear layer that is driven by regressed illumination coefficients L and the surface-based harmonic field, H(N (x)). Given S(x), the texture can then be obtained from albedo and shading images according to Eq. 8.

In practice, the normal field we estimate is not as detailed as would be needed, e.g. to capture sharp corners, while the illumination coefficients can be inaccurate. To compensate for this, we first render an estimate of the shading S render with spherical harmonics parameters L and normal maps N and then use a U-Net to refine it, obtaining S adapted .

Given that the shading-albedo decomposition is an illposed problem, we further use a combination of losses that capture increasingly detailed prior knowledge about the desired solution. First, as in [START_REF] Shu | Neural face editing with intrinsic image disentangling[END_REF] we employ intrinsic imagebased smoothness losses on albedo and shading, L smooth shading = λ shade ∇S adapted 2 , and L albedo = λ albedo ∇A 1 , where ∇ represents the spatial gradient, which means that we allow the albedo to have sharp discontinuities, while the shading image should have mostly smooth variations [START_REF] Samaras | Variable albedo surface reconstruction from stereo and shape from shading[END_REF]. In our experiment, we set λ shade = 1×10 -4 and λ albedo = 2×10 -6 . Second, we compute a deterministic estimate L of the illumination parameters and penalize its distance to the regressed illumination values, L L = L -L 2 . More specifically, L is based on the crude assumption that the face's albedo is constant, Â(x) = 0.5, where we treat albedo as a grayscale. Even though clearly very rough, this assumption captures the fact that a face is largely uniform, and allows us to compute a proxy to the shading in terms of Ŝ = T Â where denotes Hadamard division. We subsequently compute the approximation L from Ŝ and the harmonic field H(N ) using least squares. For face images, similar to [START_REF] Shu | Neural face editing with intrinsic image disentangling[END_REF], L serves as a reasonably rough approximation of the illumination coefficient and is used for weak supervision via L L .

Finally, the shading consistency loss regularizes the U-Net, and is designed to encourage the U-Net based adapted shading S adapted to be consistent with the shading rendered from the spherical harmonics representation S rendered -L consistency shading = Huber(S adapted , S rendered ),

where we use Huber loss for a robust regression since S rendered can contain some outlier pixels due to an imperfect 3D shape.

Disentangling Expression, Identity and Pose

We consider that a face shape as observed in an image is the composite effect of camera pose, identity, and expression. Without some guidance, the parameters controlling shape can be mixed -for instance accounting for the effects of camera rotation through non-rigid deformations of the face.

For a given identity we can understand expression-based shape variability in terms of deviation from a neutral pose. We can consider that a reasonable approximation to this consists in using a separate linear basis B I for identity and another for expression B E , which amounts to following model-

S i (s I , s E ) = B 0 i + I s=1 s I s B I,s i + E s=1 s E s B E,s i . ( 10
)
Even though the model is still linear and is at first sight equivalent, clearly separating the two subspaces means that we can control them through side information-for instance, by imposing specific losses on these subspaces depending on the property. Here we use the MultiPIE [15] dataset to help disentangle the latent representation of a person's identity, facial expression, and pose (camera). Multi-PIE is captured under a controlled environment and contains image pairs acquired under identical conditions with differences only in (1) facial expression, (2) camera position, and (3) illumination conditions.

We use facial expression disentangling as an example, and follow a similar procedure for pose and camera disentangling. Given an image I exp with known expression exp, we sample two more images from the dataset. The first, I + exp has the same facial expression but different identity, pose, and illumination conditions. The second, I - exp , has a different facial expression but the same identity, pose and illumination condition as I exp . We use siamese training to encourage I exp and I + exp to have similar latent representations for facial expression, and a triplet loss to ensure that I exp and I + exp are closer in expression space than I exp and I - exp :

L expression = L similarity expression + L triplet expression , where ( 11)

L similarity expression = f exp (I exp ) -f exp (I + exp ) 2 , ( 12
)
L triplet expression = max(0, 1 + f exp (I exp ) -f exp (I + exp ) 2 -f exp (I exp ) -f exp (I - exp ) 2 ). ( 13
)
Following a similar collection of triplets for the remaining sources of variability, we disentangle the latent code for shape in terms of camera pose, identity, and expression. With MultiPIE, the overall disentanglement objective for shape is hence

L disentangle = L expression + L identity + L pose , (14)
where L identity and L pose are defined similarly to L expression .

In our experiments, we used the scaling parameter for this loss, λ disentangle = 1.

Complete Objective

We further control the model learning with a regularization loss given by

L reg = λ scale K k=1 σ k 2 + λ shape K k=1 S s=1 s k s B s 2 , (15)
where σ k is the scaling parameter in Eq. 4 and S s=1 s s B s i is the non-rigid deviation from the mean shape, B 0 . We use λ scale = 0.01, and λ shape = 0.1 in all our experiments. Combining this with the reprojection loss, L 3D , defined in Eq. 7, we can write the complete objective function, which is trained end-to-end:

L total = λ 3D • L 3D + λ disentangle • L disentangle + λ scale • L scale + λ shape • L shape . (16)
In our experiments, we used the scaling factor for the 3D reprojection loss, λ 3D = 50. This relatively high scaling factor was chosen so that the reprojection loss is not overpowered by other losses at later training iterations. A similar hyperparameter setting was also used by the authors of [START_REF] Kanazawa | Learning category-specific mesh reconstruction from image collections[END_REF].

For training the LAE-Lux, we also add the albedoshading disentanglement losses, summarised by

L lux = L smooth shading + L consistensy shading + L albedo + L L . (17)
Figure 4: Visualizations from various yaw angles of the learned 3D shapes without weak supervision. Our reconstructions respect prominent face features, such as the nose, forehead and checks, allowing us to rotate an object reconstruction in 3D.

Figure 5: Reconstruction from learnt shape trained with weak supervision and photorealistic refinement. Starting from an image reconstruction by an LAE (centre), weak supervision learns a much better shape, and an adversariallytrained refinement network adds details to increase the photorealism of a face (right).

Experiments

Architectural Choices

Our encoder and decoder architectures are similar to the ones employed in [START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF], but working on images of size 128 × 128 pixels instead of 64 × 64. We use convolutional neural networks with five stridedConv-batchNorm-leakyReLU layers in image encoders, which regress the expansion coefficients ss. Image decoders consist similarly of five stridedDeconv-batchNorm-ReLU layers.

Datasets

We now note the face datasets that we used for our experiments. Certain among them contain side information, for instance, multiple views of the same person, or videos of the same person. This side information was used for expression-identity disentanglement experiments, but not for the 3D lifting part. For the reconstruction results, our algorithms were only provided with unstructured datasets unless otherwise noted.

Qualitative Results

We show examples of the learned 3D shapes in Figure 4. The figure also visualises reconstructed faces from various yaw angles using a model that was trained only on CelebA images. We see that the model learns a shape that expresses the input well, as well as captures prominent facial features. The reconstructions are, however, weak for side poses. This is further refined using weak supervision from the Multi-PIE dataset.

Face manipulation results

In this section, we show some results of manipulating the expression and pose latent spaces. In Figure 6 (b), we

Quantitative Analysis: Landmark Localization

We evaluate our approach quantitatively in terms of landmark localization. Specifically, we evaluate on two datasets-the MAFL test set for 2D landmarks, and the AFLW2000-3D for 3D shape. In both cases, as we do not train with ground-truth landmarks, we manually annotate, only once, the necessary landmarks on the base shape as linear combinations of one or more mesh vertices. That is to say, each landmark location corresponds to a linear combination of the locations of several vertices. We use five landmarks for the MAFL test set, namely the two eyes, the tip of the nose, and the ends of the mouth. Similarly to [START_REF] Thewlis | Unsupervised learning of object landmarks by factorized spatial embeddings[END_REF][START_REF] Thewlis | Unsupervised learning of object frames by dense equivariant image labelling[END_REF][START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF], we evaluate the extent to which landmarks are captured by our 3D shape model by training a linear regressor to predict them given the locations of the mesh vertices in 3D.

We observe from Table 1 that our system is able to perform at-par with the DAE, which is our starting modeland as such serves as the upper bound on the performance that we can attain. This shows that while being able to successfully perform the lifting operation, we do not sacrifice localization accuracy. The small increase in error can be attributed to the fact that perfect reconstruction of a system is nearly impossible with a low-dimensional shape model. Furthermore we use a feedforward, single-shot camera and shape regression network, while in principle this is a problem that could require iterative model fitting techniques to align a 3D deformable model to 2D landmarks [START_REF] Pavlakos | 6-dof object pose from semantic keypoints[END_REF].

We report localization results in 3D on 21 landmarks that feature in the AFLW2000-3D dataset. As our unsupervised system is often unable to locate human ears, the learned face model does not account for them in the UV space. This makes it impossible to evaluate landmark localization for points that lie on or near the ears, which is the case for two of these landmarks. Hence, for the AFLW2000-3D dataset, we report localization accuracies only for 19 landmarks. Furthermore, as an evaluation of the discovered shape, we also show landmark localization results after rigid alignment (without reflection) of the predicted landmarks with the ground truth. We perform Procrustes analysis, with and without adding rotation to the alignment, the latter giving us an evaluation of the accuracy of pose estimation as well.

Table 2 also demonstrates the gain achieved by adding weak supervision via the Multi-PIE dataset. We see that the mean NMEs for LAEs trained with and without the Multi-PIE dataset increase as the yaw angle increases. This is also visible in our qualitative results shown in Fig. 5, where we visualize the discovered shapes for both of these cases.

Conclusion

In this work we have introduced an unsupervised method for lifting an object category into a 3D representation, allowing us to learn a 3D morphable model of faces from an unorganized photo collection. We have shown that we can use the resulting model for controllable manipulation and editing of observed images.

Deep image-based generative models have shown the ability to deliver photorealistic synthesis results with substantially more diverse categories than faces [START_REF] Brock | Large scale GAN training for high fidelity natural image synthesis[END_REF][START_REF] Karras | Progressive growing of GANs for improved quality, stability, and variation[END_REF] -we anticipate that their combination with 3D representations like LAEs will further unleash their potential for controllable image synthesis.

Figure 2 :

 2 Figure 2: Lifting AutoEncoders (LAE) bring Non-Rigid Structure from Motion (NRSfM) into the problem of learning disentangled generative models for object categories. We start from a Deforming-AutoEncoder (DAE) that produces dense correspondences between an observed and a template image, we train an LAE by minimising an NRSfM-based reprojection objective.

Figure 3 :

 3 Figure 3: Texture decoder for LAE-lux: disentangling albedo and illumination with 3D shape and Spherical Harmonics representation for illumination.

1 .

 1 CelebA [26]: This dataset contains about 200,000 inthe-wild images, and is one of the datasets we use to train our DAE. A subset of this dataset, MAFL[START_REF] Zhang | Facial landmark detection by deep multi-task learning[END_REF], was also released which contains annotations for five facial landmarks. We use the training set of MAFL in our evaluation experiments, and report results on the test set. Further, as MAFL is a subset of CelebA, we removed the images in the MAFL test set from the CelebA training set before training the DAE. 2. Multi-PIE [15]: Multi-PIE contains images of 337 subjects of 7 facial expressions, each of which is captured under 15 viewpoints and 19 illumination conditions simultaneously. 3. AFLW2000-3D [58]: This dataset consists of 3D fitted faces for the first 2000 images of the AFLW dataset. In this paper, we employ it for evaluation of our learned shapes using 3D landmark localization errors.

Figure 6 :

 6 Figure 6: Changing Pose with LAE. Given input face image (a), LAE learns to recover the 3D shape (c), with which we can manipulate the pose of the faces (b). With the additional refinement network, we can enhance the manipulated face image by adding facial details (d) that better preserve the characteristic features of the input faces.

Figure 7 :

 7 Figure 7: Changing expression with LAE. We interpolate between source (a) and target (c) images and show results of synthesizing new faces (b). We also show the result of extrapolation.

Figure 8 :

 8 Figure 8: Lighting manipulation using LAE-lux. We demonstrate the result of synthesizing new images by moving the lighting direction.

Table 1 :

 1 2D landmark localization results for the proposed LAEs compared with other state-of-the-art approaches. All numbers signify the average error per landmark normalized by the inter-ocular distance, over the entire dataset.

	5.45

Thewlis et al. (2017) [42] 6.67 Thewlis et al. (2018) [41] 5.83 Jakub et al. (2018) [17] 2.54 Shu et al. (2018), DAE, no regressor [36] 7.54 Shu et al. (2018), DAE, with regressor

[START_REF] Shu | Deforming autoencoders: Unsupervised disentangling of shape and appearance[END_REF]

Table 2 :

 2 Mean 3D landmark localization errors, after Procrustes analysis, normalized by bounding box size and averaged over the entire AFLW2000-3D test set. The number in brackets for the LAEs refers to the dimension of the latent space for the rigid and non-rigid components of the deformable model. The second column specifies whether rotation is included in the Procrustes analysis. We also note the training dataset used for training each LAE.

	Method	Rotation		Yaw angle
			[0, 30]	(30, 60]	(60, 90]	All
	3DDFA [58]	Y	4.25±0.95	4.34±1.04	4.39±1.35	4.28±1.03
	(supervised)	N	12.51±6.40	23.20±5.92	32.55±3.85	17.31±9.30
	PRNet [10]	Y	4.88±1.24	6.94±2.83	10.51±5.31	6.01±3.08
	(supervised)	N	7.17±3.45	10.96±5.00	16.34±8.91	9.11±5.66
	3D-FAN [6]	Y	2.73±1.38	2.48±2.24	3.74±2.95	2.84±1.92
	(supervised)	N	7.51±2.21	7.06±3.94	8.75±4.53	7.61±3.10
	LAE (64) CelebA	Y N	6.86±1.07 9.29±4.90	9.01±1.07 20.98±7.74	10.91±1.37 37.62±7.50	7.89±1.89 15.85±11.89
	LAE (128) CelebA	Y N	6.02±1.04 8.41±4.96	7.91±1.04 19.56±7.97	9.58±1.32 36.31±7.78	6.92±1.73 14.80±11.80
	LAE (128) MultiPIE	Y N	6.85±0.85 9.80±4.88	7.94±0.97 13.87±6.51	9.02±1.26 24.19±8.72	7.39±1.25 12.78±7.83
	LAE (128)	Y	6.83±0.96	8.41±1.15	9.83±1.65	7.59±1.60
	CelebA+MultiPIE	N	9.11±4.54	13.60±6.08	24.62±8.37	12.33±7.84

Acknowledgement. This work was supported by a gift from Adobe, NSF grants CNS-1718014 and DMS 1737876, the Partner University Fund, and the SUNY2020 Infrastructure Transportation Security Center.


Yael Fleischmann
email: yael.fleischmann@math.upb.de

Tobias Mai
email: tmai@math.upb.de

Rolf Biehler
email: biehler@math.upb.de

Alexander Gold
email: alexander.gold@math.upb.de

Digital learning materials in traditional lectures and their evaluation at the example of a voluntary pre-university bridging course

Keywords: transition, bridging course, evaluation design

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Description of the research topic

This contribution documents an excerpt from a conducted research study concerning the integration of digital learning materials into a voluntary, attendance-based bridging course for freshmen majoring in different engineering subjects at the University of Paderborn, Germany. The focus of the study was to evaluate the acceptance of digital learning materials by the students during the lectures and self-regulated learning days of the four-week course. Based on the use of an audience response system (ARS), we developed an instrument to collect feedback from the students immediately after elements of blended learningsuch as students' usage of interactive applets and digital, self-evaluating tasks with solutions and different elements using the ARSwere implemented during the lectures of this course to obtain students' reactions as instantaneous as possible.

The digital material was provided in form of the studiVEMINT-online course, which can be found on the website www.studiport.de as free-access material (only available in German yet). This learning material was designed and implemented by our research team during the years 2014 to 2016 and was created as an independent online course in mathematics that can be used for free by any person who wishes to prepare her-or himself for university mathematics (e.g., Biehler, Fleischmann, Gold, & Mai, 2017;Colberg, Mai, Wilms, & Biehler, 2014; studiVEMINT homepage: go.upb.de/studivemint).

Additionally, we prepared work assignments for this digital course that the students were asked to work on during the self-regulated learning days with the intention to support them in repetition and deepening their knowledge.

The study was conducted to answer (among others) the following research questions:

1. Do students appreciate the integration of digital learning materials in the lecture, i.e. do they experience it as a support for their learning process and do they find pleasure in using the materials? 2. How much time do the students invest to work with the provided learning materials on the self-regulated learning days?

Theoretical background and method

ARSs can support feedback from learners and provide a means to collect data for research purposes. Ebner, Haintz, Pichler, and Schön (2014) suggest a distinction between front-channel (direct feedback during lectures) and back-channel (asynchronous feedback during and out of the lecture) of those systems. They further distinguish these into qualitative and quantitative forms of feedback.

The focus of the study is the evaluation of the presented teaching scenario in mathematics with its elements of blended learning (e.g., Bernard, Borokhovski, Abrami, Schmid & Tamim, 2014) via the ARS.

We used the ARS functionality regarding quantitative front channel feedback (Ebner et al., 2014) during the lectures to collect reactions to the use of the digital elements as instantaneous as possible. Furthermore, we asked the students how much time they spent working (as they were requested to do) with the digital learning material on the self-regulated learning days in each subsequent lecture. Supplementary data was collected by a pen and paper questionnaire issued at the first and the last day of the course.

Results and discussion

The feedback concerning the integration of the digital learning materials into the lectures was largely positive. However, while looking at the results more closely, we found that more students appreciated the support of the digital materials for their learning processes than reported having pleasure while working with those materials. Another result was that the time students spent on their self-regulated learning days decreased dramatically during the four weeks of the course, even though students reported that the work on the tasks for these days was helpful for their individual learning and understanding. Further and more detailed results from the study have been presented at the conference.

  Didactics of Mathematics in Higher Education as a Scientific Discipline -Conference Proceedings. (pp. 466-470). Kassel: Universität Kassel: Khdm-report 17-05. Ebner, M., Haintz, C., Pichler, K., & Schön, S. (2014). Technologiegestützte Echtzeit-Interaktion in Massenvorlesungen im Hörsaal. Entwicklung und Erprobung eines digitalen Backchannels während der Vorlesung. In K. Rummler (Ed.), Lernräume gestalten -Bildungskontexte vielfältig denken. (pp. 567-578). Münster: Waxmann.

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