Christiane Kuklinski
email: kuklinski@idmp.uni-hannover.de

Elena Leis

Michael Liebendörfer

Reinhard Hochmuth

Rolf Biehler²

Elisa Lankeit²

Silke Neuhaus²

Niclas Schaper²

Mirko Schürmann²

Evaluating Innovative Measures in University Mathematics -The Case of Affective Outcomes in a Lecture focused on Problem-Solving

Keywords: Transition to and across university mathematics, Novel approaches to teaching, Teacher education, Motivational developments, Beliefs

The transition from high school to university mathematics has proven to be difficult for many students but especially for pre-service secondary teachers. To support these students at mastering this transition, various universities have introduced support measures of various kinds. The WiGeMath project developed a taxonomy that makes it possible to describe and compare these measures concerning their goals as well as their frame characteristics. We exemplify the use of the taxonomy in the description of one specific innovative measure that was part of the WiGeMath evaluations. Moreover, we present first results concerning the goal-fulfilment of this measure concerning affective characteristics of the student cohort and their predominant beliefs.

BACKGROUND

In German mathematics teacher education, pre-service teachers first study at university before they enter a practical training. In this first phase, there is a strong focus on mathematical content, in particular in higher secondary teacher education where students mostly attend the same courses as mathematics major students. In these shared lectures, many internationally known problems of the secondary-tertiary transition arise [START_REF] Gueudet | Investigating the secondary-tertiary transition[END_REF], in particular, motivational problems and drop-out are often reported. There is a substantial decline in students' mathematical interest in the first semester with Cohen's d around 0.4 (Rach und Heinze, 2013, 2016), a decline in their mathematical self-concept with Cohen's d ranging from 0.5 to 0.7 (Rach und Heinze, 2013, 2016), and a strong dominance of controlled motivation over autonomous motivation (Liebendörfer, in press) in the terminology of [START_REF] Ryan | Identity Development, Self-Esteem, and Authenticity[END_REF] self-determination theory. Consequently, many pre-service students experience their university courses as a necessary evil rather than a helpful qualification towards their aspired job [START_REF] Kalesse | Change in Mathematical Views of First Year University Students II[END_REF]Liebendörfer, in press).

THE WIGEMATH PROJECT

To counteract the negative effects which for many students seem to occur at the transition between school and university mathematics, many universities have introduced support measures of various kinds. Even though university internal evaluations of these measures mostly exist, a framework that helps to facilitate the comparison of design and outcomes of different measures has until recently been lacking. The WiGeMath project (Wirkung und Gelingensbedingungen von Unterstützungsmaßnahmen für mathematikbezogenes Lernen in der Studieneingangsphase; Effects and success conditions of mathematics learning support in the introductory study phase) [1], which is a joint research project of the Universities of Hannover and Paderborn [START_REF] Colberg | Wirkung und Gelingensbedingungen von Unterstützungsmaßnahmen für mathematikbezogenes Lernen in der Studieneingangsphase[END_REF] led by Biehler, Hochmuth and Schaper, has developed a framework for goal dimensions and frame conditions of mathematics learning support in universities (Liebendörfer et al., in press) that aims at such a comparison. Moreover, the project has used the framework in first evaluations of various support measures at different universities in Germany. Some exemplary results for one representative of the category of redesigned lectures, which is one type of support measure that was evaluated in the project, is presented below.

THE WIGEMATH TAXONOMY

The aim of the WiGeMath project is to develop and exemplify in use a taxonomy that categorizes features and goals of Projects of Mathematics Learning Support (PMLS) and to use this taxonomy to evaluate different support measures at German universities. All projects that fall under the category of PMLS have in common that they try to support students at the beginning of their university studies in mastering the critical transition to university mathematics. They are innovative insofar as they deviate from the standard format of lectures and tutorials that is encountered in university mathematics even though the way how they do this differs. In the WiGeMath project, different PMLS are subsumed under one of four categories, namely bridging courses, mathematics support centres, support measures that parallel courses and redesigned lectures. Due to space limits, this text focuses only on redesigned lectures; a description of the other types of PMLS is given in (Liebendörfer et al., in press).

Redesigned lectures are lectures that offer particular support to students that have been shown to have higher risks at failing mathematics courses or focus on very specific learning goals in a non-traditional way. We examined both redesigned lectures that address preservice secondary teachers, who often show the greatest problems with the transition from school mathematics to more abstract mathematical content, and redesigned lectures which address engineering students who had already failed a compulsory mathematics test of some kind. All redesigned lectures have in common that new mathematical content is not the focus of teaching.

Different PMLS have different aims some of which are explicit but some of which stay hidden even to the teaching staff until they are inquired about them by an outsider. WiGeMath aimed at evaluating different PMLS based on their own assumptions following a program evaluation approach [START_REF] Chen | Theory-driven evaluations[END_REF] as well as comparing them on grounds of an encompassing taxonomy that should categorize descriptive (non-normative) goals in the sense of criterions that the PMLSs set out to meet in their conception, features and conditions of PMLSs.

The taxonomy was constructed in a two-fold process. A first draft was developed by the project members by means of a document analysis, taking into account documents provided by project partners. The WiGeMath project collaborates with 11 partner universities in Germany at which PMLS have been implemented. The draft for the taxonomy was then used as a guiding thread for guided interviews with teaching staff of eight PMLS, two of each category. The interviews were taped and transcribed and afterwards the draft for the taxonomy was tested by trying to fit mentioned goals into the draft's categories. This led to minor refinements and reformulations of categories and yielded the final WiGeMath taxonomy. This final WiGeMath taxonomy consists of three main categories, namely frame conditions, measure categories/ characteristics and goal categories. The frame conditions include various sub-categories, which help to characterize the student cohort addressed by a PMLS, the way it came about and developed, its embeddedness in the university course system, organisational characteristics that may pertain to it, characteristics of the room where it is held, financial and staff conditions and lastly characteristics of the learning culture. Measure categories/ characteristics serve to describe certain elements that characterize the PMLS in its structure, its didactical elements and its teaching staff. Finally, goal categories encompass various subcategories of goals that either regard the individual learner or goals that the university may have in implementing the PMLS as a broader organization as well as a subcategory that allows to describe the quality of the goals. Not every sub-category has to be relevant in the description of every PMLS and some aspects of a PMLS may pertain to more than one category but the use of these categories aims to give an allencompassing description of a PMLS's characteristics.

In the following, the use of the WiGeMath taxonomy shall be exemplified by applying it to one of the redesigned lectures.

CHARACTERIZATION OF REDESIGNED LECTURES

All in all, six redesigned lectures at five German universities were evaluated in the WiGeMath project. Out of these, four addressed preservice secondary teachers and two addressed engineering students. To reach a characterization of each lecture, an interview guided by the WiGeMath taxonomy was held with the teacher before the start of the semester. It was audiotaped and transcribed and the transcript was then used to name the measure's characteristics and sort them into the right categories of the WiGeMath taxonomy. What follows is the exemplary description of one of the evaluated lectures along the lines of the taxonomy's categories and sub-categories.

Frame conditions

Concerning the characteristics of the student cohort, those students that attended the given redesigned lecture were preservice secondary teachers in their first semester meaning that most of them were less than 20 years old and had just graduated from high school. The lecture has had round about 200 participants in each of its turns.

The development of the lecture officially started in 2011. Since then, there has not been a strict script, which is followed each year, but the different lecturers who have been responsible focused on different aspects. Nonetheless, the basis of the lecture always is the book by [START_REF] Grieser | Mathematisches Problemlösen und Beweisen[END_REF] which deals with problem-solving strategies and proofs.

As to the embeddedness of the lecture in the wider system of university lectures, it is compulsory for preservice secondary teachers and voluntary for mathematics majors in their third semester. For preservice secondary teachers the lecture has substituted Linear Algebra as a first semester lecture though they still have to attend Linear Algebra in their second semester.

Staff conditions have been marked recently by problems to find qualified tutors to give the tutorials that support the lecture. As noted before, though there is only one lecturer per semester, it is not always the same one.

Finally, the learning culture is characterized by a strong focus on the students being active in their learning. They are supposed to try new methods and solve tasks during the lectures as well as during the tutorials. The concept of a "thinking pause" is very much enforced in the problem solving process. The lecturer gives some new input at the beginning of each class and collects and discusses results after the students have worked on problems or proofs.

Measure categories/characteristics

As to the structural characteristics, the measure consists of a lecture of two times 90 minutes per week with a tutorial of 90 minutes per week. One cycle of the measure starts at the beginning of each winter semester and finishes at its end (October through January).

The didactical elements include weekly homework and tutorial work of three to four exercises. All exercises may be worked on in groups. The solutions are discussed in the tutorials but no exemplary solutions are handed out. During the lecture there are phases of teacher talk, partner work and individual work. The book by [START_REF] Grieser | Mathematisches Problemlösen und Beweisen[END_REF] is named as a reference text and can be accessed online on campus. At the end of the semester, a written exam concludes the course.

Concerning the characteristics of the teaching staff, the lecturer has his PhD in mathematics and is responsible for the contents of the lecture as well as the tutorials, the exercises and the final exam. The tutorials are given by six tutors who are students in higher semesters. These same tutors also have to correct the exercises which are handed in by students. As mentioned before, the selection of tutors proved difficult due to a small number of qualified applicants.

Goal categories

In the category of individual learning goals the measure focuses on activity-oriented rather than on knowledge-related goals: Both mathematical working strategies, like problem-solving strategies or use of examples and counter-examples, and learning strategies shall be improved. Attitudinal goals play a major role, as well: The measure aims at strengthening process beliefs and weakening toolbox beliefs in the sense of [START_REF] Grigutsch | World views of mathematics held by university teachers of mathematics science[END_REF] and wants to introduce the students into the mathematical professional community. Affective characteristics are to be influenced insofar as anxieties shall be lessened, interest and motivation shall be strengthened and the students shall gain a higher mathematical self-efficacy. Moreover, the measure wishes to let students recognize the relevance of its contents for further university studies.

The system-related goals include the preparation of the participants for their further university studies and the decrease of the number of dropouts. Besides, the measure wants to increase the quality of the feedback that students receive during their studies.

The quality of the goals as understood in the WiGeMath taxonomy is not to be understood in a normative sense but rather as a description of their substantiality. With this aim in mind, goals are examined concerning how specific, measurable, accepted, realistic and time-phased they are. Such a description of the quality of a goal would be done for every goal individually in a thorough analysis but due to space restrictions we will only focus on one specific goal in this paper to illustrate the point: One of the measure's goals is to improve affective characteristics of the participants, i.e. to lessen maths anxiety, increase motivation and interest and improve the participants' mathematical self-efficacy, in the course of the semester. This goal is specific to the point that it explicates what shall be achieved when by whom. It remains unspecific in naming why the goal is important, who holds the responsibility to reach it and which preconditions or limitations possibly exist. The goal is indirectly measurable through a survey directed at the students and it is accepted as it was named by the lecturer as a goal he wants to achieve rather than a goal he has to achieve due to orders given from above. The goal seems to be realistic to the point that the affective characteristics of the students seem to change for worse quite fast at the beginning of their university studies so it seems plausible that they may be changed for better even within a single semester. Still, this has to be checked as will be shown below. Finally, the goal is time-phased as it shall be reached within a limited time, namely the duration of the measure.

In this paper we will evaluate the following research question: To what extent was the described redesigned lecture successful in achieving the last-mentioned goal of influencing affective characteristics of the student cohort and in how far was the lecture successful in changing beliefs away from toolbox beliefs towards process beliefs?

METHODS

To measure the extent to which the above mentioned goals were reached, two questionnaire surveys were conducted with the participants of the described lecture. The first survey (t1) took place in the second week of the winter semester 2016 and the second one (t2) was conducted in the second to last week of the same semester. For each survey a questionnaire was developed, each one laid out to take about thirty minutes to complete. These questionnaires were handed out at the end of a lecture so that only those who were present that day could participate and participation was voluntary which the students were informed about. Moreover participation was anonymous but students used an individually constructed code so that the results of each participant in the first survey could be compared and contrasted to the results in the second survey. 163 students participated in the first survey and 103 in the second. We analyze the data of the 76 participants who answered both questionnaires.

We used adopted versions of the scale of [START_REF] Schiefele | Der "Fragebogen zum Studieninteresse" (FSI)[END_REF] to measure interest, the scale of [START_REF] Schöne | Skalen zur Erfassung des schulischen Selbstkonzepts: SESSKO[END_REF] to measure mathematical self-concept, a translation of the instrument of [START_REF] Longo | Measuring Need Satisfaction and Frustration in Educational and Work Contexts: The Need Satisfaction and Frustration Scale (NSFS)[END_REF] to measure the experience of competence, autonomy and social relatedness, an adopted version of the PISA 2000 instrument for self-efficacy [START_REF] Kunter | PISA 2000: Dokumentation der Erhebungsinstrumente[END_REF], a shortened version of the scales by [START_REF] Grigutsch | World views of mathematics held by university teachers of mathematics science[END_REF] to measure beliefs (application, process, system and toolbox) and a translated version of the academic motivation scale [START_REF] Vallerand | The Academic Motivation Scale: A measure of intrinsic, extrinsic, and amotivation in education[END_REF] to measure different types of motivational regulation (intrinsic, identified, introjected and extrinsic). We used Likert scales ranging from 1 to 4 for self-concept, self-efficacy and beliefs, from 1 to 5 for motivational regulation, from 1 to 6 for interest and from 1 to 7 for the experience of competence, autonomy and social relatedness. All reliability coefficients (Cronbach's alpha) are acceptable or better, compare Table 1.

For each scale and each survey, a descriptive data analysis was conducted in order to get an overview of the results. Though the entirety of scales included more than the ones mentioned above, we concentrate on these only as our focus is to check to what extent affective characteristics of the student cohort, their experience of competence and their attitude towards different beliefs was changed in the course of the semester.

RESULTS

Table 1 shows the changes in mean values during the semester and effect sizes (Cohen's d) as well as p-values of paired t-tests. We see a substantial decline in interest and in the experience of competence, whereas students' mathematical self-concept and self-efficacy did not change significantly. We can also see that the objective of reducing toolbox beliefs was clearly achieved, but not the objective of strengthening process beliefs. The mean values of motivational regulation show that intrinsic and identified regulation are dominating although they are decreasing in the course of the semester.

Scale

DISCUSSION

The observation of a decline in interest is similar to the results showing a decline in traditional courses; however, student's mathematical self-concept does not change significantly, which is a major difference (Rach und Heinze, 2013, 2016). Although in our tests we were not able to show that the course could raise student's self-efficacy, it did not reduce it significantly either, which may still be an achievement. The dominance of intrinsic and identified motivation is a positive result as a study in traditional courses found extrinsic and introjected regulations to be dominant (Liebendörfer, in press). Thus, although students' interest in university mathematics and their intrinsic motivation may reduce, they do not seem to develop a stronger feeling of being inadequate for studying mathematics in the newly designed lecture. The decline in interest as well as intrinsic and identified regulation may be explained by a decline in the experience of competence. The change in students' toolbox beliefs is remarkable as beliefs are rather stable by definition and toolbox beliefs did not change in other studies in the first year of lower secondary or primary mathematics teacher education [START_REF] Kolter | Mathenein danke? Interesse, Beliefs und Lernstrategien im Mathematikstudium bei Grundschullehramtsstudierenden mit Pflichtfach[END_REF][START_REF] Liebendörfer | Interest development during the first year at university: do mathematical beliefs predict interest in mathematics?[END_REF].

These results show that a specifically designed lecture may reduce problems of the secondary-tertiary transition in mathematics. Nonetheless, the question remains whether such lectures prepare the students for their further studies just as well as traditional teaching does, considering that the course covered fewer mathematical topics.

Moreover, the analysis that has been done to this point cannot ensure that the results obtained were produced by the innovative measure alone. First of all, the lecturer's personality has an influence on the measure's outcomes that could not be separated from the outcomes of the measure itself in our study. A possible further effect may be caused by the change in order of other lectures, in explicit the postponement of Linear Algebra to a later semester: Whereas students usually experience their low competency in both Analysis and Linear Algebra in the first semester, in this case it is only one lecture.

In order to test this hypothesis, a next step in the WiGeMath project will be to distribute the same questionnaire that was used in the investigation described above to a different innovative measure at a university where different courses are attended simultaneously, as well as to a traditional lecture. This will make comparisons more explicit.

As to the taxonomy that was developed by the WiGeMath project, this in part resembles the objectives of other taxonomies [START_REF] Krathwohl | A Revision of Bloom's Taxonomy: An Overview[END_REF] though with a different focus. Whereas other taxonomies are mostly concerned with individual learning outcomes, the WiGeMath taxonomy targets a description and ensuing comparison of innovative measures as a whole. Though other taxonomies exist which classify systems of higher education institutions (for example the Carnegie Classification of Institutions of Higher Education, described in [START_REF] Bartelse | The European Higher Education Classification: Objectives and Concepts[END_REF], the perspective taken by WiGeMath to interpret such characteristics as goals is a new one.

So far, the taxonomy is only laid out to serve innovative measures and even in this area will have to be adapted as measures develop and improve. Traditional lectures have not been taken into consideration so far but we propose that these would also benefit from a similar taxonomy in terms of communicating frame conditions and learning goals. In our interviews with lecturers, we found that often even to them goals remained implicit until they were asked about them specifically. This might even more be the case in traditional lectures that have "worked" for a long time.

As mentioned above, in many cases goals stay hidden until a framework like the one developed by the WiGeMath project provides a common language to talk about them.

Even though lecturers have specific intentions when they design a course with specific learning goals that a student cohort with certain characteristics shall achieve in a setting framed by staff conditions, learning culture, etc., they often lack guidelines to arrange these in a way that is comprehensible for others. Yet, only if they can explicate their ideas, can an evaluation be successful and show strengths as well as possible weaknesses of the designed course. In our example, the lecturer had his PhD in mathematics and had hardly been in contact with didactical theories and frameworks until the point of the WiGeMath evaluation. Hence, he had would not speak in terms of mathematical beliefs, for example. When the concept was explained to him, though, he clearly saw that one intention of the lecture was to change students' beliefs but to that point he simply lacked the vocabulary to explain this intention.

Our taxonomy will help to communicate goals between universities, staff and students as it provides a frame of reference and a common language as has been shown for one example in this text.

Table 1 : Scales and their Cronbach's alphas, means, effect sizes of changes between the two surveys and p-values for a significant change.

 1

		Number	Cronbach's α Mean value Cohen's	p-
		of items	t1	t2	t1	t2	d	value
	Interest for mathematics	9	.83	.83	4.23 3.96	0.32 .001
	Mathematical self-concept	3	.81	.81	3.03 2.96	0.12 .149

NOTES

1. The WiGeMath project is funded by the German Federal Ministry of Education and Research (BMBF), grant identifiers 01PB14015A and 01PB14015B.


Sinéad Breen

Maria Meehan²

Ann O'shea
email: ann.oshea@mu.ie

Tim Rowland

An Analysis of University Mathematics Teaching using the Knowledge Quartet

Keywords: Knowledge Quartet, teacher knowledge, university mathematics teaching

We analyse accounts written by three mathematics lecturers on their practice using the Knowledge Quartet framework. This framework has been used to study how a teacher's knowledge of mathematics and mathematics pedagogy influences his/her

INTRODUCTION

The first three authors of this paper are mathematics lecturers at three universities in Ireland, who also engage in mathematics education research. Over the course of two years, they wrote accounts of incidents which occurred during their teaching as part of a professional development project using the Discipline of Noticing [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF]. In this paper, we report on our more recent use of a different theoretical framework, the Knowledge Quartet framework [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: the Knowledge Quartet and the case of Naomi[END_REF], to analyse these accounts. The Knowledge Quartet categorizes situations from classrooms where mathematical knowledge surfaces in teaching. There has been one previous attempt to use the framework to analyse university mathematics teaching [START_REF] Rowland | Beliefs and actions in university mathematics teaching[END_REF]. The focus of that paper was the knowledge-grounded foundation beliefs of the university lecturer, about mathematics and about teaching and learning.

The purpose of our current study is twofold. Firstly, we are interested in whether the Knowledge Quartet framework could be applied to study teaching at university level. Secondly, we would like to know what features of university teaching are highlighted when our set of accounts are analysed using the Knowledge Quartet. Previously, the first three authors had analysed their accounts to study the many decision points that arose while teaching and in O'Shea, Breen and Meehan (2017) these decision points and their triggers were categorised. We were interested to see if using the Knowledge Quartet framework would draw attention to other aspects of teaching in the accounts.

In this article, we will first of all consider the literature on teacher knowledge, especially at university level. We will then expand on the Knowledge Quartet framework, and give some results from our analysis using this lens. Finally, we will discuss our findings and suggest some future avenues for research.

LITERATURE REVIEW

The Knowledge Quartet is a theoretical tool for observing, analysing and reflecting on actual mathematics teaching. [START_REF] Ball | Content knowledge for teaching: What makes is special?[END_REF] also studied mathematics classrooms to develop a theory of mathematical knowledge for teaching that built on the work of [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF]. This resulted in the identification of an important subdomain of content knowledge -'specialized content knowledge'. This is distinct from 'common content knowledge' and is unique to the work of teaching.

Independently, [START_REF] Ainley | The role of attention in expert classroom practice[END_REF] suggested that experienced teachers draw on 'attention-dependent knowledge' in addition to subject knowledge and pedagogical knowledge (both general and subject-specific). Few research studies have been concerned with the knowledge employed in university mathematics teaching. [START_REF] Mcalpine | Reflection: Issues related to improve professors' teaching and students' learning[END_REF] conducted a research study with six professors considered exemplary in their teaching and found that all the professors drew on pedagogical knowledge, pedagogical content knowledge, content knowledge and knowledge of learners (following [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF]) while monitoring their own actions and making decisions during lectures. This was despite the fact that three of the professors were mathematicians who had no pedagogical training (while the remaining three were mathematics educators or trained teachers). [START_REF] Mcalpine | Reflection: Issues related to improve professors' teaching and students' learning[END_REF] hypothesised that the mathematicians constructed this knowledge largely through experience and reflection, and that their lack of training led them to depend more on their experience than the mathematics educators did.

On the other hand, [START_REF] Wagner | Beyond mathematical content knowledge: A mathematician's knowledge needed for teaching an inquiry-oriented differential equations course[END_REF] examined the knowledge, other than content knowledge, required by a mathematician teaching an undergraduate course. They reported that he was unable to anticipate how students would respond to particular activities and how the content or sequence of individual classes contributed to the instructional goals of the entire course. The authors claim these findings lend support to the assertion that there is knowledge particular to teaching that is distinct from, and not easily constructed from, knowledge of content. [START_REF] Speer | Knowledge needed by a teacher to provide analytic scaffolding during undergraduate Mathematics classroom discussions[END_REF] focussed on whole-class discussions and examined the nature of the knowledge that a mathematician could employ to make effective use of undergraduates' mathematical contributions in a way that furthered the goals for the class. Their analysis focussed on the role of (a lack of) pedagogical content knowledge and specialized content knowledge in the difficulties experienced by the instructor in scaffolding student learning while orchestrating such discussions.

THEORETICAL FRAMEWORK The Knowledge Quartet

The Knowledge Quartet is a 'theory' in the sense that it proposes a way of thinking about mathematics teaching in the usual institutional settings (lessons/classes), with a focus on the disciplinary content (mathematics) of the lesson.

The Knowledge Quartet (KQ) was the outcome of empirical research at the University of Cambridge, UK [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: the Knowledge Quartet and the case of Naomi[END_REF], in which 24 mathematics lessons were videotaped and scrutinised. The research team identified aspects of the teachers' actions in the classroom that could be construed as being informed by their mathematics subject matter knowledge or pedagogical content knowledge. This inductive process initially generated a set of 18 codes (later expanded to 21), subsequently grouped into four broad, super-ordinate categories or dimensions.

The first dimension of the KQ, foundation, consists of teachers' mathematics-related knowledge, beliefs and understanding. The second dimension, transformation, concerns knowledge-in-action as demonstrated both in planning to teach and in the act of teaching itself. The third dimension, connection, concerns the ways by which the teacher achieves coherence within and between sessions. The final dimension, contingency, is witnessed in classroom events that were not envisaged in the teachers' planning. Essentially, it is the ability to "think on one's feet".

Conceptualising the Knowledge Quartet

The concise conceptualisation of the KQ which now follows is a synthesis of the characteristics of its four dimensions.

Foundation

The first member of the KQ is rooted in the foundation of the teacher's theoretical background and beliefs. It concerns their knowledge, understanding and ready recourse to what was learned in preparation (intentionally or otherwise) for their role in the classroom. The key components of this theoretical background are: knowledge and understanding of mathematics per se; knowledge of significant tracts of the literature and thinking which has resulted from systematic enquiry into the teaching and learning of mathematics; and espoused beliefs about mathematics, including beliefs about why and how it is learnt. The remaining three categories focus on knowledge-in-action as demonstrated both in planning to teach and in the act of teaching itself.

Transformation

At the heart of the second member of the KQ is Shulman's observation that the knowledge base for teaching is distinguished by " … the capacity of a teacher to transform the content knowledge he or she possesses into forms that are pedagogically powerful" (Shulman, 1987, p. 15, emphasis added). This dimension picks out behaviour that is directed towards a student (or a group of students), and which follows from deliberation and judgement informed by foundation knowledge. The choice and use of examples has emerged as a rich vein for reflection and critique, and one of the most prevalent codes observed in practice [START_REF] Rowland | The purpose, design and use of examples in the teaching of elementary mathematics[END_REF].

Connection

The next dimension concerns the coherence of the planning or teaching displayed across an episode, lesson or series of lessons. Our conception of connection includes the sequencing of topics of instruction within and between lessons, including the ordering of tasks and exercises. To a significant extent, these reflect deliberations and choices entailing not only knowledge of structural connections within mathematics, but also awareness of the relative cognitive demands of different topics and tasks, and the implementation of strategies to remove (or lessen) obstacles to learning.

Contingency

Our final dimension concerns the teacher's response to classroom events that were not anticipated in the planning. This dimension of the KQ is about the ability to 'think on one's feet': it is about contingent action. Whilst the teacher's intended actions can be planned, the students' responses cannot. The teachers' response to students' unexpected contributions is one of the most low-inference codes of the KQ.

Many moments or episodes within a session can be understood in terms of two or more of the four units; for example, a contingent response to a student's suggestion might helpfully connect with ideas considered earlier. Furthermore, the application of content knowledge in the classroom always rests on foundational knowledge.

The KQ is a lens through which the observer 'sees' classroom mathematics instruction. It offers a four-dimensional framework against which mathematics lessons can be discussed, with a focus on their subject-matter content, and the teacher's related knowledge and beliefs.

This framework has been used in different contexts and for different purposes. For instance, Rowland (2012) used the KQ to examine situations in which mathematical knowledge surfaces in primary and secondary mathematics. He concludes that elementary mathematics teaching poses challenges which are qualitatively different from those confronting secondary mathematics teachers. However, the mathematics knowledge primary mathematics teachers must possess is neither less profound nor easier to acquire than that of secondary teachers. [START_REF] Turner | The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers' Mathematics Knowledge[END_REF] describe a project in which the framework was used to guide pre-service teachers in a process of personal reflection on their teaching. The participants found that the KQ helped them to focus more effectively on the mathematical content of their lessons and its enhancement. The authors reported that this enhanced focus on mathematical content knowledge had a positive influence on its further development. There was also evidence that the KQ helped the participants to develop a more learner-centred view of teaching and one in which conceptual understanding rather than procedural fluency was emphasised. Other recent studies using the KQ have focussed on contingent moments in the classroom (e.g. [START_REF] Rowland | Contingency in the mathematics classroom: Opportunities taken and opportunities missed[END_REF].

METHODOLOGY

The accounts which form the data for this study were written using the Discipline of Noticing [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF]. This advocates that practitioners write 'brief-but-vivid' accounts of incidents that they have noticed in their practice. [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF] defines a brief-but-vivid account as one which readers readily find relates to their experience. Brevity is obtained by omitting details which divert attention away from the main issue. The aim is to locate a phenomenon, so the less particular the description, the easier this is, without becoming so general as to be of no value….Thus description is as factual as possible. (p.57) He advises that these accounts should also avoid justification of incidents or actions, and should therefore be 'accounts of' rather than 'accounting for' a particular situation. The first three authors of this paper had written brief-but-vivid accounts of their teaching over a two-year period. These focused on notable incidents that occurred while they were teaching, but are not reflections or descriptions of a whole lecture. For more details, see O'Shea, [START_REF] O'shea | Decision points in mathematics lectures[END_REF].

For this paper each of the three lecturers chose one of their modules; only the accounts relating to that module which contained references to mathematical knowledge were analysed (20 accounts for Lecturer 1, 29 for Lecturer 2, and 38 for Lecturer 3). Lecturer 1 chose a one-semester Introduction to Analysis module for 27 second-year students (this module was delivered separately to 7 Pure Maths students and 20 Science students), Lecturer 2 also chose a one-semester Introduction to Analysis module for a group of 75 second-year students, while Lecturer 3 chose a year-long Differential Calculus module for a group of 49 first-year students. All three lecturers aimed to foster dialogue in their classrooms, perhaps because of their interest in educational research and the relatively small class sizes in these modules.

When coding the data we compared our accounts with the descriptions of each of the 21 codes associated to the KQ framework, with reference to the examples available at www.knowledgequartet.org. We began the coding process by first coding a small set of accounts together. Then each lecturer coded her own set of accounts and passed on her analysis to the other two lecturers in turn. They coded the accounts independently before comparing their analysis with that of the original instructor. All three discussed any discrepancies and agreed on the final coding.

During the coding process, we felt that the names of a few of the codes did not fully reflect the terminology used in teaching mathematics at the university level. We interpreted the code Teacher Demonstration (to explain a procedure) to also encompass teacher demonstration to explain a proof. We chose to use the code Choice of Example (CE) to include particular instances of an abstract concept or a general procedure; and, as the rehearsal of a procedure or 'exercise' [START_REF] Rowland | The purpose, design and use of examples in the teaching of elementary mathematics[END_REF], and also for non-routine tasks. We also applied the code Responding to Students' Ideas (RSI) from the Contingency Dimension to encompass instances where the lecturer had to respond to a lack of students' ideas.

RESULTS

A summary of the number and percentage of codes found in each of the four categories of the KQ for each author is given in Table 1 below. While a number of codes could be applied to some events, the one which we judged to be predominant was what was counted in this table. ) in the Foundation Dimension frequently appear for all three lecturers. Lecturer 3, who was simultaneously conducting a research project on mathematical tasks, was often explicit about the pedagogical rationale behind a given task. Consequently, another significant code for her accounts was Theoretical Underpinning of Pedagogy (TUP) in the Foundation Dimension.

KQ Dimension

By way of example, the following is an account from Lecturer 1, coded as RSI. She struggles to understand what the student is asking but still feels she has to respond:

A student asked a question in the middle of a complicated proof. I didn't understand the question and asked him to ask it again. He tried but I still couldn't understand. So I explained the proof again as best I could paying attention to what I thought he had had problems with. However I realised I had made a choice. I could have continued probing until I figured out what he was asking. I decided not to do that so as not to embarrass him, but maybe I didn't really answer his question in the end.

While the majority of accounts were on incidents during lectures, some relate to preparation of tasks and lessons, or conversations with students outside of class. The following is an example of an account by Lecturer 2 coded as CE, which describes a task given to students to work on during the second lecture of the semester.

I handed out the first Inclass Exercise of the module. It contained the following statement: There exists a university in the world, where every Analysis student achieves a final mark of at least 90% in the module. The instructions were as follows: Write down what you would need to do to prove that Statement B is false. At the end of the class, a student came up to me and said that suppose there were infinitely many universities in the world, then you couldn't actually disprove the statement because you wouldn't be able to get around to all of them to check the Analysis grades. I was impressed with how he extended the statement.

Given that CE was the most frequently occurring code for all three lecturers, the following account provides another example of a task given, this time by Lecturer 1, to help students propose conjectures about the relationship between bounded and convergent sequences.

I was talking about bounded sequences with the class today. I got them to come up with some bounded and some unbounded sequences. I tried to get the class to make conjectures by asking them to guess what the next theorem would be, or what it definitely wouldn't be. They immediately realised that there would be no theorem that said that every bounded sequence converges and then conjectured that every convergent sequence is bounded. They seemed to enjoy the process.

Next we present an account from Lecturer 3, coded as AP. She is explicit in her intentions to engage students in mathematical sense-making and on challenging students' views of mathematics as a set of rules to be learned and applied.

Today I continued with sketching graphs of functions and the students to draw the graph of f(x)=1/x on its natural domain, among others. I circulated the room as they were doing this and noticed that a number of what I had considered to be the more able students were drawing the graph incorrectly (possibly confusing f(x)=1/x and g(x)=1/x^2). I have been trying to put across the idea of Calculus as a 'science' from the point of view that 'experiments/trials' can be undertaken to check 'hypotheses', results can be 'replicated' and so on, but it appears some students are disregarding this and still regard it as a collection of facts to be learnt and remembered.

Finally, we present an account from Lecturer 3. Her pedagogy is underpinned by having students take a guided-discovery approach as a classroom community (TUP).

I tried to use a 'guided-discovery' approach to facilitate students' realization that the graph of a function and its inverse are mirror images of each other in the line y=x. However, each step of this took a lot longer than I envisaged. Moreover, I wasn't convinced at the end that the students would retain this particular piece of information longer or understand it better for having discovered it themselves as a class community.

DISCUSSION

In this paper we have used the KQ to analyse a set of accounts written as part of a professional development project that involved engaging with the Discipline of Noticing [START_REF] Mason | Researching Your Own Practice: The Discipline of Noticing[END_REF]. This is not the usual type of data that has been used in previous KQ studies. Typically, the researchers in those studies had access to classrooms (of either pre-service or experienced teachers), and have been able to record and analyse entire lessons. Our data is different in two key ways. Firstly we do not have recordings of entire lessons but the brief-but-vivid accounts of the instructor herself on some aspect of the class, which was memorable to her. This is a limitation because we may have chosen not to include some relevant aspects of our classes, or of our students' experience and reactions, but the accounts do shed some light on the 'attention-dependent knowledge' of the instructor [START_REF] Ainley | The role of attention in expert classroom practice[END_REF]. We did not write our accounts in order to give a representative view of our teaching, rather we concentrated on aspects which were troublesome to us. However, we do have accounts from almost every lecture in the modules considered whereas previous studies have data only from a very small number of classes with a given teacher.

In his KQ analysis of university mathematics teaching, [START_REF] Rowland | Beliefs and actions in university mathematics teaching[END_REF] refers to only one lecture. The analysis homes in on the foundation dimension and in particular on the beliefs of the lecturer (about mathematics and pedagogy), but does not explore the other three dimensions. Our analysis has shown that all four dimensions were present in our data. It should be noted that all three lecturers pursued an interactive approach in their classes, and perhaps the same spread of codes would not be present in an analysis of a more stereotypical university lecture.

On the other hand, the prevalence of the use of the responding to student ideas code for the accounts discussed here suggests that the traditional image of a lecture (in which a lecturer delivers from a pre-prepared script, rarely deviating from it, and interacts minimally with students) is not always accurate and highlighted this element of our practice for us.

In addition, given our previous focus on decision points in these accounts (O'Shea, Breen & Meehan, 2017), we may have expected the contingency dimension to be dominant but this was not the case. The KQ highlighted the importance of the other three categories in our accounts, especially the transformation dimension in the choice of examples. We found the framework provided a lens through which the knowledge brought to bear in the preparation and teaching of lessons could be viewed in a coherent and comprehensive manner.

Each of the first three authors is a mathematician and while none has any formal pedagogical training, all three conduct research in mathematics education. Many of the accounts suggested an awareness of purpose on the lecturers' behalf or a theoretical underpinning to the pedagogy used when teaching. Perhaps this is a consequence of their familiarity with the research literature. However, the fact that the instructors often contrasted student learning in the lectures for which accounts were written with that of other cohorts lends some support to the hypothesis of [START_REF] Mcalpine | Reflection: Issues related to improve professors' teaching and students' learning[END_REF] that a teaching mathematician can construct knowledge of learners and pedagogy through experience and reflection.

In several accounts the three lecturers highlighted what they noticed about student thinking on a given task and reflected on this after the lecture. These reflections could be said to inform their knowledge about mathematics pedagogy, particularly their knowledge of content and students and knowledge of content and teaching [START_REF] Ball | Content knowledge for teaching: What makes is special?[END_REF], which is a component of the Foundation Dimension. Although we chose not to code these reflections since they occurred after lectures, they point to a growth of knowledge as a consequence of reflection on teaching. It seems that use of the KQ as a reflection tool could afford mathematicians (with no formal pedagogical training) an opportunity to develop pedagogical knowledge. It is interesting to compare this with [START_REF] Turner | The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers' Mathematics Knowledge[END_REF] finding that the KQ afforded preservice primary teachers (typically non-specialists in mathematics) an opportunity to develop mathematical content knowledge, illustrating the usefulness of KQ to mathematics teachers of a variety of backgrounds.

Some KQ codes did not appear in our analysis. For example there were no accounts coded as displaying behaviour such as adherence to a textbook or concentration on procedures. This may be because of the nature of the mathematics taught in the given modules. We also found very few references to use of mathematical terminology and overt display of subject knowledge, which is not to say that the lecturers did not use terminology or show their subject knowledge during classes but that they did not talk about it in their accounts (perhaps because it was normal and not problematic). We used the code identifying student errors sparingly, even though many accounts contained instances of a lecturer noticing a problem with student understanding. In our accounts the lecturers seemed to focus more on how to respond to a student rather than being able to tell when a piece of mathematics was wrong, and so we coded these episodes using the responding to student ideas code. We also used this code when the lecturer was faced with a lack of student ideas, for instance when she asked a question but received no replies. It may be that this is a situation that occurs more frequently in university than in school, where the size of classes can result in unwillingness to take part in discussions. We found that the type of specialist knowledge required to teach abstract mathematics at university was accounted for in the KQ with many of the codes mentioned earlier as well as others such as choice of representation, recognition of conceptual appropriateness and making connections between concepts or representations.

Even though there are some differences in the prevalence of codes at school and university level, we believe that the KQ offers a useful lens with which to study undergraduate teaching. It has drawn our attention to the importance of different facets of lecturers' mathematical knowledge which we may otherwise have overlooked. It would be interesting to explore the relationships between the four dimensions of the Quartet, for example how the underpinning dimension of foundation knowledge influences the lecturers' choices made in the other three dimensions, and how it is in turn influenced by knowledge generated by the lecturer in a contingent moment. We used the KQ to code reflective accounts written by mathematics lecturers as they reflected on their teaching. However, we suggest it could also be used to guide the reflective process and the writing of the accounts. It would be interesting to explore whether such an approach would lead to a change in the lecturers' perspectives on teaching similar to those described by [START_REF] Turner | The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers' Mathematics Knowledge[END_REF].

Table 1 .

 1 Number and percentage of codes in each KQ dimension for each lecturer On coding the accounts it became apparent that all three lecturers frequently wrote accounts about giving a task to the class or instigating a whole class discussion around a task, recording some students' responses (or lack of responses) in relation to the task, and noting what the lecturer thought or learned about student thinking.

		Lecturer 1	Lecturer 2	Lecturer 3
	Foundation	10 (20.83%)	4 (11.11%)	41 (34.74%)
	Transformation	14 (29.17%)	17 (47.22%)	32 (27.12%)
	Connection	7 (14.58%)	8 (22.22%)	24 (20.34%)
	Contingency	17 (35.42%)	7 (19.45%)	21 (17.8%)
	Total	48 (100%)	36 (100%)	118 (100%)

Therefore it is not surprising that the code Choice of Example (CE) in the Transformation Dimension was the most frequently occurring code for all three lecturers, while Responding to Students' Ideas in the Contingency Dimension was in each of their top three most frequently occurring codes. In many accounts, the lecturer contrasted student learning on a task with learning on the same task the previous year or with students in a different class, often noting what students found easy or difficult. The tasks were usually designed and planned by the lecturer with specific aims for student learning in mind, thus the codes Anticipation of Complexity (AC) in the Connection Dimension and Awareness of Purpose (AP


A university lecturer's role and interactions in a flipped classroom

Erlend Sten Syrdalen University of Agder,Norway,erless15@student.uia.no This poster explores the role of the lecturer in a university level mathematics class utilizing the flipped classroom approach to teaching. The role of the lecturer is investigated using the Systematic Classroom Analysis Notation (SCAN) tool and the theoretical framework of instrumental orchestration.

Keywords: Novel approaches to teaching, lecturers' practices at university level, flipped classrooms, instrumental orchestration, analysis of lecturers' actions

NEW RULES OF INTERACTION IN FLIPPED CLASSROOMS

University lecturers traditionally presents new mathematical topics in class, while students work on related tasks at home or in colloquiums arranged by the university. The Flipped Classroom approach (FC) flips this around, having the students watch pre-recorded videos at home, while working in a more collaborative and studentcentred manner while in class. This flipping of the classroom changes the rules of the classroom drastically, as the lecturer now has to prepare videos for the students' outof-class to "prime" them for an in-class active phase of learning (Fredriksen, Hadjerrouit, Monaghan & Rensaa, 2017). To explore the new role of the lecturer in a FC, I will look at a lecturer's role using the framework of instrumental orchestration, and analysing the in-class interactions of the lecturer using a modified version of the Systematic Classroom Analysis Notation (SCAN) (Beeby, Burkhardt & Fraser, 1979).

THEORETICAL FRAMEWORK

To conceptualize the lecturers' new role as a result of flipping the classroom I use the framework of instrumental orchestration (Trouche, 2004). The term is defined as the lecturer's intentional and systematic organization and use of the various available artifacts in a learning situationin this case both out-of-class and in-classin order to facilitate students' instrumental genesisthe process during which an artifact or object is transformed into the psychological construct of an instrument, the combination of artifacts and schemes to be used for different specific types of task. Within instrumental orchestration it is possible to distinguish between three elements: a didactic configuration, an exploitation mode and a didactical performance (Drijvers, 2010). The didactic configuration is the arrangement of the teaching setting and the artifacts involved in it. The exploitation mode is the manner in which the lecturer uses a given didactic configuration for the benefit of his/her didactic intentions. The didactic performance takes into consideration the decisions the lecturer must make on the fly while performing in the given didactic configuration with its didactic intentions. Using these three elements of orchestration, it should possible to explore the lecturers' new role as a result of FC.

RESEARCH QUESTIONS

The research questions address both the in-class and out-of-class aspects of the lecturer's new role in a FC.

1. How does the lecturer in a FC interact with the students in the lesson? 2. How does the lecturer perceive his new role, and how does this compare to his observed interactions while in-class?

METHODOLOGY

To look into these questions, I have observed and interviewed a lecturer utilizing the FC approach in a mathematics course for engineers. During the course of a week, four 90 minute in-class sessions were videotaped. These sessions were part of a module on mathematical series. Short videos, out-of-class preparation for problemsolving activities in-class, facilitated the learning of Taylor series and its applications and tests of series convergence. The study employs an interpretative research paradigm with qualitative research methods. The videotapes of sessions will be analyzed utilizing SCAN (Beeby et al., 1979) to code the teacher's interactions with students. SCAN is based on "time-slicing", and works on three different time-scales -"event", "episode" and "activity". A prominent feature is a set of qualifiers for each activity, which evaluates the depth of demand and level of guidance in lecturer interactions. The interview will be analyzed using open coding.

  Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1 -5, 2017). Dublin, Ireland: DCU Institute of Education and ERME. Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), 281-307.

	Beeby, T., Burkhardt, H. & Fraser, R. (1979) Systematic Classroom Analysis
	Notation for Mathematics Lessons. Nottingham: Shell Centre for Mathematical
	Education.
	Drijvers, P., Doorman, M., Boon, P., Reed, H., & Gravemeijer, K. (2010). The
	teacher and the tool: Instrumental orchestrations in the technology-rich
	mathematics classroom. Educational Studies in Mathematics, 75(2), 213-234.
	Fredriksen, H., Hadjerrouit, S., Monaghan, J., & Rensaa, R. J. (2017) Exploring
	Tensions in a Mathematical Course for Engineers utilizing a Flipped Classroom
	Approach. In Dooley, T. & Gueudet, G.. (Eds.)


J S Lintuvuori

K Stratford

M E Cates

D Marenduzzo

Mixtures of blue phase liquid crystal with simple liquids: elastic emulsions and cubic fluid cylinders

We investigate numerically the behaviour of a phase-separating mixture of a blue phase I liquid crystal with an isotropic fluid. The resulting morphology is primarily controlled by an inverse capillary number, χ, setting the balance between interfacial and elastic forces. When χ and the concentration of the isotropic component are both low, the blue phase disclination lattice templates a cubic array of fluid cylinders. For larger χ, the isotropic phase arranges primarily into liquid emulsion droplets which coarsen very slowly, rewiring the blue phase disclination lines into an amorphous elastic network. Our blue phase/simple fluid composites can be externally manipulated: an electric field can trigger a morphological transition between cubic fluid cylinder phases with different topologies.

Composite materials based on complex fluids, such as liquid crystalline droplets or emulsions [START_REF] Martinez-Gonzalez | Proc. Natl. Acad. Sci[END_REF][START_REF] Sec | [END_REF][3][4][5] and colloidliquid crystal mixtures [START_REF] Ravnik | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Ravnik | [END_REF][8][9], have received a lot of attention of late. This is due both to their rich and often unexpected physical behaviour, and to their potential as soft materials with unusual optical, flow and mechanical properties. These features may lead to technological applications. For example, cholesteric droplets can be utilised as tunable microlasers [10] or in optofluidics [11,12]. Likewise, photopolymerised blue phase disclination networks make intriguing switchable electrooptic devices [13].

From a fundamental viewpoint, the richness in behaviour of many such composites derives from the competition between different length scales (e.g., droplet size, cholesteric pitch, defect size), and from the contest between interfacial and elastic energies [14][15][16]. The free energy landscape is often glassy and possesses multiple metastable structures, where energy barriers dwarf thermal energies. Structural arrest in this landscape may pave the way to energy-saving multistable devices which retain memory of their state in the absence of an external (electric or magnetic) field [17].

Here, we study computationally a binary mixture of a blue phase (BP) liquid crystal and an isotropic liquid, and characterise the resulting behaviour and dynamics. BPs are remarkable liquid crystals (LCs) displaying a 3D network of disclination lines [18]. Without external fields, BP disclinations may have either cubic symmetry (blue phases I and II, BPI and BPII), or be amorphous (blue phase III [19]). The typical length scale of the network is close to the pitch of the cholesterics which form the BP, normally a few hundred nm. Dispersing colloidal nanoparticles or polymers inside BPs leads to a dramatic increase in their range of thermodynamic stability, as the energetically costly disclinations are covered by these inclusions [START_REF] Ravnik | [END_REF]20].

Our simulations reveal a number of striking physical properties for composites of BPI with simple liquids. If the interfacial tension between the two, σ, is sufficiently small, then the isotropic component arranges into very long liquid tubes surrounding the BP disclinations, creating what we call a "cubic fluid cylinder phase" -this is similar to the emulsified blue phase first theorised in [21], although in our case no surfactants are required. Just as polymers or nanoparticles, such liquid tubes can stabilise the network thermodynamically. In turn they are stabilized against the Rayleigh instability by the BP elasticity.

For larger σ, the phenomenology is completely different: there is a transition to a regime where quasi-spherical isotropic droplets grow within the BP matrix. Coarsening does not proceed indefinitely, but is arrested before full phase separation is reached to create an amorphous, elastic emulsion. This transition can be understood via a simple mean field theory which identifies an elastic capillary number as the key control parameter. Finally, we show that these composites are reconfigurable: the topology of the cubic fluid cylinder phase can be altered by an electric field. Simulation methods: To simulate a BP mixture we consider a free energy functional F = f dV . Its density consists of two parts, f = f φ + f Q , describing local mixture composition and LC ordering respectively. The first part can be written in terms of a compositional order parameter field φ as

f φ = - a 2 φ 2 + b 4 φ 4 + κ 2 (∇φ) 2 (1)
where a = b and κ are positive constants. For a cholesteric LC, f Q can be expressed in terms of a traceless and symmetric order parameter tensor Q,

f Q = A0 2 1 -γ(φ) 3 Q 2 αβ -A0γ(φ) 3 Q αβ Q βγ Q γα + A0γ(φ) 4 (Q 2 αβ ) 2 - a 12π E α Q αβ E β (2)
+ K 2 ∇ β Q αβ 2 + K 2 αγδ ∇ γ Q δβ + 2q 0 Q αβ 2 ,
where repeated Greek indices (denoting Cartesian components) are summed. In Eq. 2, A 0 gives the energy scale, K is the elastic constant and q 0 = 2π/p fixes the equilibrium pitch length p. We include an electric field E (with associated dielectric constant anisotropy a ), which will be considered at the end of this Letter. The quantity γ(φ) = γ 0 +δ(1+φ) controls the order. We choose γ 0 and δ such that γ(-1) is below the threshold for isotropiccholesteric transition, whereas γ(+1) is above it. The full parameter list is given in the Appendix together with details of initial conditions, and a mapping between simulation and physical units in [START_REF] Assuming | = 10pN and λ = 300nm, a value of σ = 1 in simulations maps to ∼ 20mN/m, giving σ ≈ 10 -5 N/m for the cubic fluid (Fig. 1) and σ ≈ 10 -3 N/m for the elastic emulsion in Figure[END_REF]. We highlight that the chirality, κ LC = 108Kq 2 0 /(A 0 γ), and the reduced temperature, τ = 27(1 -γ/3)/γ were chosen as κ LC ≈ 0.7 and τ ≈ -0.25 throughout, to favour BPI thermodynamically [START_REF] Henrich | [END_REF]. We do not include an explicit surface anchoring term; we expect its addition to further enrich the possibilities for composite design [9].

The evolution of the LC order Q is given by the Beris-Edwards equation [START_REF] Beris | Thermodynamics of Flowing Systems with Internal Microstructure[END_REF]

(∂ t + u α ∂ α ) Q -S (u, Q) = ΓH, (3)
where S (u, Q) accounts for flow-induced LC rotation (see Appendix), Γ is related to the rotational viscosity and the molecular field H = -δF δQ -1 3 ITr δF δQ . The velocity field u α satisfies a Navier-Stokes equation where the stress tensor includes elastic contributions [START_REF] Marenduzzo | [END_REF] (see Appendix). The equation of motion for φ is

∂ t φ + ∂ α (φu α ) = M ∇ 2 µ ( 4
)
where M is a constant mobility and µ = δF δφ is the chemical potential. We solve the equations of motion using a hybrid lattice Boltzmann algorithm [START_REF] Marenduzzo | [END_REF]. To minimize discretization artefacts we include thermal noise in the Navier-Stokes equation [26]; this is set at a level too weak to change qualitatively the structural dynamics on the length scales of interest here [27]. Simulations were carried out in a 128 × 128 × 128 cubic periodic simulation box, where the BPI unit cell size λ = p/ √ 2 [START_REF] Henrich | [END_REF] is either 32 or 64 lattice units -the defect core ξ and interfacial width ξ φ are both much smaller, and of the order of a few lattice units (see Appendix).

Results: All the simulations presented have the isotropic component as the minority phase. Figure 1 shows the behaviour of a 90:10 LC:isotropic mixture with a low value of the interfacial tension (σ = 5.4 × 10 -4 in simulation units, or ∼ 10 -5 Nm -1 in physical units [START_REF] Assuming | = 10pN and λ = 300nm, a value of σ = 1 in simulations maps to ∼ 20mN/m, giving σ ≈ 10 -5 N/m for the cubic fluid (Fig. 1) and σ ≈ 10 -3 N/m for the elastic emulsion in Figure[END_REF]). The components rapidly demix so that the the isotropic fluid is templated by the cubic symmetry of the emerging BPI disclination pattern (Suppl. Movies 1 and 2), in agreement with theoretical predictions [21]. We name the resulting structure a cubic fluid cylinder phase, because the topology of the isotropic component is templated by that of the cubic BPI. The elastic character of the latter then stabilizes the fluid cylinders against the Rayleigh instability that would cause them to break into droplets if surrounded instead by a second fluid.

To explain why a cubic fluid cylinder phase forms, we propose a simple mean-field argument. The elastic free energy cost per unit length E el associated with a cylinder of radius r containing a straight disclination line (with charge s = -1/2, relevant for BPs [18]) of core radius ξ can be estimated by assuming that the defect core is isotropic, as follows,

E el = E 0 (r/ξ) 2 for r < ξ (5)
E el = πK 4 log(r/ξ) + E 0 for r ≥ ξ.
In Eq. 5, E 0 /ξ 2 ∼ A 0 is the uniform free energy density associated with the melting of the isotropic defect core. (In our simulations, and in reality, the core is not fully melted but weakly biaxial; this would change only prefactors above.) This needs to be compared with the interfacial free energy per unit length associated with the formation of a cylindrical tube of isotropic fluid of radius r, E in = 2πσr. The fluid tube forms if δE = E in -E el < 0, in which case the selected radius, r * , is found by minimising δE, giving r * = K/(8σ). This solution only holds for , (e) t = 9.6 × 10 6 . (Simulations performed using parameter set B (see Appendix).) (f-j) Dynamics of an 80:20 BP mixture with higher interfacial tension (χ ≈ 1.9). Snapshots correspond to: (f) t = 1.8 × 10 6 , (g) t = 2.1 × 10 6 , (h) t = 2.2 × 10 6 , (j) t = 6.4 × 10 6 . (Simulations performed using parameter set C (see Appendix).) σ < σ * , where σ * is given by the formula

(f) (g) (h) (i) (j) (c) (b) (d) (e) (a)
σ * ξ K = 1 8 exp 1 -4E0 πK . (6)
As previously mentioned, E 0 ∼ A 0 ξ 2 , whereas the core size ξ ∼ K/A 0 [28], so the right hand side of Eq. 6 is a constant. Therefore within our mean field theory the physics is determined by the dimensionless parameter χ ≡ σξ/K, controlling the ratio between interfacial and elastic properties -we call this an inverse elastic capillary number. Eq. 6 leads to the expectation of a discontinuous transition between a regime where the isotropic component forms cylinders wetting the disclination network (for σ < σ * ) and another regime where such cylinders disappear (for σ > σ * ). Anchoring with finite strength W (units N/m) would introduce another dimensionless parameter, w = W/σ; reasoning as in [29] suggests that anchoring effects are unimportant if W r * /K < 1 (w < 8).

Residual effective anchoring due to implicit couplings between Q and ∇φ is estimated to be small from inspection of the director field profiles (Suppl. Movie 2). Thermodynamically, the cubic fluid cylinder phase should be stable when the LC:isotropic ratio is ∼ p/r * but coexist with an excess of one or other pure phase otherwise. Indeed, in Figure 2(a-e) (and Suppl. Movie 3) we show that the dynamics for an 85:15 emulsion with χ ≈ 0.019 is significantly different from that of Figure 1 (for a 90:10). A liquid tube network still forms early on, but is unstable: the structure later on twists and rearranges leading to the formation of isotropic fluid domains of irregular shape (Fig. 2(b-d)). These slowly coarsen until they reach a seemingly arrested "elastic emulsion" state at late times (Fig. 2(e)).

When χ is increased (χ ≈ 1.9, Fig. 2(f-j) and Suppl. Movie 4, for an 80:20 mixture) the mixture morphology changes again. Following a deep quench into the demixed phase, fluid droplets coarsen and interact with the BPI disclination network, forming an emulsion of spherical droplets (Fig. 2(f) and Suppl. Movies 4 and 5). Droplets are connected to each other by disclinations, creating elastic interactions between them. Nearby droplets can coalesce (see highlighted dotted region in Fig. 2(g-i)) and small subcritical droplets shrink in favour of larger nearby ones, through a process akin to Ostwald ripening (see highlighted dashed region in Fig. 2(f,g)). At late times, the dynamics slows down until we again find a seemingly arrested metastable structure (Fig. 2(j), and Suppl. Movie 4). The arrest of Ostwald ripening is expected whenever the continuous phase has a yield stress [30]; this may apply here despite the rewiring of BPI into an amorphous disclination network [31]. For this value of χ the droplet-disrupted defect network is reminiscent of structures observed with hard colloids in BPs [START_REF] Ravnik | Proc. Natl. Acad. Sci. USA[END_REF][START_REF] Ravnik | [END_REF]9]. However, the dynamics of formation of these structures is very different in the two cases. Similar dynamics and arrest as shown in Figure 2(f-j) is observed for different compositions, including a 90:10 emulsion (Suppl. Movie 6).

Experimentally, σ, ξ and K should be ∼ 1 mN/m, ∼ 10 nm and ∼ 10 pN respectively, yielding a typical χ ∼ 1, corresponding to arrested droplets (Fig. 2(f-j), see also [START_REF] Assuming | = 10pN and λ = 300nm, a value of σ = 1 in simulations maps to ∼ 20mN/m, giving σ ≈ 10 -5 N/m for the cubic fluid (Fig. 1) and σ ≈ 10 -3 N/m for the elastic emulsion in Figure[END_REF]). It might though be possible to decrease χ to reach the cubic fluid cylinder regime, for instance by employing as the second component a mesogenic liquid that is chemically similar to the chiral LC but has a lower clearing temperature so that it remains isotropic within the thermal stability range of the BP. The latter could give a 50-fold reduction in σ [START_REF]The interfacial tension between the isotropic and nematic phases of the same mesogen[END_REF].

We next consider the time-dependent domain size L(t) = 2π S(k, t)dk/ kS(k, t)dk, where k = |k| [33].

Here S(k, t) = φ(k, t)φ(-k, t) is the structure factor at time t. We compare L to the BP unit cell size, λ. Figure 3 shows plots of L(t) for all cases presented hitherto. For low χ, corresponding to self-assembly of the cubic fluid cylinder phase (Fig. 1(b)), L(t) rapidly settles to L ∼ λ 4 (for a 90:10 mixture with χ ≈ 0.019, dot-dashed line in Fig. 3). When the amount of isotropic fluid is increased, the dynamics of L(t) reflects the events visualised in Figure 2(a-e) (or Suppl. Movie 2). Initially L(t) ∼ λ 4 as a cubic fluid forms. This structure is only metastable due to the excess of isotropic fluid, so later on L(t) shows a rapid growth which appears to saturate at late times. While the curve does not become completely flat in Figure 3, the dynamics is many orders of magnitude slower than observed with binary simple fluids [33]. For the case favouring spherical droplets (Fig. 2(f-j), and Suppl. Movie 3), the domain size shows an immediate rapid growth corresponding to coarsening. Later on, L(t) settles to a value ∼ 1.2λ (solid and dashed lines in Fig. 3).

We now demonstrate that our BP-based elastic emulsions can be manipulated and controlled with an electric field. In Figure 4 (and Suppl. Movie 5), we show the dynamics of a 90:10 emulsion with low χ ≈ 0.019 under a strong electric field along the z axis. Without field, the mixture self-assembles into a cubic fluid network. Under the applied field the isotropic fluid follows closely the reorganising disclination network (Fig. 4(a-c)), including an intermediate state with square lattice of disclinations perpendicular to the field (Fig. 4(b)). The time develop- Conclusions: We have shown that a mixture of a blue phase (BPI) and an isotropic fluid leads to a range of elastic emulsions with fascinating dynamical and phase behaviours. We have identified a key control parameter, χ = σξ/K, which determines the relative importance of interfacial and elastic forces, and which, together with the mixture composition, determines most of the observed physics. For small values of χ, and sufficiently small concentration of the isotropic component, we observe the self-assembly of a cubic fluid cylinder phase, where the isotropic fluid arranges into cylinders which replace the disclination lines of the BPI network. Unusually, this structure contains unbranched, straight fluid cylinders extending in all three spatial directions with potential applications in materials templating [34]. If the fraction of isotropic component is increased, this cubic fluid structure still forms initially, but is subsequently un-stable and collapses to yield irregular isotropic domains connected by disclination lines. For larger values of χ the isotropic fluid arranges into droplets, which slowly coarsen via a combination of coalescence and an Ostwaldtype process. At very late times, we find an apparently arrested metastable structure whose typical size is comparable with the BP unit cell. The cubic fluid cylinder phase found at low χ can additionally be manipulated with an external field, and we have shown that in this way we can switch between cubic fluids with different symmetries and/or network topologies.

Acknowledgements: This work was funded by EU intra-European fellowship 623637 DyCoCoS FP7-PEOPLE-2013-IEF, IdEx Bordeaux junior chair to JSL and UK EPSRC grant EP/J007404/1. Part of this work was performed using HPC resources from GENCI-IDRIS (Grant 2017-20907383). MEC is funded by the Royal Society.

APPENDIX Simulation parameters and thermodynamic stability of blue phase I

To simulate the blue phase I (BPI) and isotropic fluid mixture, we consider a total free energy F = f dV . The free energy density f consists of two parts, f = f φ + f Q , where the f φ describes a binary fluid and f Q describes the liquid crystal (LC) ordering. These are given in Eqs.

(1-2) in the main text.

The binary fluid part (Eq. (1) in the main text) depends on three positive constants: a, b and κ, which can be used to define an interfacial width ξ φ = 2κ/a and an interfacial tension σ = 8κa 3 /(9b 2 ). All simulations use the same value of ξ φ ≈ 1.3, while the interfacial tension σ was varied (see below for details). Additionally, a = b was chosen, so that φ = ±1 corresponds to a fully orderded LC phase, and an isotropic fluid, respectively.

For the chiral liquid crystal component, the free energy functional is expressed in terms of an order parameter tensor Q and given by Eq. ( 2) in the main text. In the equation 2, A 0 fixes the energy scale, K is the elastic constant penalising distortions and q 0 = 2π/p sets the equilibrium pitch length p of the chiral liquid crystal. γ(φ) is a temperature like parameter, which controls the order

γ(φ) = γ 0 + δ(1 + φ). (7)
The parameters γ 0 = 2.586 and δ = 0.25 were chosen such that for the isotropic phase γ(-1) = 2.586 is below the order-disorder transition at γ c ∼ 2.7, while for the ordered phase γ(+1) = 3.086. The thermodynamic state of the chiral LC is determined by the reduced chirality

κ LC = 108Kq 2 0 /(A 0 γ) (8)
and the reduced temperature

τ = 27 (1 -γ/3) /γ. (9)
The same values of κ and τ was used for all simulations.

Using the values for the ordered phase (φ = +1), these are κ LC ≈ 0.7 and τ ≈ 0.25, giving a thermodynamically stable BPI [START_REF] Henrich | [END_REF]. We also define a typical defect core size as ξ = K/A 0 -this is analogous to the nematic correlation length defined in [18] but misses out the dimensionless prefactor (which in our case would depend on γ, hence on φ).

All the simulations were carried out in cubic simulations box spanning 128 × 128 × 128 lattice points, with periodic boundary conditions. Throughout the simulations the fluid viscosity was set to η = 0.5 and the rotational viscosity of the LC Γ = 0.5 (in Eq. ( 3) of the main text).

In Eq. ( 3), the S (W, Q) term, accounting for flowinduced LC rotations, is explicitly given by [35]

S(u, Q) = (νD + Ω) (Q + I/3) + (Q + I/3) (νD -Ω) -2ν (Q + I/3) T r (QW) , (10)
where D = (W + W T )/2 and Ω = (W -W T )/2 are the symmetric and anti-symmetric parts of the velocity gradient tensor W αβ = ∂ β u α respectively, and I is the identity matrix. The quantity ν is the flow aligning parameter -we choose ν = 0.7 which corresponds to a flow-aligning LC.

The stress tensor of the Navier-Stokes equation, Π αβ , has the following explicit form [35],

Π αβ = -P 0 δ αβ + 2ν Q αβ + 1 3 δ αβ Q γµ H γµ (11) --νH αγ Q γβ + 1 3 δ γβ -ν Q αγ + 1 3 δ αγ H γβ - ∂f ∂∂ β Q γµ ∂ α Q γµ + Q αν H νβ -H αν Q νβ ,
where Greek indices denote Cartesian coordinates and summation over repeated indices is implied.

Regarding the rest of the parameters: a = b, κ and mobility M for the binary fluid as well as A 0 , K and q 0 (as well as the field E when present) for the LC, were varied, so that the conditions of constant interfacial width, and the thermodynamic stability of BPI were retained (as explained above) -values for each of the Figures are given below For all simulations reported, the the Q was initialised with the high-chirality limit solution corresponding to BPI (see [START_REF] Henrich | [END_REF] for details). This is appropriate as the values of κ LC and τ are such that BPI is the thermodynamically stable phase for the liquid crystal component. In Figures 2(a-e) and 4, the compositional order parameter was initially set to φ 0 = (A -B)/(A + B) with small noise, where A and B denote the fraction of LC and isotropic phase respectively. This corresponds to an initially demixed configuration. In Figures 1 and2(f-j), the isotropic fluid was initialised into tiny droplets to trigger phase separation (in view of the large asymmetry between fractions of A and B). Such an initialisation may also be used in Figures 2(a-e) and 4, yielding analogous results as shown in the main text. (see main text for details on the physical meaning of χ, note the relevant value of γ is that in the ordered phase, at φ = +1). We have verified that qualitatively indistinguistable results are obtained when including a "redshift" r = 0.83 for BPI, so that λ = p/(r √ 2) -which leads to a lower free energy in single-component BPI [START_REF] Henrich | [END_REF]. and M = 0.5 for the binary fluid. These give ξ ∼ 2.5 and χ ≈ 0.019. Note that as our definition of ξ does not include dimensionless prefactors, a defect core may in practice still be resolved by a few lattice sites in our simulations even when ξ ∼ 1 (see Suppl. Movies 2 and 4). These give ξ ∼ 2.5 and χ ≈ 0.019. The electric field is E = [0, 0, 0.04], and the dielectric anisotropy is a = 41.4.

Simulation parameters for the Figures in the main

Mapping to SI units

Assuming typical values for chiral liquid crystals K ∼ 10pN and λ ∼ 300nm, we can map an interfacial tension value of σ = 1 in simulations to σ = 20 mN/m in physical units. This gives a typical interfacial tension σ ≈ 10 -5 N/m for the cubic fluid (Figure 1 in the main text), while the elastic emulsion (Figure 2(f-j)) is observed when σ ≈ 10 -3 N/m.

FIG. 1 :

 1 FIG. 1: Formation of a cubic fluid cylinder phase. (a) BPI disclination network (blue ribbons). (b) In a 90:10 BP:isotropic mixture with low σ (χ ≡ σξ/K ≈ 0.019), the isotropic fluid self-organises to form tubes following the cubic symmetry of the BPI disclination network. (c) The free energy of the system as a function of time t. (Simulations performed using parameter set A (see Appendix).)

FIG. 2 :

 2 FIG. 2: Dynamics of BP mixtures with different values of χ and LC:isotropic composition. (a-e) Dynamics of a low interfacial tension system (χ ≈ 0.019), with 85:15 composition, starting from an initially mixed state. Snapshots correspond to: (a) t = 2 × 10 5 (simulation steps); (b) t = 7 × 10 5 , (c) t = 1 × 10 6 , (d) t = 2.5 × 10 6, (e) t = 9.6 × 10 6 . (Simulations performed using parameter set B (see Appendix).) (f-j) Dynamics of an 80:20 BP mixture with higher interfacial tension (χ ≈ 1.9). Snapshots correspond to: (f) t = 1.8 × 10 6 , (g) t = 2.1 × 10 6 , (h) t = 2.2 × 10 6 , (j) t = 6.4 × 10 6 . (Simulations performed using parameter set C (see Appendix).)

9 FIG. 3 :

 93 FIG. 3: Time evolution of the domain length scale L(t) for different values of χ and emulsion compositions.

FIG. 4 :

 4 FIG. 4: Dynamics of a 90:10 mixture of low σ (χ ≈ 0.019) in a strong electric field along the z axis (into the page). The strength of the field is quantified by the dimensionless quantity e = 27 aE 2 /(32πA0γ): in our case e ≈ 0.24 (given the mapping in [22], this may be realised with E ∼ 25 V/µm for a LC with a ∼ 1.8 × 10 -10 F/m). (a) Snapshot of the starting configuration at t = 0 (no field). When the field is on, the isotropic fluid (red) follows the rapidly reorganising defect network (blue). (b,c) Snapshots corresponding to t = 5 × 10 4 (b), and to t = 2 × 10 5 (c). (d) Plots of the domain lengthscale L(t)/λ as a function of time. (Simulations performed using parameter set D (see Appendix).)

  (key dimensionless parameters are also reported in the Figure captions in the main text.

  text

Figure 1

 1 Figure 1 in the main text uses parameter set A: K = 0.14, A 0 = 0.05, lattice parameter for the BPI λ = p/ √ 2 = √ 2π/q 0 = 64 for the LC and a = b = 6.25 × 10 -4 , κ ≈ 5.3 × 10 -4 and M = 5 for the binary fluid. These correspond to an inverse capillary number χ = σξ/K ≈ 0.019, where ξ = 27K A0γ

Figure 2 (

 2 Figure2(a-e) in the main text uses parameter set B: K = 0.07, A 0 = 0.1, lattice parameter for the BPI λ = 32 for the LC and a = b = 6.25 × 10 -4 , κ ≈ 5.3 × 10 -4 and M = 0.5 for the binary fluid. These give ξ ∼ 2.5 and χ ≈ 0.019. Note that as our definition of ξ does not include dimensionless prefactors, a defect core may in practice still be resolved by a few lattice sites in our simulations even when ξ ∼ 1 (see Suppl. Movies 2 and 4).

Figure 2 (

 2 Figure 2(f-j) in the main text uses parameter set C: K = 0.14, A 0 = 0.05, lattice parameter for the BPI λ = 64 for the LC and a = b = 6.25 × 10 -2 , κ ≈ 5.3 × 10 -2 and M = 0.05 for the binary fluid. These give ξ ∼ 4.9 and χ ≈ 1.9.

Figure 4

 4 Figure 4 in the main text uses parameter set D: K = 0.07, A 0 = 0.1, lattice parameter for the BPI λ = 32 for the LC and a = b = 6.25 × 10 -4 , κ ≈ 5.3 × 10 -4 and M = 0.5 for the binary fluid.

Captions for Supplementary Movies

Suppl. Movie 1: Movie of the dynamics corresponding to Figure 1 in the main text.

Suppl.

Movie 2: Slices of the director profile through the system (planes perpendicular to the z axis), for the final configuration in Figure 1. The director field is defined for each non-zero value of the Q tensor, but the associated order is very weak within the isotropic phase (blue). It can be seen that the effective anchoring if present is small, as there is not a well-defined angle between the director field and the surface normal.

Suppl. Movie 3: Movie of the dynamics corresponding to Figure 2(a-e) in the main text.

Suppl. Movie 4: Movie of the dynamics corresponding to Figure 2(f-j) in the main text.

Suppl.

Movie 5: Slices of the director profile through the system (planes perpendicular to the z axis), for the configuration in Figure 2j. Similar considerations apply as for Suppl. Movie 2 -the effective anchoring though appears normal in this case.

Suppl. Movie 6: Movie of the dynamics corresponding to a system with the same parameters as in Figure 2(f-j) (i.e., parameter set C), but with 90:10 composition.

Suppl. Movie 7: Movie of the dynamics corresponding to Figure 4 in the main text.