Fionnán Howard
email: fionnan.howard@dcu.ie

Siún Nic Mhuirí

Maurice Oreilly
email: maurice.oreilly@dcu.ie

"I don't like Maths as a subject but I like doing it

Keywords: Mathematical identity, transition, higher level education, thematic analysis

This paper presents a thematic analysis methodology which uses a hybrid coding process to understand how science and engineering students in higher level education relate to mathematics. This process utilises and builds on previous research on mathematical identity amongst student teachers by using deductive coding while continuing to be grounded in the new data through inductive coding. Many authors have written on both or either of these methods. Most treat it was a simple choice between one or the other. Few have addressed best practice in integrating these approaches.

Introduction

To create an effective learning environment for science and engineering undergraduates, an awareness of how they relate to mathematics is vital. Investigating mathematical identity can help educators better understand this relationship and help determine potential issues with pedagogy and the learning experience of students. Such issues may contribute to marginalisation and thus impact students' relationship with mathematics and their decision to continue, or not, their mathematical studies [START_REF] Grootenboer | Identity as a lens to understand learning mathematics: Developing a model[END_REF][START_REF] Solomon | Not belonging? What makes a functional learner identity in undergraduate mathematics[END_REF]. Reflecting on their mathematical identity can help students engage more effectively as mathematics learners while they transition to higher level education [START_REF] Kaasila | Using narrative inquiry for investigating the becoming of a mathematics teacher[END_REF]. In the same vein, Sfard and Prusak (2005, p. 16) suggest that "identity talk makes us able to cope with new situations in terms of our past experience and gives us tools to plan for the future." Mathematical identity has been investigated in the context of Initial Teacher Education in Ireland since 2008. A series of previous studies developed an instrument for exploring mathematical identity of pre-service teachers which has been adapted for this new context. The 'Mathematical Identity of Student Teachers' (MIST) study used grounded theory among 9 students to develop two open-ended questions: a broad opening question to allow participants make responses that are "indicative of their personal mathematical identity" and a follow-up question which included some prompts "to balance the need for some direction" (Eaton & OReilly, 2009, p. 2). 'Mathematical Identity using Narrative as a Tool' (MINT) involved 99 students and not only converted the questionnaire to an online tool but also added a third question about self-reflection which had become evident as a key element of mathematical identity in MIST [START_REF] Eaton | Developing an instrument to explore mathematical identity: A study of students from several third level institutions in Ireland[END_REF].

Although mathematical identity has enjoyed increased attention in recent years within education [START_REF] Darragh | Identity research in mathematics education[END_REF], a lack of cohesion and agreement on ontological, epistemological, and methodological issues has produced a fractured body of research [START_REF] Kaspersen | Measuring STEM students' mathematical identities[END_REF]. Despite this new attention, [START_REF] Hannula | Identifying subgroups of CERME affect research papers[END_REF] found that identity had featured in just four of 100 papers in this working group (TWG8) between CERME4 and CERME9. At CERME10 Kaspersen, Pepin, and Sikko (2017a) in TWG20 focused on the link between mathematical identity and grades much like [START_REF] Cobb | An interpretive scheme for analyzing the identities that students develop in mathematics classrooms[END_REF] did. A study by [START_REF] Craig | Conceptions of mathematics and student identity: Implications for engineering education[END_REF] took a narrower view of identity, thinking only of procedural versus conceptual learning, ignoring the many other facets of identity that interact with this specific strand. In Ireland, MIST/MINT [START_REF] Eaton | Who am I and how did I get here? Exploring the mathematical identity of student teachers[END_REF][START_REF] Eaton | Developing an instrument to explore mathematical identity: A study of students from several third level institutions in Ireland[END_REF] focused on developing an instrument to collect narrative data, but the process of analysing this data requires some clarification in order to be reliably used in other contexts or amongst larger groups of students. Although some studies have investigated engineering identity, and incidentally addressed mathematics, studies dealing with the mathematical identity of engineering students are rare. This paper will outline how we adapted and integrated inductive and deductive techniques to facilitate a thematic analysis. This addresses the need to incorporate the knowledge gained from previous work on mathematical identity in Ireland while also continuing to be grounded in the new data. We refined the MIST/MINT coding process by deconstructing the themes they developed amongst student teachers (which may not accurately represent the mathematical identity of this new cohort) into codes and constructed new themes based on the current data. We will demonstrate how the coding process aided the thematic analysis, already established in MIST/MINT, for cohorts of students not previously considered there. We expect this to advance the drive toward an effective process for accessing students' mathematical identities. [START_REF] Sfard | Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity[END_REF] and Kaspersen et al. (2017a) both noted that researchers in mathematics education had yet to agree upon a working definition of identity while Cobb et al. (2009, p. 41) conceded that it is "vague and ill-defined." Authors often operate from different viewpoints: the sociological viewpoint where identity is an action, constructed and reconstructed over time, which has long characterised identity at CERME [START_REF] Hannula | Identifying subgroups of CERME affect research papers[END_REF], and the psychological viewpoint where identity is a core stable phenomenon [START_REF] Cobb | An interpretive scheme for analyzing the identities that students develop in mathematics classrooms[END_REF]. [START_REF] Sfard | Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity[END_REF] detailed the similarities between identity and other concepts such as attitudes, conceptions and beliefs but pinpointed the constantly changing and evolving nature of identity as the key power behind the concept. The sociological viewpoint of identity acknowledges that it is ever-changing and communicated best through narratives authored by the participants. They can communicate mathematical experiences they consider to be influential and reflect on them as they write often leading to realisations or re-interpretations of the experiences.

Background and Theoretical perspective

This study is conducted under a social constructivist paradigm and mathematical identity is defined as the "multi-faceted relationship that an individual has with mathematics, including knowledge, experiences and perceptions of oneself and others." (Eaton & OReilly, 2009, p. 228; See also [START_REF] Grootenboer | Identity as a lens to understand learning mathematics: Developing a model[END_REF] We propose that empirical information about experiences is not enough to reveal mathematical identity and we aim to establish the meaning of these experiences from the views of participants (Creswell, 2009, p. 16). We follow the narrative approach detailed by Creswell (2009, p. 13) whereby the narrative produced by the research represents a repackaging of the experiences and stories communicated by the participants. The resulting narrative represents a combination of these experiences with those of the researcher, positioning the participants as coresearchers who shape the research process (Cohen, Manion, & Morrison, 2007, p. 37).

Methodology

We identified 16 cohorts of science and engineering students in DCU who study mathematics in their first year. They represent a significant portion of the undergraduate population and of students taking mathematics modules, but they have not previously been included in research on mathematical identity. Participants were recruited in lectures where the lead author explained the study, distributed plain language statements and returned one week later to voluntarily sign up students who were interested in taking part. An online questionnaire (adapted from MIST/MINT) consisting of three open-ended questions was employed with each question appearing on a separate page. A pilot study conducted in Winter 2017 involved 18 respondents (10 science and 8 engineering students). There were 32 respondents to the main study (22 science and 10 engineering students), contributing a total of 8000 words. The first two questions are the focus of this paper and are presented in Figure 1. Note that third level is an Irish term for higher level education.

Q1. Think about your total experience of mathematics. Tell me about the dominant features that come to mind.

Q2. Now think carefully about all stages of your mathematical journey from primary school to university mathematics. Consider: To analyse the open-ended responses to the online questionnaire we used a hybrid process of inductive (data-driven) and deductive (theoretical) coding adapted predominantly from [START_REF] Fereday | Demonstrating rigor using thematic analysis: A hybrid approach of inductive and deductive coding and theme development[END_REF] with influence from [START_REF] Braun | Using thematic analysis in psychology[END_REF], [START_REF] Boyatzis | Transforming qualitative information[END_REF] and [START_REF] Crabtree | Doing qualitative research[END_REF]. The latter authors noted that codes "can be constructed a priori, based on prior research or theoretical perspectives or created on preliminary scanning of the text" and that "...some initial codes are refined and modified during the analysis process." (p. 167) Their claim that constructivists tend to lean towards co-created codes fits well with Braun and Clarke's categorisation of thematic analysis as "grounded theory 'lite'" (Braun & Clarke, 2006, p. 81) since this research builds on the grounded theory research of MIST/MINT.

Our approach consisted of seven stages (of which, the first four will be described further on): Stages 3-7 broadly align with the five steps suggested by [START_REF] Braun | Using thematic analysis in psychology[END_REF] who treated inductive and deductive coding as separate techniques. Stages 1-5 correspond to steps from Fereday and Muir-Cochrane ( 2006) but with a significantly expanded fourth stage as well as stronger interplay between the data analysis stages. [START_REF] Crabtree | Doing qualitative research[END_REF] have a simple four step model (with stages 2-4 combined) which omits the testing stage although they do go on to discuss such a step (p. 168). All authors participated in stage 2 coding but stage 3 and 4 were conducted by the lead author.

Although most authors recommend an iterative line-by-line coding process where codes are developed, applied and immediately examined for groupings without reference to the data [START_REF] Braun | Using thematic analysis in psychology[END_REF], we realised that the density of these responses (on average every 9 words resulted in a code) meant that it was considerably more difficult to extract meaningful codes from the data in a single round of coding without including multiple perspectives [START_REF] Strauss | Basics of qualitative research: Techniques procedures for developing grounded theory[END_REF]. We thus developed stage 4 significantly beyond what is described by other authors. The remainder of this section will describe stages 1-4, which have been completed, with particular emphasis on stage 4. It is intended that Stages 5-7 will follow the recommendations given by [START_REF] Braun | Using thematic analysis in psychology[END_REF].

Stage 1

MINT/MIST participants' responses were used to develop seven main categories: self-reflection, influential people, ways of working in mathematics, comparing other subjects, nature of mathematics, right and wrong and mathematics as a rewarding subject (Eaton, Oldham, & OReilly, 2011, p. 32). These categories were expanded into codes using the examples and discussion from five MINT papers, three MIST papers (one unpublished) and two papers from an interim 'bridging' study. Codes that did not clearly fit in any of these categories were simply left uncategorised.

The pilot study helped to further demarcate the codes under each category. We followed Fereday and Muir-Cochrane (2006, p. 85) and [START_REF] Boyatzis | Transforming qualitative information[END_REF] by naming the codes, providing a description, some keywords (if possible) and giving at least one example from the pilot study of an instance of the code. From the start we adopted the use of an extra miscellaneous code for items which do not fit any existing category. This step is used to check the comprehensiveness of the codebook and to ensure nothing relevant is overlooked the coding process because it is difficult to categorise. For example:

ID86

I think its good that I'll be practicing something I'm bad at.It means I'll be eventually not so bad at it.

Stage 2

A codebook consisting of 37 codes was applied to a stratified sample of 8 from the 32 respondents, based on grouping, cohort, gender, word count and age. All three authors coded this data separately.

We discussed the results with reference to the codebook and adapted the codebook by splitting, combining or creating codes as appropriate changing the number of codes to 47. The definitions were also updated, keywords added and examples from the sample included in line with the relevant codes. Our initial main coding principles were derived from the sample coding at this stage. They are given here with examples to show their effectiveness:

1. Code in-line by including a number at or near each instance. Codes can run across sentences, overlap and a single statement may require multiple codes.

Pay attention to context between sentences or along entire paragraphs.

ID 118

Physics is about explaining everything essentially, and making sense of everything. That's why maths is interesting to me... This student talks about making sense of everything but the second sentence makes clear that mathematics also plays a role either intrinsically or in tandem with physics.

2.

Codes should be broad enough to allow a range of responses to be grouped. Especially where a continuum exists, e.g., mathematics is exciting/boring or hard/easy.

3.

Often participants will express a combination of both positive and negative, e.g., elements they like and elements they don't like or parts they find hard or easy: ID124 I enjoy certain parts of it.

ID78

Certain aspects came easier to me than others...

ID98

Leaving a hard topic and starting to learn a harder one makes the first hard topic seem very easy.

They may give a 'middle-ground' response: ID96 Generally ok experience... ID120 I found mathematics okay.

Students sometimes mitigate their opinions: ID40 I don't like Maths as a subject but I like doing it.

Stage 3

The entire dataset was read through for "close contact and familiarity" (Boyatzis, 1998, p. 45) and to make the important first step towards understanding the narratives therein. To accomplish this the data was read through repeatedly, "taking notes or marking ideas for coding" (Braun & Clarke, 2006, p. 87) as well as noting the key points made by participants (Fereday & Muir-Cochrane, 2006, p. 86). These notes were used later for comparison with coding.

Stage 4

To enact the principles stated at the beginning of this section, the coding process consisted of several steps (ways of looking at the data):

(a) Apply codes using a fine-grained, line-by-line approach.

(b) Write a summary of each participant's response and compare with the codes allocated.

(c) Compare with original notes from broad reading of the data.

(d) Code each 'piece of interest' by removing chunks of text and re-reading them in isolation.

(e) In tandem:

i. Pick each code and read through data for possible extra occurrences of the code.

ii. Use keywords to identify possible extra occurrences of each code.

(f) Interrogate data coded as miscellaneous to determine if re-categorisation is possible.

This resulted in the creation of 20 inductive codes to be reviewed at the next stage. Each step shown above added to the exhaustiveness of the coding by illuminating more opportunities for codes to be applied but was also useful for adding new ideas as notes under each participants' response.

Step (a) arose because we found that students' responses contain a lot of information even in short pieces of text:

ID 55 I generally did well in maths exams [good at maths] compared with other subjects [comparing subjects], which made me enjoy the subject more [I enjoy maths], and gave me confidence and a belief that I could do well [origin of interest in maths] in the field [confidence in maths].

In step (c) almost every initial note had been encapsulated in the coding. Of the remainder, most notes were now either obviously unfounded or involved too high a level of interpretation to be included: ID 118 ...explaining the reasoning behind what we were learning. We just learned for the sake of learning... This was initially considered as "I like to understand the content rather than learn it off" which requires a leap of interpretation. It fit much better in the inductive code "Learning/doing maths without knowing why."

In step (e) we discovered that searching for keywords often resulted in hits from the summaries in part (b). This was useful for identifying via a paraphrasing whether the correct codes had been applied. Searching for 'pace' brings up my comment "Pace. Quicker (because I was good?)" Original statement was "I would be finished far ahead of everyone in my class" (ID69).

In (f) the miscellaneous code proved useful for collecting statements that were hard to categorise or would require a new code to be classified. Of the 36 statements coded, 28 were reclassified leaving only 8 statements remaining as miscellaneous.

Discussion

This paper outlined a hybrid inductive and deductive methodology to facilitate a thematic analysis which incorporated the knowledge gained from previous work on mathematical identity in Ireland while also continuing to be grounded in the new data from this study. We built on the previous grounded theory research by MIST/MINT while acknowledging that we do not yet have a fully validated and reliable process for accessing mathematical identity. The categories developed by MIST/MINT were particularly helpful to determine whether a code might be present in a statement. Adopting a finer, detailed coding approach at stage 4 with many steps acknowledges that students' narratives are a link or window to their identities rather than being their putative 'actual' identities.

The ever-changing nature of identity necessitated a semantic approach where one does not look "beyond what a participant has written" when coding. The interpretative stages happen during analysis after coding has taken place. This is in line with the advice of Braun and Clarke (2006, p. 98) but contrary to that of [START_REF] Boyatzis | Transforming qualitative information[END_REF]. All steps of coding proved useful for combating discounting of evidence in uncoded text (Crabtree & Miller, 1999, p. 171) particularly coding 'pieces of interest' and using keywords. Using a miscellaneous code represented 53 of the 116 (or 46% of) code changes made and thus was the most effective way to ensure that meaningful responses did not go undetected. The small sample size of 32 facilitated this multi-step coding.

We expect this methodology to advance the drive toward an effective process for accessing students' mathematical identities. We see this approach as providing insights for students to improve their learning experiences and for educators to ease the transition to higher education for such students.



  Your feelings or attitudes to mathematics  Influential people  Critical incidents or events  Specific mathematical content or topics  How mathematics compares to other subjects  Why you chose to study a course which includes mathematics at third level With these and other thoughts in mind, describe some further features of your relationship with mathematics over time.

Figure 1 :

 1 Figure 1: First two questions of the online questionnaire which were presented on separate pages

Stage 1

 1 Development of codebook from literature review of MIST/MINT and pilot study. Stage 2 Code stratified sample of main data. Stage 3 Broad reading and summarising of entire data set. Stage 4 Code data using inductive and deductive codes. Elaborated in detail below. Stage 5 Group codes to develop thematic map. Stage 6 Review themes and use the data to check for internal and external homogeneity. Stage 7 Define and name themes.


Patricia Eaton
email: p.eaton@stran.ac.uk

Christine Horn
email: christine.horn@iadt.ie

Elizabeth Oldham
email: eoldham@tcd.ie

Maurice Oreilly
email: maurice.oreilly@dcu.ie

Reflections on the development and application of an instrument to access mathematical identity

Keywords: Mathematical identity, personal narratives, online survey

Mathematical identity is understood in this paper as the multi-faceted relationship that an individual has with mathematics, including knowledge, experiences and perceptions of oneself and others. The researchers developed a protocol for accessing mathematical identity, initially designed for use with initial teacher education (ITE) students. The protocol has been refined and its use extended to students of mathematics in other contexts in Ireland. This paper reviews the iterations of the protocol, its development into an instrument, and the context and use of such an instrument. It addresses the issue of robustness of the instrument when applying it to contexts within and outside ITE.

Introduction

Mathematical identity has been a topic of interest especially over the last twenty years, and many studies have addressed the mathematical identity of teachers or students in initial teacher education (ITE) programmes. The work described in this paper is intended to contribute further to exploring students' mathematical identity. Originally focused on ITE students, it now includes other cohorts.

In 2009, as part of the project MIST (The Mathematical Identity of Student Teachers), a protocol was developed that scaffolded students in writing about their encounters with mathematics. It was used with two groups of ITE students in the island of Ireland [START_REF] Eaton | Who am I and how did I get here? Exploring the mathematical identity of student teachers[END_REF]. The aim was to encourage them to reflect on their mathematical identity with a view to facilitating discussions around how that identity may impact on their own classroom practice. The students' narrative responses allowed certain themes to be identified that spanned the responses of both cohorts. These themes also sufficed to describe the results of using a very similar protocol with undergraduate mathematicians taking modules on mathematics education [START_REF] Eaton | Looking back to look forward: The mathematical identity of prospective mathematics teachers[END_REF]. For the next iteration, in the study Mathematical Identity using Narrative as a Tool (MINT), the protocol was developed into an instrument that could be administered online, with the aim of accessing further cohorts with greater ease and efficiency. It was used with six cohorts of students in both jurisdictions in Ireland, within and outside ITE [START_REF] Eaton | Developing an instrument to explore mathematical identity: A study of students from several third level institutions in Ireland[END_REF][START_REF] Eaton | From learner to teacher: Using reflection to develop mathematical teacher identity[END_REF]. Details of these cohorts, and of the development of the instrument, are given below.

The main aim of this paper is to address the usefulness of the instrument in a variety of settings. The paper first situates the work of the projects MIST and MINT within traditions of research on identity as distinguished in recent reviews of that research, and then traces the evolution of the original protocol into the instrument used in MINT. In particular, the authors focus on how the themes identified initially in MIST apply to other cohorts (even outside ITE), and reflect on how robust the outcome themes appear to be, regardless of the cohort of students addressed.

Literature review and outline of a theoretical framework

Research in mathematical identity has attracted, and received, overviews and critical analyses for at least two decades. Earlier syntheses are discussed and expanded in some more recent surveys [START_REF] Darragh | Identity research in mathematics education[END_REF][START_REF] Goldin | Attitudes, beliefs, motivation and identity in mathematics education: An overview of the field and future directions[END_REF][START_REF] Hannula | Affect and mathematical thinking: Exploring developments, trends, and future directions[END_REF][START_REF] Lutovac | Future directions in research on mathematics-related teacher identity[END_REF]. From these overviews, several aspects of relevance to this paper can be distinguished; they provide a framework within which the studies of mathematical identity can be addressed.

A notable aspect is that there is still no agreement on the definition of mathematical identity. [START_REF] Darragh | Identity research in mathematics education[END_REF] highlights and critiques the differing and indeed conflicting versions, or, in some papers, the absence of any clear definition. In [START_REF] Lutovac | Future directions in research on mathematics-related teacher identity[END_REF] overview, the uncertainty created by multiple definitions is regarded as a given; however, following [START_REF] Sfard | Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity[END_REF], they demand that studies should provide at least a working definition.

Approaches can also be categorized as broadly social or broadly psychological. Many of the studies on mathematical identity draw on [START_REF] Wenger | Communities of practice: Learning, meaning, and identity[END_REF] work on communities of practice. In this respect, they tend to follow what [START_REF] Lerman | The social turn in mathematics education research[END_REF] calls the social turn in mathematics education research that has been a prominent feature especially from the mid-1980s [START_REF] Goldin | Attitudes, beliefs, motivation and identity in mathematics education: An overview of the field and future directions[END_REF]. However, [START_REF] Lutovac | Future directions in research on mathematics-related teacher identity[END_REF] argue that "neglecting the individual, i.e. how one thinks and feels and who one is, is at odds with the core concept of identity itself" (p. 767); they advocate for a more balanced psychosocial approach that allows for a focus on individuals' thinking and feeling.

Methodology constitutes another aspect. Qualitative approaches have dominated, with sample sizes typically being small, and just a few studies using coding and counting techniques [START_REF] Darragh | Identity research in mathematics education[END_REF][START_REF] Goldin | Attitudes, beliefs, motivation and identity in mathematics education: An overview of the field and future directions[END_REF][START_REF] Hannula | Affect and mathematical thinking: Exploring developments, trends, and future directions[END_REF]. The use of narrative has been particularly important; in a seminal paper, [START_REF] Sfard | Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity[END_REF] actually take the narratives as constituting identities, hence opening the way to approaches requiring "close attention to the words used" (Darragh, 2016, p. 25).

The mathematical identity of ITE students has been a topic attracting considerable focus. The value of research in the area is highlighted, for example by adverting to "some evidence that encouraging pre-service teachers to narrate their own or listen to their peers' personal experiences with the subject makes them cope better, which may lead to the development of a more suitable identity for mathematics teaching" (Goldin et al., 2016, p. 17). [START_REF] Lutovac | Future directions in research on mathematics-related teacher identity[END_REF] also note the process of change in ITE students' mathematical identity, for example by emphasizing affective facets.

Methodology and evolution from MIST to MINT

The definition of mathematical identity formulated for the studies considered here is the following: Mathematical identity is considered as the multi-faceted relationship that an individual has with mathematics, including knowledge, experiences and perceptions of oneself and others.

(See Eaton et al., 2014, for references.) This situated the work primarily in the broadly social tradition, with some psychological aspects. The methodology has chiefly involved students writing narrative accounts; analysis of the resulting qualitative data has been done by coding and counting as described below. Most work has involved ITE students; extension to other cohorts is a focus of this paper.

The protocols and instrument, student cohorts, and establishment of themes

The first study (of three), MIST, considered the identity of ITE students (pre-service primary teachers) who were at one of two institutionsone in Northern Ireland and one in the Republic of Irelandand were specialising in mathematics (see, for example, [START_REF] Eaton | Who am I and how did I get here? Exploring the mathematical identity of student teachers[END_REF]. Introductory demographic questions and Likert items were used along with a two-part protocol: P1. Students were asked to respond in writing to the prompt: "Think about your total experience of mathematics. Tell us about the dominant features that come to mind." P2. They were then offered a more structured prompt: "Now think carefully about all stages of your mathematical journey from primary school (or earlier) to university mathematics. Consider:

 Why you chose to study mathematics at third level  Influential people  Critical incidents or events  Your feelings or attitudes to mathematics  How mathematics compares to other subjects  Mathematical content/topics With these and other thoughts in mind, describe some further features of your relationship with mathematics over time."

The narrative responses were used to provide guidelines for focus group discussions, one in each institution. Discussion transcripts, together with the original narratives, were analysed to identify common threads and themes [START_REF] Clandinin | Narrative inquiry: Experience and story in qualitative research[END_REF]. Using an approach informed by grounded theory, seven themes were identified; their main features can be summarized as follows:

T1. Students' self-reflection on learning and teaching relates to how students' exploration of their mathematical identity leads them to deepen their insight into learning and teaching mathematics. T2. The role played by key figures in the formation of mathematical identity focuses not only on teachers and family members, but also on peers and society at large. T3. Ways of working in mathematics explores what students find effective in learning mathematics and why, either through individual endeavour or through collaboration. T4. How learning mathematics compares with learning in other subjects considers the particular characteristics of learning mathematics, usually at school, that distinguish the process from learning in other subject areas. T5. The nature of mathematics draws from a broad range of students' perceptions touching on the philosophy of mathematics and on what doing mathematics is about.

T6. "Right" and "wrong" in mathematics concerns students' perception that what is important in mathematics is to find the correct answer, and also a more general notion around the unambiguous nature of mathematical truth. T7. Mathematics as a rewarding subject examines the extent to which students enjoy the subject, often relating to how they persist with it or to significant moments of insight. (Eaton et al., 2011, pp. 156-157) These reflect both social and psychological aspects. For instance, T4 involves both aspects, and T7 in particular includes affective issues, as highlighted by [START_REF] Lutovac | Future directions in research on mathematics-related teacher identity[END_REF].

In the second study, the students were undergraduate mathematicians at another institution, had elected to take modules in mathematics education, and thus were considered as "prospective" teachers despite not being in an ITE programme. Students wrote responses to P1 and P2 in their own time as part of the module assessment. Analysis was based on the MIST themes, which were found to be adequate for describing these narratives [START_REF] Eaton | Looking back to look forward: The mathematical identity of prospective mathematics teachers[END_REF][START_REF] Eaton | From learner to teacher: Using reflection to develop mathematical teacher identity[END_REF].

In its third iteration, MINT, the research team of three peoplethe two from MIST, one from the second studywas augmented to five, with new members coming from two institutions in the Republic of Ireland. The protocol was extended [START_REF] Eaton | From learner to teacher: Using reflection to develop mathematical teacher identity[END_REF] to include a third part:

P3. What insight, if any, have you gained about your own attitude to mathematics and studying the subject as a result of completing the questionnaire?

The items, including introductory demographic questions and Likert items as used for MIST, were gathered into one instrument and made available online to six cohorts of students using SurveyMonkey, allowing students to respond at their chosen time and location. The students in four of the cohorts were in Initial Teacher Education (ITE) programmes. The other two cohorts were not; one comprised first year students of entrepreneurship (involving a one-year business mathematics module), while the other consisted of fourth year students of applied psychology (taking research methods and statistics-related modules in all of their four years of study). For the purposes of this paper, these two cohorts were denoted as Non Teacher Education (NTE).

Coding the MINT data

Of the 99 respondents in the MINT study, 86 provided data (narrative responses) for at least one of the three open questions, P1, P2 and P3; detail relevant to this discussion is provided in Table 1 below. The five members of the research team were assigned respondents, ensuring that the data from each respondent was studied by three people; each person coded the data using the MIST themes and also sought to identify any other themes. No other themes were found. Several meetings were held to compare the coding and reconcile differences; this entailed clarifying the scope of some of the themes to obviate confusion and/or overlap, as elaborated below. Consensus was reached, so the question of inter-coder reliability did not arise. Overall, the seven themes provided adequate descriptions and coverage of the data, though themes T4 and T6 were less prominent than the others. Once the coding was complete for each element (respondent and question, P1-P3) of data, the instances of each theme (T1-T7) were counted. Analysis of data coded T1 (students' selfreflection on learning and teaching) from the 56 ITE students is reported by [START_REF] Eaton | From learner to teacher: Using reflection to develop mathematical teacher identity[END_REF].

As regards clarification, when the MINT research team discussed the responses that were potentially to be coded T1, for example, the theme was refined as follows: T1 (i) applied when students explicitly reflected on their mathematical journey with reference to "then" and "now"; (ii) was extended to include reference to how students see themselves teaching and learning into the future; (iii) did not apply to data that comprised lists without context. There was also some development concerning potential overlap of themes, particularly T3 (ways of working in mathematics) and T5 (the nature of mathematics). The research team agreed that T3 applied to (i) what students did as they learned mathematics, inside or outside "the classroom", and (ii) what students find effective in learning mathematics and why, either through individual endeavour or through collaboration. Moreover, it was agreed that T3 would include "rote learning" and "learning for understanding" where the learner is the agent (rather than experiencing teaching for rote learning or understanding, for example). In the summary description of T3, it was agreed that "effective" would be interpreted as anything on the spectrum of effectiveness (including ineffective); T5 would include (i) the "relevance versus abstraction" dimension and (ii) applications in the real world. After clarifying such matters in relation to coding, it was judged that the MINT instrument was appropriate for exploring characteristics of mathematical identity among different cohorts of students or in different settings.

Extending the instrument to NTE students

The application of the instrument to NTE students in MINT is considered in this section, and a comparison is made between the data from this cohort and the ITE data. It was found that the same themes could be applied satisfactorily across the entire MINT study; however, more detailed analysis is of interest.

Cohort (ITE or NTE); questions from protocol (P)

Number of respondents

Percentage of students in each cohort coded in each theme T1 T2 T3 T4 T5 T6 T7 A feature of the protocol (for both MIST and MINT) is that respondents did not see P2 until after they had responded to P1. The importance of this open-ended and non-directive initial question is discussed in [START_REF] Eaton | Who am I and how did I get here? Exploring the mathematical identity of student teachers[END_REF]. Of the 56 ITE students who answered at least one of the three open questions, 52 responded to P1. Likewise, 27 out of 30 NTE students answered P1. On examination of the responses from all six cohorts (four ITE and two NTE), Table 1 summarizes the coding for the themes (T1-T7) for ITE and NTE students, drawing attention to the frequencies of the codes in responses to P1 and to all three parts (P1-P3) of the protocol combined, as percentages of the number of respondents.

The table reveals that themes T6 and T4 are weakest for each cohort; their frequency is especially low in response to P1. The theme most related to affect, T7, is by far the strongest for each cohort.

With the prompts offered by P2 and P3, the codes for each theme are more frequent, as might be expected. However the profile of the two cohorts differs between themes and according to the prompts given. From a qualitative perspective, we should expect that the narrative around mathematical identity would depend on the cohort to which students belong, reflecting factors such as programme focus, teaching culture or maturity of students. To test if there are any significant differences between the ITE and NTE cohorts, a Fisher's exact test was conducted for each of the themes. When all three questions in the protocol were considered together, the only statistically significant difference was for theme T5. NTE students were found to capture this theme significantly more frequently than did ITE students (with p = .013 in the Fisher test). For all other themes there was no statistically significant difference (p > .15 in all cases). However, when the responses to P1 alone are considered, the contrast between the two cohorts in relation to T5 is more stark (p = .003), while significant differences between the cohorts in relation to T2 and T3 are apparent (with p = .027 and p = .032, respectively). This indicates that prospective teachers are more likely to draw attention to the influence of others (such as their own teachers) and to the importance of ways of working with mathematics than are NTE students.

Focusing on T1, it was noted that ITE students' narrative relevant to this theme emphasized how the evolution of mathematical identity from second to third level education was intertwined with their commitment to teaching mathematics [START_REF] Eaton | From learner to teacher: Using reflection to develop mathematical teacher identity[END_REF]. Naturally, such a perspective was found to be absent from the NTE cohort. However, significant evolution across the same educational transition was intertwined with the new focus that students appreciated regarding the applicability of the subject to the "real world". It is exemplified in the following quotation from one respondent:

Whereas in Maths in secondary once the problem was solved that was the end of the matter, in college we use the solution after its completion … and apply them to everyday/business/social scenarios … My experience of maths in 3rd level has given me a new opinion on the subject, I can clearly see its uses, and an understanding of where to apply them. (P1, NTE)

As far as T5 (the nature of mathematics) is concerned, the importance of application of mathematics to real-world problems comes out strongly for 17 of the 20 NTE students whose responses were coded for this theme. In summary, the data show that the NTE students in MINT emphasise the practical nature and importance of mathematics and its relevance to business or psychology. It is reasonable to argue that precisely this aspect gives rise to the much higher instance of T5 among the NTE students than among ITE students.

Robustness of the instrument

Two aspects of robustness are considered here. They are robustness of the themes (their adequacy in describing data for different cohorts), and inter-coder reliability in using the instrument.

The seven themes identified in MIST, using a grounded approach following the steps of thematic analysis, were re-evaluated and basically endorsed in MINT by the members of the larger research team. The argument so far has shown that these themes appear to be robust in facilitating access to the mathematical identities of differing cohort of students: ITE students in different types of ITE programme, and in two different (yet neighbouring) countries; and also NTE students who had some element of mathematical work in their courses. No new themes have emerged. While variations are evident in the frequency with which the themes figured for the two cohorts, these could be related to the backgrounds and interests of the cohorts and are of intrinsic interest. It can be noted that placing the instrument online for MINT, using SurveyMonkey, reduced the influence of the researcher and strengthened anonymity for the respondents (as well as allowing them to respond where and when suited them); it also facilitated data collection for the research team.

With regard to ensuring inter-coder reliability, considerable work was needed. Some of this involved clarification of the borderlines between themes, as described above. Future coders might need training, as well as time to grasp the definition or description of each theme. However, while the "coding and counting" approach was important for research, the narrative data elicited can, in any case, be of value for lecturers in appreciating their students' preoccupations. As judged by answers to P3, it is also of value to at least some of the respondents themselves, in line with the original aims for the studies (and perhaps aligning with the work of [START_REF] Goldin | Attitudes, beliefs, motivation and identity in mathematics education: An overview of the field and future directions[END_REF], cited above). In fact (though details are outside the scope of this paper), two of the authors have used the instrument with several cohorts of students, and have found that their own teaching was enriched by what students wrote and that many of them reported finding the exercise helpful.

Conclusion

The importance of mathematical identity is well established, particularly in teacher education, notwithstanding the challenges in establishing clear definitions and a shared understanding in the academic community. This paper has drawn attention to the development of one protocol and instrument that have proved stable when working with a range of students both within and outside teacher education and in different countries. The online instrument used has elicited both social and psychological aspects of identity. It can be easily administered to students, and allows teachers and even students themselves to gain insight into mathematical identity, which can be beneficial for learning and teaching mathematics. The value of the data collected, for research and also for teaching and learning, promises well for further use in both similar and different contexts.

Table 1 : Coding of themes in response to open questions

 1

	ITE; P1	52	29 44 58	4	17	6	83
	ITE; P1, P2 & P3 combined	56	48 84 63 36 38 23 98
	NTE; P1	27	22 19 30	4	52	4	89
	NTE; P1, P2 & P3 combined	30	63 70 47 33 67 10 97

  Here is an example of what one student wrote:

	Mathematics, in particular statistics is vital within the area of Psychology. I really enjoyed it
	during third level because I could see what the analyses were being used for. … Also, while I
	really enjoy maths now, I feel that the style of teaching through primary and secondary school
	had given me a complete misconception into what maths really is. (P2, NTE)

Acknowledgment

The authors acknowledge the funding received from SCoTENS for this research, and thank all the contributors to the studies, especially Miriam Liston as a member of the original MINT team.


Bertrand Petit
email: bertrand.petit@inria.fr

Manuel Serrano
email: manuel.serrano@inria.fr

Composing and Performing Interactive Music using the HipHop.js language

Keywords: Interactive Music, Web programming, Reactive synchronous programming CCS Concepts Human-centered computing → Interaction design process and methods, Collaborative content creation, Applied computing → Sound and music computing

Skini is a platform for composing and producing live performances with the audience participating using connected devices (smartphones, tablets, PC, etc.). The music composer creates beforehand musical elements such as melodic patterns, sound patterns, instruments, group of instruments, and a dynamic score that governs the way these basic elements behave according to events produced by the audience. The platform allows to control the musical quality of the work, even if during the concert or the performance, the audience interacts with the system giving birth to an original music creation.

Skini scores are expressed in terms of constraints on events that control which musical elements are accessible to the audience and when they are available. Constraints may be instantaneous, for instance one constraint may disable violins while trumpets are playing. They may also be temporal, for instance, one constraint may prevent the piano to play more than 30 consecutive seconds.

, a reactive synchronous DSL, for implementing the music scores. The HipHop.js constructs, which consist of temporal operators such as parallel executions, sequences, awaits, synchronization points, and preemption, form the core implementation language for expressing Skini musical constraints.

This paper presents the Skini platform and It reports about live performances and an educational project.

INTRODUCTION

Licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Copyright remains with the author(s). NIME'19, June 3-6, 2019, Federal University of Rio Grande do Sul, Porto Alegre, Brazil.

In the 60's, the philosopher Umberto Eco, and musicians such as K. Stockhausen, K. Penderecki, and L. Berio wondered about the relationships between composers, musicians, and the audience [START_REF] Eco | The Open Work[END_REF]. In particular, Eco has shown that following the evolution of physics from Copernicus to Einstein, the perception of the world has evolved from being static to being more dynamic. This has impacted 20th century art where some musicians have tried to express this dynamics through works where the performer and the audience had a concrete impact on the musical result. Eco called these endeavors open work or moving work, in the sense that the final musical result is not strictly known in advance.

Recent technology improvements, mainly broad network coverage and device availability, have opened up new opportunities for interactive music, which has become an active field [START_REF] Arango | The Smartphone Ensemble. Exploring mobile computer mediation in collaborative musical performance[END_REF][START_REF] Lee | Distributing Mobile Music Applications for Audience Participation Using Mobile Ad-hoc Network (MANET)[END_REF][START_REF] Weitzner | massMobile -an Audience Participation Framework[END_REF]. A tremendous effort has been devoted to improve the technical aspects of systems allowing individual interaction, see Swarmed [START_REF] Hindle | Swarmed: Captive portals, mobile devices, and audience participation in multi-user music performance[END_REF] for instance. These technological challenges have attracted most attention, so frameworks dedicated to music composition of interactive performances following the moving work, proposing a clear composition scheme and the way to make it work, are rare. Skini is such a framework. It is meant for composing and executing interactive performances.

This paper present Skini with an organization that is as follows. Section 2 presents the related work. Section 3 presents the Skini key principles and concepts for creating and performing musical pieces. Section 4 focuses on musical interactive composition and orchestration. Section 5 presents the Skini software architecture and gives a taste of HipHop.js score programming. Section 6 presents two actual musical projects already executed with Skini. Section 7 presents directions for future work.

RELATED WORK

Taylor et al. overview the domain of collaborative music [START_REF] Taylor | A History of the Audience as a Speaker Array[END_REF]. They present several developments aim to break the unidirectional communication, well established in musical performances, by involving the public in the process of musical creation. Three main directions emerge: social behavior, technology, and impact of interaction. We think that there is a fourth direction that is still not clearly addressed: composition issues for interactive music. Here are some examples. Social behavior: In [START_REF] Arango | The Smartphone Ensemble. Exploring mobile computer mediation in collaborative musical performance[END_REF] the authors studied Computer Mediated Musical Collaboration by exploring how to use smartphone mediation in musical contexts. They report on the activities of the Smartphone ensemble at the University of Caldas. This study focuses on the design of the graphical interface and social behavior of an audience. In the same vein, [START_REF] Oh | Audience-participation techniques based on social mobile computing[END_REF] focuses on the level of interaction necessary for good public participation, based on the experiences of the Stan-ford Mobile Orchestra. The results of these experiments are valuable inputs for Skini. Technology: In [START_REF] Lee | Distributing Mobile Music Applications for Audience Participation Using Mobile Ad-hoc Network (MANET)[END_REF], the authors introduce a way to distribute mobile applications using mobile ad-hoc network. This study mostly focuses on telecommunications technology. Similarly, [START_REF] Hirabayashi | Sense of Space: The Audience Participation Music Performance with High-Frequency Sound ID[END_REF] proposes a method of communication by using DTMF (Dual Tone Multiple Frequency) with high frequency sound. Skini uses web technologies, hence it is network support agnostic. Impact of interaction: Opened band [START_REF] Stolfi | Open band: Audience Creative Participation Using Web Audio Synthesis[END_REF] proposes a system based on multi-user chat where text entries are translated into sounds. Similarly, [START_REF] Gimenes | Frontiers: Expanding Musical Imagination With Audience Participation[END_REF] presents two sound installations performed at the Peninsula Arts Contemporary Music Festival 2016, at the University of Plymouth, focusing on the experimentation of collaborative music using voting system, movement tracking and indoor positioning. More recently, the Cosima project at IRCAM [START_REF] Lambert | Synchronisation for Distributed Audio Rendering over Heterogeneous Devices[END_REF] explores the relationship between the body, media and space through new interfaces and collaborative creation tools. Cosima is a good example of using smartphones and Web technologies to control and to generate sound using smartphone speakers.

MassMobile [START_REF] Weitzner | massMobile -an Audience Participation Framework[END_REF] and Open Symphony [START_REF] Stolfi | Open band: Audience Creative Participation Using Web Audio Synthesis[END_REF] are architectures close to Skini. They present mass audience participation tools based on client-server systems dedicated to the production of live shows intended for a medium and large audience. In Open Symphony, improvisations by performers are controlled by audience members who specify collectively playing modes by a voting system. MassMobile is connected to Max/MSP and can be deployed in a variety of performance contexts as far Max/MSP is used. This platform is a complete interactive solution, and does not implement a particular composition process as skini proposes. Musical Composition issues: More in line with our musical composition issues, and especially regarding how to achieve an engaging experience for the audience members, Jieun Oh and Ge Wang in 2011 [START_REF] Oh | Audience-participation techniques based on social mobile computing[END_REF] described the challenges a composer faces when the audience participates in a music performance, such as controlling uncertainty factor, and defining roles between the audience and the composer/performers.

SKINI BASIC CONCEPTS

Skini organizes the music in short musical sequences called patterns. Playing the music consists in selecting some of these patterns. This is the role of the audience during the show that must select patterns from a set of availabilities that depend on previous audience selections and on the musical path described in the composer score. In order to help the participants select patterns they like, they can first listen to them using their smartphones, and then activate the ones they like. To ensure the musicality of the work, the composer controls which patterns are proposed to the audience by creating coherent groups. The art of managing the availability of the groups of pattern is the orchestration, in reference to the common use of this word in orchestral music.

Patterns

A pattern can be any part of a score, a MIDI sequence, or a pre-recorded sound. For the sake of interaction, patterns are generally short (few seconds), as longer patterns would monopolize the instruments, lengthen the waiting time of sequences to be played by one instrument, and thus reduce the interaction with the audience. When activated, a pattern can be played as is, or it can be changed in real time depending on how the composer imagines the show's behavior. For instance, a MIDI sequence pattern can be transposed to another tonality. Moreover, we are currently using an interface where patterns can also be created by the audience using a dedicated designer tool.

As patterns are activated by the audience, the composer cannot controlled in advance the result of the interpretation of his score. He must manage the uncertainty introduced by the interaction by thinking about the work in terms of group of pattern's architecture, in the sense of managing the available groups of patterns according to the aesthetic project.

Using pattern activation to create original music is not new. Twentieth century classical composers such as Stockhausen (Klavierstück XI ), Henri Pousseur (Scambi), and Boulez (Troisième Sonate pour piano) have used similar concepts. The originality of Skini is the delegation to the audience for the activation. Since it is no longer the musicians on stage that play the music, the composer is forced to give up on the traditional way of creating music. The interaction with the audience must be inherent to the creativity process. This gives a new dimension to U. Eco's openness.

Orchestration

In the common musical language, orchestration is the temporal description of the instruments actually played. The instruments are arranged in groups: violins, violas, cellos, double basses, horns, percussion, etc. The composer chooses the groups for each specific part of his work. In Skini, the orchestration consists in grouping patterns and defining constraints in between them. The orchestration governs the interaction with the audience but the audience reacts to what it is listening. This is an open work according to U. Eco's definition.

The composer defines the orchestration evolution according to various constraints and parameters such as time, and audience behavior. The audience behavior is defined by the activated pattern combinations. At a given time, the orchestration establishes the status of the groups of pattern (available or not available) for groups of people in the audience. Such a system, reacting to events (activation and time) to reach a defined state of the orchestration, is modeled by automata. The HipHop.js language has been precisely created for automata programming. We will see how it fits Skini architecture.

Interaction

Skini distinguishes three different roles people in the audience may play. 3. Conductors: are those who give their opinion or propose a global evolution of the performance. Their web interface is not shown here.

The software component that controls the orchestration is the automaton. It executes the rules and satisfies the constraints defined by the composer. In other words, the automaton runs the score. The complexity of the score all depends on the composer will. It can implement complex automatic strategies based on the behavior of the audience or it can be very lightweight and delegates most of the control to a controller that manually supervises the show.

COMPOSITION PROCESS

Whatever the kind of music to be created, the steps to be followed by the composer are the same: 1) design all the patterns, 2) create the initial orchestration, 3) simulate the audience behavior, and 4) modify the orchestration and repeat step 3 until the result is nice enough.

Pattern Design

The only constraint Skini imposes on patterns is their duration: all the patterns available simultaneously must have the exact same duration. This guarantees the rhythmic coherence of the piece. On the other hand, the sound of a pattern is totally free. For instance, a pattern can be a fragment of a melodic sentence. It can use a basic rhythm, and the composer can build the others on variations of this rhythm. It can be part of a family of patterns created by transposing an initial one. A pattern can also be a complex synthesized sound, or a sound sample, etc. Figures 2,3 The main difficulty for the composer is to imagine patterns adapted to all, or most, of the orchestral scenarios that will be created in the second step.

Orchestration Design

The composer designs the evolution of the orchestration that must anticipate audience interactions. For that, the orchestration must accommodate different paths through the events generated by the acting audience. How to design this orchestration and how to represent it are open questions. For the sake of the example, Figure 5 shows an early representation on the paper we used for one of our first piece. This representation is close to what is usually used for traditional musical scores. On the left (Y axis) are the groups of pattern. On the X axis is a timing view of the orchestration. We use the following symbols to represent the activation of groups of pattern: denotes a forced group activation, without waiting for an action by the audience; is denotes a forced group deactivation; • means that a group can be activated if one or several previous events linked by arrows occur; × means that a group can be deactivated if, one or several previous events linked by arrows, occur. Additionally, arrows are labeled with numbers to count the number of events and horizontal dash lines show the availability period for a group of pattern. This representation is imprecise. It lacks many details that only the actual HipHop.js program can express. However, in our experience, this sort of graphical representations have proved themselves to be particularly useful.

Simulation

It is difficult to imagine how the implementation of the orchestration will actually work, but it is basically the art of composing music to deal with this uncertainty and openness. This is why patterns and orchestration need simulation tools. We have created one for Skini. It behaves like an audience that would select patterns randomly. The simulator is controlled by three parameters: the minimum response time of people in the audience, the maximum response time, and the waiting time for a pattern to be played. Although simple, these three parameters are sufficient to simulate realistic rehearsals, at least, sufficient for the com-poser to figure out the flavor and shape of the music he is composing.

SKINI ARCHITECTURE

Skini relies on a web architecture (Figure 6) close to the one used by the Open Symphony [START_REF] Stolfi | Open band: Audience Creative Participation Using Web Audio Synthesis[END_REF]. The core system is implemented in Hop.js, a multi-tier JavaScript programming language. The orchestration is implemented in HipHop.js that implements the core synchronous reactive primitives of Esterel [START_REF] Berry | The Esterel v5 Language Primer[END_REF]. In this section we present the main part of the system architecture, and we introduce the orchestration programming.

System

The individual Skini components are conceptually distributed over several computers but of course, a single computer to run them all if the complexity of the performance is not too CPU demanding. The components are:

Pattern Player FIFO management

Hop.js

Orchestration Automaton

Hiphop.js

Synthetizers (Musicians)

Daw, Ableton Live

Web Server

Hop.js Web Server: it is in charge of all the interactions with the audience and it controls all the other components of the platform. The orchestration automaton runs inside the web server.

Actor and Controller: they are dynamic HTML5 clients connected to the Hop server, communicating using HTTP and websockets. Simulator: this is a Hop.js client using the same protocols as the actor. It behaves like an audience by generating automatically pattern activation. Parameters allow to dimension the size of the simulated audience. Pattern Player: the Pattern Player manages the requests coming from the audience. It uses waiting queues that accumulate the patterns that cannot be played instantaneously when all the synthesizers or musicians are busy. Orchestration Automaton: this is the software component that runs the composer score. It reacts to audience events and selects which patterns are available. Synthesizers: this is the component in charge of playing the actual music. It communicates with the other components using Open Sound Control (OSC) commands. A Digital Audio Workstation (DAW) such as Reaper or Ableton Live runs the synthesizers. Designer Interface and Distributed Sequencer: The Designer component is an HTML5 client, which communicates with the Hop.js server and the Distributed Sequencer using HTTP and websockets. The Distributed Sequencer plays the sequences of the Designers and is synchronized with the Pattern Player. The Designer client display is synchronized with the Distributed Sequencer. Display System: it displays various information to the users, for instance, who is selecting what in the audience. It communicates with the rest of the application using HTTP protocols.

HipHop.js Flavor for Orchestration

We motivate the usage of the HipHop.js language for implementing orchestrations with a simple example. We show the implementation of an orchestration that initially proposes to the audience cellos, and violins playing staccato patterns. After five ticks (where ticks are defined by the music tempo), percussion patterns are made available and then, after the first percussion pattern has been played, the cellos can play five patterns and stop. In parallel violins should stop after 4 cello events or after 10 ticks. Figure 7 shows the time events of that orchestration. This example is sufficiently complex to illustrate the main benefit of HipHop.js. We left to the reader the exercise to implement it using traditional sequential programming languages.

A HipHop.js execution is split in a succession of reactions that are triggered by external events associated with user interactions. The purpose of a reaction is to select the next expression Hop.js will execute. An HipHop.js program is organized as a list of modules that are loaded into a reactive machine. A module specifies the signals it can receive and emit. The module implementing our example is as follows: The module orchestration first emits two output signals ViolinStacatto and Cello with a value true. Then, it waits for five ticks before emitting the signal Perc with a value true. These emissions will be received by the Hop.js web server that will modify the status of the orchestration accordingly. The value true means that the patterns in a group of pattern (here Cello, Perc and ViolinStacatto) can be activated by the audience. These emissions will also update the web interface of all connected participants in the audience. A value false does the opposite. It deactivates a group of pattern. Activating a pattern in the audience generates a playing signals corresponding to the group the pattern belongs to. For instance, we someone selects a percussion, the signal PercPlaying is emitted. It enables the HipHop.js to proceed by executing the fork/par constructs that follows.

hiphop
The fork/par control structure runs its branches in parallel and waits for all of them to complete. The branches can communicate and synchronize with each others by broadcasting signals that are instantaneously delivered. During a reaction, all branches of all parallel constructs receive the exact same information. The HipHop.js execution is synchronous and deterministic.

HipHop.js can block on complex predicates. For instance, in the second branch of the fork/par construct, it waits for the first condition of 4 CelloPlaying events or 10 Tick. HipHop.js imposes no constraint on conditional expressions. They can mix arbitrary temporal expressions and JavaScript expressions.

This example, although simple, illustrates the benefit of HipHop.js. The constraints of the musical orchestration are mapped directly into constructs of the language without extra bookkeeping or encoding techniques. This greatly alleviate the composer/programmer task as it dramatically reduces the distance between the representation of his score and its actual executable implementation.

MUSICAL PERFORMANCES

Skini has already been used in different contexts. We present two experiences here. The first one is a show performed in 2017 during the MANCA festival of contemporary music in Nice, France. The second one is the Fabrique à Musique, a project funded by the SACEM (Music Composer Society).

Golem, MANCA 2017

The Golem show is based on the legend of Golem in Prague who was created in the 16th century in order to protect the Jewish community. On stage it involved a calligrapher and an actress/singer. The calame (pencil) of the calligrapher was used as a percussion producing sounds which were processed in real-time. The show lasted 45 min, along with two interactive scenes of about five minutes. There is a short video of the show available at: https://www.youtube.com/ watch?v=MuZfpSgsIPo.

We ran the show twice, with two different categories of spectators. On 8 December 2017, with two classes of young pupils, and on 9 December 2017 with a more classical panel of 150 spectators. Before the show, some instructions were given in order to explain the audience how to access the private WiFi network, how to get connected to the server, and the main features of the interface. The role of the audience in the story was explained and emphasized.

Golem: 8 December 2017

There were 42 pupils of 12 years old, highly motivated by the idea of using a smartphone during a live show. Though the smartphone is an important device for them, generally it is forbidden in schools. The idea of using smartphones certainly contributed to their active participation both to the show, and also to the "question and answer" session that followed. According to the system logs, 42 users tried to access the system, that is, all the pupils. This was not surprising as the pupils were informed and prepared by their teachers. During this show we had two sessions of interactions of 4 and 5 minutes each. This generated 4890 events (2400 in session 1; 2490 in session 2), 689 patterns were activated by the pupils (228 in sessions 1; 461 in session 2) which corresponds to 1.3 pattern activation per second.

Golem: 9 December 2017

There were 150 people, mostly the MANCA festival public who were aged between 30 and 70 years, with the mean age of around 40-50 years, and thus very different than the previous day. During both sessions, around 90 people accessed the system. That is 40 % of the spectators did not interact, or did not succeed in using the application for several reasons, such as handset incompatibility or misunderstanding of the process. During this show, we had two interactive sessions of 3 and 6 minutes each. This generated 4099 events (2171 in session 1; 1928 in session 2), 754 patterns were activated by the public, which makes roughly a mean of 1.4 pattern activations per second.

Take Away from Golem Shows

The first observation is that the audience actively participated in the show. This is demonstrated by the level of activity during all the interactive sessions. This shows that the concept of the interaction has been well received by different public. Unsurprisingly we notice that the pupils were in proportion much more connected than the standard public with a much better rate of connection (100% against 60%). Surprisingly, we found that the distribution of activity among the public is very similar. This means that the proportions of active, low active, or inactive people did not depend on the type of audience. We expected a more stable average activity per individual with the young population than with the public. For both populations, the diversity of behaviors we have observed led us to vary the type of interface by developing the designer interface in particular.

La Fabrique à Musique 2018/2019

The Skini platform and a composer were selected by the SACEM for the creation of a show by a class of 23 young pupils (12 years old). The goal was to create a complete musical work which was presented officially in a concert hall of the Nice Conservatory of Music. (See in French at https://www.youtube.com/watch?v=Zp0eUvIFqvk)

This project is much more than a performance. It is a complete educational approach. The Skini platform is not only used for playing music created by a single composer, but also for creating collaborative music. This means that the pupils have to first understand what is a pattern and an orchestration. They had to learn how to manipulate the designer interfaces of figure 1 using tablets. The creation of the patterns was done in small groups of 6 pupils maximum, in five sessions. The orchestration was done by the composer and validated by the class. The show took place with the pupils activating the musical work in real time.

NEXT STEPS

The simulator used during the composition process can be used as an automatic music generator by merely recording a simulation session. The extracted score is complete and can be played as is by musicians. However, the simulator is currently overly simple and the music it generates is just good enough for testing but not as a final product. The evolution of the simulator into a realistic generator is a research topic in itself that we plan to explore.

We usually run our own dedicated WiFi network for the performances, but we have also ran conclusive experiments with plain internet connections. Skini requires a very limited bandwidth as the amount of data exchanged in the communication is low. We plan to implement Skini on the Cloud as this will allow performances to be run in public contexts such as in the streets, public places, museums and stations.

Currently, Skini synchronizes the patterns created by the composer and those created by the designers, but patterns created in real-time can only be played by the designers. We would like to remove that constraint in order to let patterns created by some members being integrated in real time into the current musical piece and played by other participants. This would make it possible to achieve performances where the musical piece would be entirely created and manipulated by the audience.

In our experiments the motivation for the audience to participate in the music was high. In Golem, probably because the audience actions were part of the story telling and because the audience knew that it had something to do during certain periods of the show. For the other performances, we suppose that the novelty of the system was enough to attract attention. This interest might not last too long... In the current implementation, the actor interface is very simple. This is a conscious design decision, but we believe that this interface, being very simple, is not engaging enough because it is too disconnected from the music being played. Thus, the actors influence is not intuitive enough to feel the real impact on the musical narration. We are investing new ways to display the actions of each participants in order to provide a graphical feedback that could complement the impact on the music being played. This will certainly be possible by finding different ways of staging the score, by displaying its evolution in real time using graphical representations on a large screen. Another possibility comes with the pattern model which offers the opportunity to link non musical information to the patterns. In the first version of Skini, the simple actor received some code names for the patterns, and used it to listen to the patterns and activate them. We imagine that we can bring new ideas at this level. Instead of giving code names to the audience we can give sentences, with real meaning, and use this sentences to generate a text in real time.

CONCLUSION

Skini is a platform for interactive performance compositions. It takes into account an uncommon dimension in the composition process: the audience behavior. By contrast with classical music, where scores are used to define the musical discourse according to time, and sometimes according to the decision of the performers, in an interactive score the composer has to think about the audience behavior, the type of interactions he wants to occur according to the aesthetic of the project.

Skini is the result of the search for a solution that combines the constraints of interaction, and those of a musical discourse using scores. Skini offers a solution that is based on a few basic concepts, but whose musical implementation has proven to be complex. Indeed, we have chosen to implement our performances based on groups of musical patterns whose access is offered to the audience according to different criteria essentially related to the behavior of this audience.

Following the composition and performances of various musical pieces using Skini, we have demonstrated that our basic concepts allow to produce music pieces where listeners feel involved in the production of a coherent musical structure.

In addition, the first attempts to use Skini in music education are promising. Skini gives the opportunity to progress step by step, in a group and in interaction with a composer or teacher while following a simple methodology. This allowed to discuss with schoolchildren various musical subjects ranging from the design of simple musical phrases to the design of a complete musical discourse, in order to produce a real musical performance.

Figure 1 :

 1 Figure 1: From left to right, (i) the Actor interface, (ii) the Skini sequencer.

  , and 4 show three representions of the same simple pattern which can be played in three different ways: Figure 2 by a synthesizer, Figure 3 by a musician, Figure 4 by a sound file player.

Figure 2 :

 2 Figure 2: Example of a simple MIDI pattern

Figure 4 :

 4 Figure 4: Example of a pattern sound

Figure 5 :

 5 Figure 5: An orchestration on paper

ControllerFigure 6 :

 6 Figure 6: Logical view of Skini.

Figure 7 :

 7 Figure 7: Simple example.