Embodied enactment of a hypothetical scenario in an English medium instruction secondary mathematics classroom: A translanguaging approach

Adopting a language other than the first as the medium of instruction in teaching mathematics can create significant problems for students’ mathematical learning. However, there is little research on the ways in which bi/multilingual mathematics teachers can make use of diverse multilingual and multimodal resources to keep students engaged in learning abstract mathematical knowledge. This article aims to explore how an English-medium-instruction (EMI) mathematics teacher creates different hypothetical scenarios through embodied enactments to enhance students’ mathematical understanding and motivate students’ interest in learning mathematics. The data of this study is obtained from a focused classroom observation in a Hong Kong EMI secondary mathematics classroom. Multimodal conversation analysis is employed to analyse the classroom interaction data, and the analysis is triangulated with video-stimulated-recall interviews which are analysed with interpretative phenomenological analysis. This article aims to extend the notion of embodied enactment as a form of translanguaging to facilitate mathematical learning and develop students’ motivation in learning mathematics in the EMI setting. Particularly, this article argues that embodied enactment creates a translanguaging space for the EMI teacher to achieve mathematical teaching.


I Introduction
Learning mathematics through a second language (L2) presents additional and unique challenges. A perpetual concern has been how mathematics teachers can facilitate classroom discussion in and through an L2, particularly when the students are still developing their L2 mathematical literacies and their L2 English proficiency and are concurrently learning abstract and unfamiliar mathematical knowledge (Robertson and Graven, 2020;Tai, 2022a). Motivation and engagement are important. So is gaining understanding through the L2. Do multilingual learners know the mathematical concept, fact, method, or reasoning already in their first languages (L1s)? If yes, can we make good use of the knowledge they already have in their L1s in learning through the English medium? Another concern raised by scholars is about how mathematics teachers can use different resources to bridge the differences between the students' everyday life culture and the cultures of school mathematics (e.g. Bozbiyik and Morton, 2022;Tai & Li, 2020;Teo, 2008). Bridging such a gap will assist teachers to make the mathematical knowledge more relatable and relevant to the student's everyday life experiences (Hand, 2012;Tai & Li, 2020). The present study aims to show how a bilingual teacher in an Englishmedium-instruction (EMI) mathematics class in a secondary school in Hong Kong (HK) draws on diverse linguistic and multimodal resources to construct different hypothetical scenarios which, in turn, allows the teacher to facilitate students' learning of mathematical knowledge and promote students' interest in learning mathematics.
EMI refers to the teaching and learning of content subjects through English as an L2 (Macaro, 2018). EMI is still maintained in post-colonial regions, such as HK where the present study is based, and is increasingly popular in emerging economies where access to English is deemed to be an important step towards globalization and internationalization (Lo, 2014). Recent research on EMI classroom interaction has examined the effective role of translanguaging in supporting classroom participants to use multilingual and multimodal resources to facilitate mathematical teaching and learning (e.g. Lin & He, 2017;Mazak & Herbas-Donoso, 2015). Translanguaging scholars (e.g. Li, 2011Li, , 2018Tai, 2022a;Tai & Li, 2020, 2021a, 2021b, 2021cVallejo & Dooly, 2020) have emphasized the significance of drawing on different multilingual and multimodal resources and various sociocultural knowledge to challenge the divides between the so-called 'named languages' and the non-verbal communication cues since these resources are all part of the repertoire of meaning-and sense-making resources. Existing classroom translanguaging research (e.g. Li, 2014;Wu & Lin, 2019) has demonstrated that translanguaging offered rich affordance for improving multilingual students' metalinguistic awareness and creating a safe learning environment which built on students' funds of linguistic and cultural knowledge.
Research on mathematics education has emphasized the importance of multimodal resources in facilitating the process of teaching mathematics. Mathematics education researchers (e.g. Alibali & Nathan, 2012;Chikiwa & Schäfer, 2019;Chu & Kita, 2011) have argued that gestures are important resources to visualize mathematics, facilitate the construction of mental representations of mathematical concepts and complement the speaker's spoken language. Nevertheless, there is a lack of research that investigates specifically how and why multiple linguistic and multimodal resources are mobilized by mathematics teachers in EMI settings which do not encourage the use of students' first language (L1) (e.g. Li, 2018Li, , 2021Tai & Li, 2020). More research studies are needed to uncover how mathematics teachers use different resources to challenge the monolingual EMI policy in their efforts to make mathematical knowledge accessible to multilingual students.
Based on data collected from a 2-week focused classroom observation in HK EMI secondary mathematics classrooms, this article explicates how the mathematics teacher engages in embodied enactment which entails playing out different hypothetical scenarios, verbally and physically, in order to support students in understanding the mathematical concepts and advice, and build up students' motivation in learning mathematics. In this article, we define embodied enactment as an interactional practice which refers to the participants' use of embodied resources, such as gestures and body movement alongside verbal utterances, to represent an aspect of hypothetical events. It can be considered as an interactional phenomenon in which the teacher physically creates a hypothetical context for students to understand how the L2 can be used in everyday life communicative contexts (Tai & Brandt, 2018). Classroom video data, fieldnotes, ethnographic interviews with teachers and other stakeholders, and video recordings form the main database. Multimodal conversation analysis (MCA) is carried out on the classroom-interactional data, looking at not only different languages (Cantonese and English in this case) but also spatial repertoire, the use of objects and other facilities in the classroom space. The analyses of the classroom-interactional data are triangulated with the video-stimulatedrecall-interview data which are analysed using interpretative phenomenological analysis (IPA) in order to analyse the teacher's reflections on his pedagogical and interactional strategies.

Reconceptualizing embodied enactment from a translanguaging perspective
Translanguaging refers to the process that speakers draw on their full linguistic and semiotic repertoire to make meaning (Li, 2018). Translanguaging transcends the boundaries between different named languages and also between different modalities (e.g. speech, sign, gesture) (Li, 2018(Li, , 2020. Additionally, Li (2011) proposes the notion of 'translanguaging space' which is an interactional space enabling multilingual, multimodal, multisemiotic and multi-sensory repertoires to interact and co-produce new meanings. It is also a space for multilinguals to 'bring together different dimensions of their personal history, experience and environment; their attitude, belief, and ideology; their cognitive and physical capacity, into one coordinated and meaningful performance' (Li, 2011(Li, , p. 1223). There are a number of research studies that investigate the construction of translanguaging spaces in EMI mathematics classrooms. Tai and Li (2020) illustrate the ways in which an EMI mathematics teacher brings the students' everyday life space into the classroom in order to transform the classroom into a lived experience for students. This allows the teacher and students to bring their funds of knowledge to the forefront which makes the mathematical knowledge more relatable and relevant to the student's everyday life experiences. Additionally, Tai and Li (2021a) illuminate the potential of playful talk in transforming the EMI classroom into a translanguaging space, which allows the teacher to bring in various linguistic and multimodal resources and different kinds of knowledge to perform a range of creative acts for facilitating mathematical learning and promoting meaning communication. A recent study by Zhu et al. (2020) has investigated the role of embodied repertoires in teaching and learning. In a study of a multi-ethnic and multilingual karate club in East London, Zhu et al. (2020) have shown how a karate instructor orchestrates various multilingual and multimodal repertoires creatively and critically to support the learning of Japanese karate terms and his teaching. The authors argue that translanguaging space is where 'all the semiotic systems are integrated and orchestrated' (p. 65) to make and communicate meaning. Zhu et al. (2020) argue that the orchestration of embodied resources and spoken words and utterances is a process of translanguaging since the notion of translanguaging emphasizes the importance of going beyond the boundaries between language and other semiotic means of communication, in social interactions.
In the MCA literature on enactment, scholars have considered enactments as a distinct form of interactional and embodied practice. Wilkinson et al. (2010, p. 58) define enactment as 'the employment by participants of direct reported speech (DRS) and/or behaviour, such as the use of gesture/body movement and/or prosody to depict to recipients some aspect(s) of a reported scene or event'. By employing DRS, a speaker can create 'a version of not only what was said but how it was produced through prosody, voice quality, body movement, and linguistic selections' (Kasper & Prior, 2015, p. 244). This allows the interlocutors to mobilize non-verbal and embodied resources to recreate the authenticity and immediacy of the reported scenes or events (Holt & Clift, 2007). Nevertheless, prior studies on DRS and enactment typically examine participants' interactions related to past scenes or events. Recent studies have further reconceptualized the notion of enactment, and they take the analytical stance that enactment should be a broader category which entails that speakers utilize both linguistic and non-linguistic resources to produce past, future and imagined scenarios with a range of characters in order to demonstrate particular ideas, instead of verbally describing them (Arita, 2018;Leyland, 2016;Sert, 2017). Leyland (2016) has investigated how a Japanese English teacher and native English-speaking teaching assistant enact possible future classroom activities involving objects that the participants have access to. By enacting a vision of a forecasted future scenario, this facilitates teachers' lesson planning discussions. Arano (2020) coins a term called 'embodied solitary confirmation' which refers to the interactional phenomenon of speakers gesticulating what they learnt from upon completion of an instruction activity. Arano argues that the process of enacting what one learnt from the instruction enables the speakers to demonstrate their understanding of the instructed tasks. Such a phenomenon supports Tai and Khabbazbashi's (2019a) argument that the interactional procedure of enactment can work as a 'window' to understand the current state of speakers' knowledge in the learning process. In the context of English for speakers of other languages (ESOL), Tai and Brandt (2018) coin a notion of 'embodied enactment', and they have demonstrated how the ESOL teacher provides several hypothetical everyday scenarios to explain vocabulary items (e.g. 'excuse me'), thereby creating an environment to familiarize the students with the specific meaning of the vocabulary. In other words, embodied enactment does not consider gestures as a supplement or aid to the teacher's verbal explanations. Rather, embodied enactment involves 'more than simply using [one's] body to emphasize or add visual description of a concept' (Tai and Brandt, 2018, p. 262). As shown in Tai and Brandt (2018), the ESOL teacher physically creates a hypothetical context for students to understand how the target language can be used in specific situational contexts.
To date, these CA studies on enactment have focused on L2 learning contexts or everyday life social interactions. There is a need for such studies in the EMI context, as constructing embodied enactment is also relevant to EMI teachers' pedagogical practices. The present study aims to contribute to the current literature on translanguaging and mathematics teaching and learning by investigating the role of embodied enactment in creating hypothetical scenarios for achieving the teacher's pedagogical goals. In this article, we aim to argue that embodied enactment is a translanguaging phenomenon which creates a translanguaging space for participants to translanguage fluidly between registers, styles, languages, as well as across modalities in order to support meaningmaking processes. By doing so, it plays a role in supporting students' mathematical thinking and motivating students' interest in learning mathematics.

Use of multimodal resources in mathematics learning
Prior research on mathematics education has captured the role of multimodal resources in developing students' conceptual understanding of mathematical knowledge. As Tran et al. (2017, p. 3) argue, before abstract forms of mathematical ideas are emerged, people manipulate diverse resources to solve mathematical issues in the real world. Such an argument is further reinforced by mathematics education researchers who call for future research to adopt multimodal conversation analysis in order to investigate how mathematics classrooms are interactionally organized and how specific mathematical practices are accomplished (Abrahamson et al., 2019;Krummheuer, 2011;Ingram, 2018). As argued by Krummheuer (2011), mathematical learning is locally produced in the interaction. That is, the learning of mathematical knowledge and understanding is inherently locally constructed at particular moments of classroom interactions. Future research studies in mathematics education are encouraged to investigate how different languages and modalities have different affordances for mathematics learning, making it important for researchers to explore their potential for meaning-making. This can only be investigated through a detailed analysis of teachers' and students' actual practices.
A number of research studies in mathematics education have shown how a teacher's gestures can affect students' construction of mental representations of mathematical concepts (e.g. Alibali & Nathan, 2012;Chikiwa, 2021;Chikiwa & Schäfer, 2019;Yoon et al., 2011). Chikiwa (2021) is one of the recent studies that adopt embodied cognition as a theoretical framework to understand mathematics teachers' use of gestures to foster students' learning. Chikiwa explores two grade 11 secondary teachers teaching trigonometry for a week, and the findings illustrate how the teachers use iconic gestures to offer visual resemblances with the trigonometric concepts, including angles, triangles, height and adjacent sides. The teacher also makes use of pointing gestures to direct students' attention to the diagrams. Chikiwa argues that all forms of teacher gestures complement their use of verbal languages, and gestures can be used effectively to mediate and scaffold students' learning of mathematics. On the other hand, Yoon et al. (2011) explore how two mathematics secondary teachers in New Zealand use gestures to communicate mathematical concepts and create a physical mathematical gesture space for mathematical learning. The findings demonstrate that teachers use different gestures to visualize and make sense of the mathematical features of the anti-derivative graphs that they construct. It is argued that the creation of mathematical gestures can enable students to travel from context-embedded mathematical thinking to more abstract kinds of mathematics. A recent literature review conducted by Tran et al. (2017) highlights the impact of technology on affecting teacher's and students' embodied activities. The review concludes that the accessibility of technology, such as digital touch screens and interactive whiteboards, can improve the teacher's pedagogical methods for embodied learning of mathematics. However, teachers need to be aware of how to integrate technology effectively for enhancing students' mathematical learning since not all embodied resources are useful for teaching all mathematical concepts to all students. Such an argument is also reflected in a recent study by Tai and Li (2021c). Although the authors have demonstrated the affordance of iPad in allowing the EMI mathematics teacher to create a technology-mediated translanguaging space, they argue that 'such a technology-mediated space may not be always perceived as a safe atmosphere in the classroom' (p. 47), and, for example, an iPad must be used with clear pedagogical intentions. In the analysis of the interaction, they have shown how the teacher uses the iPad's camera function to take photos of students in order to create a humorous atmosphere. Nevertheless, it can be suggested that the teacher is creating an unsafe space for students whose pictures are being taken since such an action could dampen the students' self-esteem, which may trigger fear or depression. Hence, it is necessary for mathematics teachers to harness the available linguistic, semiotic and spatial repertoires strategically and appropriately in order to achieve his/her pedagogical goals.

EMI in HK
The nature of EMI policy goes against the notion of translanguaging, which challenges the necessity of only using one named language (L1 or L2) as the medium to teach subject matters such as mathematics. The EMI policy has been implemented in HK which restricts teachers and students to use English-only in the classroom, which may hinder students from drawing on their knowledge and skills in their L1 and restrict opportunities for teachers and students with shared linguistic and cultural backgrounds to communicate effectively (Lin, 2019;Lin & He, 2017). As Li (2021) argues, the promotion of English 'as the language of science, knowledge, and internationalization is an ideological act' (p. 178), and such a monolingual policy contributes to the institutionalization of linguistic and social inequalities.
HK is a uniquely suitable context for this study because English, being the official language of the former colonial era and current international language, is economically valued in HK's society, and there is a strong preference for EMI among parents and students because EMI schools are perceived to be more prestigious and allow students to acquire English more effectively (Choi, 2003;Tollefson & Tsui, 2014). The selection of medium-of-instruction in the educational system has been a highly controversial issue in HK, where the majority of the citizens speak Cantonese as their L1. Under the latest medium-of-instruction policy in 2010, the government implemented the fine-tuned medium-of-instruction policy by allowing Chinese-medium-instruction schools that have met certain requirements to have some approved EMI classes (Lo & Lo, 2014). As a result, lots of secondary schools are teaching at least some subjects through EMI. A rough estimation based on the Secondary School Profile in 2019-2020 indicates that around 30% of secondary schools adopt EMI throughout all the grades and approximately 40% adopt EMI for at least one academic subject.
Several research studies have suggested that adopting EMI in secondary schools provides limited opportunities for students to participate in classroom interaction. This is because the English-only rule prevents students from using their familiar languages to respond to the teacher's questions or self-initiate turns in classroom interactions. In addition to the language barrier imposed by the EMI policy, HK mathematics education researchers have identified features in the traditional mathematics teaching practices in HK EMI classrooms that limit opportunities for students to participate in classroom discussions. It is suggested that EMI teachers tend to adopt the lecture format to teach the subject (Tollefson & Tsui, 2014), and most of the teacher-talk time focuses on demonstrating solutions to mathematics problems, with little employment of multimodal resources beyond blackboard and chalk (Mok, 2019). Lo (2014) suggests that EMI mathematics lessons may not promote classroom interaction between the teacher and students. Lo argues that this is possibly because the lessons typically involve solving mathematical equations and following calculation procedures. As Mok (2019) argues, although the HK mathematics curriculum has recognized the importance of students' participation in the process of mathematics learning, mathematics teachers remain the important people in charge of the classroom events, and the teaching is often guided by the teacher and by the tasks in the lessons.
Nevertheless, recent research on EMI mathematics classrooms (e.g. Tai & Li, 2021a, 2021b, 2021c has demonstrated how EMI mathematics teachers creatively employ various linguistic and multimodal resources to transform the traditionally teacher-fronted interaction and create a space to engage students in learning mathematics and encourage students to develop positive attitudes towards mathematics learning. This, in turn, constructs a more dynamic and contingent environment to facilitate students' participation (Baynham, 2006;Bowden et al., 2022;Lee, 2010). The research findings observed by Tai and Li suggest the need for researchers to move away from the narrow view of EMI mathematics classrooms, which provide limited opportunities for students to interact with the teacher, to a perspective of a creative and socially oriented classroom which prioritizes developing students' mathematical understanding. As highlighted by Chronaki (2000), there is a growing number of students who become disinterested in learning mathematics in schools, due to the fact that learning mathematics becomes irrelevant to students' personal lives. Chronaki (1998Chronaki ( , 2000 argues that there is a need for mathematics teachers to structure their classroom interactions for empowering students to acknowledge and utilize mathematics meaningfully in their lives. In the current study, it is evident that the teacher takes mathematics as procedural and formulaic to memorize, as it is the traditional approach in HK secondary mathematics education: stressing memorization and practising mathematical skills (e.g. Leung, 1995;Lo, 2014;Mok, 2019). Nevertheless, the teacher's use of translanguaging strategies and bringing everyday life knowledge into the classroom is an important first step towards reconceptualizing mathematics teaching in this specific EMI context. To date, there are limited research studies that explicate translanguaging practices in EMI mathematics classroom interactions (Tai & Li, 2020). Hence, this article aims to fill in the research gap by showing how the EMI mathematics teacher constructs hypothetical scenarios through embodied enactments in order to create a translanguaging space that engages students in learning mathematics and going beyond the traditional emphasis of demonstrating solutions of mathematics problems.

II Data and method
The participating school is a prestigious EMI secondary school in HK and it offers education from year 7 to year 12 based on the curriculum guides set by the HK Education Bureau. The school uses EMI to deliver most of the lessons, and the school examinations are conducted in English (except Chinese, liberal studies and Putonghua). Although the school's mission statement explicitly states the aim to develop students to be multilinguals, the school language policy places heavy emphasis on the use of English on the school campus in order to create a rich English learning environment for all students. For instance, all morning assemblies and staff meetings are conducted in English. Moreover, English-for-all-day is held every Monday where everyone (all teaching staff and students) in school must use English for communication in order to prepare students to interact effectively with other individuals from other countries.
The mathematics teacher has at least eight years' experience in teaching mathematics in English. The teacher is an L1 speaker of Cantonese and previously attended an EMI school for his secondary and university education. English is his L2 and he has a limited level of Mandarin/Putonghua proficiency. During his undergraduate studies, he occasionally taught drama at several HK secondary schools. He did not receive any specific EMI teacher training while he was pursuing his education degree.
During the two-week classroom observation, the first author observed a year 9 class. There were 18 students in the class and this class was classified as an enhancement class. Students who ranked below average in their cohort in the internal mathematics examination were enrolled in this class. All students had received at least six years of primary education, where Cantonese was employed as the medium of instruction and English was taught as an L2. All students in the class spoke Cantonese as their L1 except two students. The two students spoke Mandarin/Putonghua as their L1s and Cantonese and English as their additional languages, and they were migrants from mainland China. A total of 11 40-min lessons were observed and video-recorded in April 2019.
A semi-structured interview, which lasted for an hour, was carried out with the teacher in order to understand his perceptions regarding teaching mathematics in EMI settings and his attitude towards using multiple languages in the EMI mathematics classrooms. Ten ethnographic interviews were conducted with the teacher during the two-week observational period in order to better understand the observed lessons. Each ethnographic interview lasted for approximately 10 minutes and ethnographic interviews were carried out while the first author was walking back to the staff room with the participating teacher. These ethnographic interviews are meant to complement what we observe in the EMI classrooms and provide the teacher with an opportunity to reflect on his own practices and attitudes (Rampton, 2006;Spradley, 1979). Three post-video-stimulatedrecall interviews were conducted with the teacher in order to compare his actual translanguaging practices and his interpretations of the practices.
This study integrates MCA and IPA to study the functions of translanguaging practices in EMI mathematics lessons (Tai, Forthcoming). MCA 'focuses on how social order is coconstructed by the members of a social group' (Brouwer & Wagner, 2004, p. 30) through fine-grained analysis of the social interaction. It takes an emic/participant-relevant approach (Markee & Kasper, 2004) in order to explicate the detailed process of how social activities, such as learning, are co-organized and achieved through talk-in-interaction. Using MCA to analyse mathematics classroom interactions enables us to understand what it means to learn mathematics, do mathematics and understand mathematics. The micro-analysis of the interaction also opens up opportunities for researchers to identify how mathematical learning is achieved through the mobilization of different languages and multimodal resources. The data are transcribed using Jefferson's (2004) and Mondada's (2018) transcription conventions (see Appendix 1). In the analysis, although we use conventional linguistic labels, such as L1 and L2, to describe the named languages that the participants know, the analytical focus is on empirically observable practices. Instead of demonstrating which named languages to use in a particular classroom moment, we wish to reveal how speakers use their linguistic and multimodal repertoire from which they select resources to achieve their communicative goals (Zhu et al., 2020). The first stage of analysis involved taking a stance of 'unmotivated looking' (Mori, 2004) as the guiding principle when reviewing the videorecordings. The first author watched multiple classroom videos with an open mind (i.e. without any particular interest in research focus) to discover any interesting interactional phenomena that are worthy of further exploratory analysis. After transcribing the data, both the first and second authors carried out line-by-line analyses to closely investigate various sequences-of-talk that entailed the teacher mobilizing diverse resources to facilitate students' imagination of a scenario. In order to ensure the reliability of the analysis, we made use of the next turn proof procedure and participant orientation for validating the analyst's claims. In other words, we as analysts were required to make observations based on the participants' observable orientation and understanding.
The framework of IPA was also employed to examine how the mathematics teacher made sense of his own pedagogical practices at specific moments in the interaction and how the classroom interactions were shaped by multiple sociocultural factors. IPA acknowledges the investigation of the meanings of the participants' experiences as an interpretative enterprise on the part of both researcher and participants. A dual interpretation process called 'double hermeneutic' is involved, and this requires researchers to take an emic approach in order to make sense of the participants trying to make sense of their world (Smith et al., 2013). Both authors conducted the IPA analysis together and we followed the analytical stages suggested by Smith et al. (2013), moving from a descriptive level to a more interpretative level. In order to enhance interpretative validity, iterative coding with constant comparison was conducted. This process involved us constantly checking our sense making against what the participating teacher had actually said in the interview.

III Analysis
In the analysis, we will present three examples of how the EMI teacher constructs embodied enactment for achieving his specific pedagogical goals. These extracts are representative of instances of the interaction. The analysed extracts are interrelated to illustrate the typical instances of translanguaging practices in the EMI classroom which construct hypothetical scenarios for facilitating mathematics learning and arousing students' interest in mathematics (ten Have, 1990). The goal of MCA analysis is to identify the interactional phenomenon in the social interaction, rather than justifying the best possible representative extracts (ten Have, 1990). Therefore, as long as the selected extracts can address the research questions to reveal the relevant 'orderliness' with their representative nature, it can be said, to a large extent, that the representativeness is sufficient, or the research findings can be reliable.

Extract 1: Creating a hypothetical scenario of computer war games
The mathematical question in Extract 1 requires students to determine the time when the aircraft will be detected by the radar. This mathematical question aims to assess students' ability to solve trigonometric problems involving bearings. The question presents two locations (points A and B) which are 100 km apart. The compass bearing of B from A is N75°E. The figure indicates an aircraft which departs from point B and flies at a speed of 150km/h along S50°W. The first question requires students to identify whether the aircraft will be detected within 50 km from point A. The second question requires students to determine when the aircraft will be detected after departing from point B. Students are expected to give their answers correct to the nearest second. In order to answer this question, the operative processes of addition, subtraction and division will be involved in the search for the answer.
In Extract 1, the teacher (T) read out the key information of the question in English, and he drew out the diagram on the blackboard. He then employed Cantonese to ask students to determine when the aircraft would be detected.

Extract 1. Creating a Hypothetical Scenario of Computer War Games.
In line 133, T rephrases his question in Cantonese and asks students to consider at which point the aircraft will be detected. While he is speaking, T is drawing several small 'x's along the straight line (Figure 1) in order to visually illustrate the possible moments when the aircraft will be detected. After a 0.5-second silence, T repeats his questions in English, 'so many points right?', which possibly allows students to better understand his question. However, no student responds to T's question during the 0.5-second pause in line 134. In line 135, T continues to build on the previous hypothetical scenario, which was constructed prior to Extract 1 in order to encourage students to imagine 'when the plane flies along this line'. While he is providing the English explanation, T curves his leech finger and middle finger and straightens his thumb, index and little fingers out in order to imitate the shape of an aircraft (Figure 2). He then moves his right hand along the line from low to top position in order to visually illustrate the movement of the aircraft. After T utters 'okay like this', T enacts the same gesture from lines 136-141 to visually illustrate the shape of the aircraft (Figure 3 and Figure 4). By doing so, T is multimodally creating a hypothetical scenario of an aircraft flying along the path, and it encourages students to work out at which moment the aircraft will be detected by the tower.
It is noted that no student responds to T's question (line 140), and this is possibly due to students' shyness or their unwillingness to respond to T's question in English (Lo, 2014;Mok, 2019). In order to engage students in participating in the interaction, T repeats his question in Cantonese in lines 141 and 143 which possibly aims to make his question more comprehensible to the students. T's repetition of his question does not lead to any students' response, and this motivates T to rephrase his question and encourage students to imagine themselves having a pilot, '如果你有個機師 (if you have a pilot)' (line 145). T then continues to ask students to imagine the situation when they do not wish their pilots to be detected by the tower, '唔想俾佢 detect (you do not wish the aircraft to be detected)' (line 147). Simultaneously, T points at point A on the blackboard in order to visually specify the location of the tower. After constructing the scenario, T launches a question by asking students to predict how far the pilot needs to distance the aircraft from the tower (line 147). After a 0.5-second pause, student 1 provides an answer (i.e. 51) in Cantonese (line 149) and such an answer is accepted by T in line 151.
In Extract 1, it shows that T heavily draws on the multilingual (Cantonese and English), multimodal and semiotic resources (i.e. drawings on the blackboard and gestures) to facilitate students' imagination of a pilot preventing him/herself from being detected by the tower. Although the teacher and students are supposed to use English as the main linguistic code to teach and learn mathematics in an EMI context, the teacher chooses to employ different linguistic and multimodal resources to go beyond the top-down EMI policy that promotes a monolingual ideology in order to achieve his pedagogical goal (i.e. facilitating the teaching of a complex mathematical question) (Tai & Li, 2020, 2021a, 2021b. Such an embodied enactment supports students' understanding of the mathematical problem through bringing students' own knowledge of a pilot flying an aircraft into the classroom interaction. During the video-stimulated-recall interview, the researcher is interested to understand the rationale of T's use of gesture to visually illustrate the shape of the aircraft and move his hand along the straight line on the blackboard for indicating the movement of the plane (Table 1). fingers pointing downward)) ((tr. So whenever I need to use gestures to imitate the shape of a plane, I will use this way to represent an aircraft.)) T expressed his personal interest in the plane which appeared in the cartoon.
T expressed his personal interest in this cartoon. Clearly states that this cartoon is part of T's childhood memory.
T states that the cartoon inspires him to use gestures to represent the feature of this particular plane from the cartoon. T's personal interest in this particular plane and this cartoon shape his use of gesture in representing a plane in the classroom. T's childhood memory shapes his use of gestures in representing a plane in the classroom.
The researcher first questions T's rationale for using that particular gesture (i.e. Figure  2) to illustrate the shape of the aircraft. This is because the researcher believes that using an index finger to point at the straight line on the blackboard could also draw students' attention to the straight line. T then mentions a cartoon called 'Macross Seven' that he previously watched. 'Macross Seven' is a Japanese science fiction cartoon, and the theme of this cartoon is space war. Particularly, he is impressed by the plane in the cartoon and this motivates T to use gestures to adopt the shape of the plane, '因為我鍾意呢個，啲 嗰個飛機呢 (because I really like the plane) (line 13)'. T also further comments that such a cartoon was part of his childhood memory, '細個睇卡通片嗰時成日睇 (I always watched this cartoon when I was a child)' (line 16). Hence, it can be argued that T's particular interest in the plane and his childhood memory of this cartoon inspires him to use gestures to imitate the shape of the plane. This also explains why T chooses to use this gesture to represent a plane, '我每次要用飛機我都會咁樣做呢個形式嚟顯示 (whenever when I need to illustrate a plane, I will use this gesture to represent it)'. It is observable in the MCA analysis that T makes use of his thumb, middle and last fingers to visually illustrate the shape of an aircraft, which is similar to the plane in 'Macross Seven' (Image 1). T extends his thumb and last fingers in order to mirror the wings of the aircraft, and the middle finger represents the aircraft's fuselage.

Image 1. Aircraft in 'Macross Seven'.
Thus, such creative use of gestures, accompanied by the teacher's use of multilingual resources, shapes the construction of the hypothetical scenario of a pilot driving an aircraft in order to help students to understand the complexity of the mathematical question on bearings. It can be argued that the production of an embodied enactment, involving the mobilization of a gesture of an aircraft and multilingual utterances, provides a translanguaging space for T to bring his personal interest in the cartoon and his childhood experience into the classroom. It is a translanguaging space which allows classroom participants to reveal their knowledge and experience of the social world in order to negotiate and create new meanings.

Extract 2: Creating a hypothetical scenario about securing a nation
Extract 2 is the subsequent part of the interaction in Extract 1, a day after Extract 1. Prior to this extract, T is continuing with the explanation of the same mathematical question. In this extract, T is engaging in an extended discussion with students after the students understand the answer to the question.
In lines 1-3, T reiterates the answer in Cantonese and explains that after 1,534 seconds, the plane will be detected by the radar. In line 5, T asks a rhetorical question in Cantonese to invite students to imagine themselves as secretary for national security, '如 果你係國防部長+你就要點啊 (if you are the secretary for national security, what would you do)'. This is accompanied by his enactment of a salute gesture in order to facilitate the construction of the hypothetical scenario. After a 0.7-second silence in line 5, T points at the answer on the blackboard (i.e. 1,534 seconds, Figure 5) and verbally explains to students that they have to decide within 1,534 seconds in terms of their decision of attacking the aircraft (lines 5-7). Before T further elaborates on the hypothetical scenario, he first utters '$okay?$' (line 9) in English and then switches back to Cantonese in order to ensure students understand his messages. T explains in Cantonese that students need to calculate the speed of the missile in order for it to attack the plane (line 9). Subsequently, T switches back to English to highlight the significant role of mathematics in assisting the secretary to plan a decision, 'so it's all about math okay . . . very beautiful right? ' (lines 11-13).
In lines 18-19, it is evidenced that T switches from an instructional frame to a hypothetical frame for constructing an embodied enactment. In line 18, T imagines himself as the secretary calculating how fast the missile can be launched to attack the aircraft. This is illustrated as he utters the English mathematical terms, 'cosine (0.2) sine', to pretend that he is considering the appropriate mathematical functions to solve the equation. T then switches back to Cantonese and enacts a beckoning gesture (line 18) as he pretends to seek assistance from a helper, '唔係 (.) helper 唔該 (no helper please)', in order to make the scene more dramatic.
In line 19, T mobilizes linguistic and non-linguistic cues in order to switch between the character views of a secretary and the assistant. T first imagines himself as the assistant who is offering verbal advice to the secretary: '部長你仲有十秒咋 (.) 你仲有十秒 就要計完 (Secretary, you still have 10 seconds left, you need to finish calculating within 10 seconds). Here, T deliberately repeats the Cantonese phrases '仲有十秒' (still have 10 seconds)' in order to stress the urgency for the secretary to finish his calculation for decision-making. While T is speaking, T is directing his gaze on his desk which enables him to enact the role of a secretary who is working on a mathematical solution. Afterwards, T enacts the gesture of pressing a button (Figure 7) and switches back to the instructional frame by inviting students to advise the 'assistant' in terms of making the decision to launch the missile, '就要啡唔啡佢呢 (should we attack or not)'.
Student 11 self-initiates a turn and questions whether it should be the headquarters launching the missile (line 21). S11's participation in the interaction motivates T to alter his narrative as he adopts the character view of the secretary (line 22). It is noted that T enacts the action of picking up a phone call ( Figure 8) and uses Cantonese to imaginatively request headquarters to seek an attack order within 10 seconds. Here, T's Cantonese utterances are spoken faster to further highlight the urgency for requesting the headquarters to attack the aircraft. However, in line 23, T deliberately divides his Cantonese utterances into 'chunks', as characterized by the short pauses, in order to illustrate the unsuccessful attempt in receiving the order in time from the headquarters. This is further emphasized as he moves his right hand to his left-hand side in order to visually illustrate the plane has flown past the radar (Figure 9).
In Extract 2, T mobilizes various linguistic resources (Cantonese and English) to construct different ways of speaking, including imitating the voice of a secretary of national security and the head of national security. The use of different linguistic resources affords the teacher to switch between character views in order to create various discursive identities and make the hypothetical scenario more vivid for the students. Along with the use of linguistic resources, T also draws on diverse multimodal cues, including eye gaze, gestures and speech pace, in order to dramatically illustrate the hypothetical dialogue between a secretary for national security and his assistant (see Figure 6, Figure 7 and Figure 8). By doing so, T aims to demonstrate the purpose of using bearings in trigonometry for solving real-life issues such as safeguarding national security. During the video-stimulated-recall interview, T is invited to comment on his motivation to create such a hypothetical scenario (Table 2). ((tr. Yeah, so haha. Yeah. They feel that this lesson was interesting or maybe they will think that mathematics in interesting. I think they will have greater motivation to learn mathematics.)) 07 K: um! 08 T: 係啦，即係就算讀得唔好但係起碼都，即係佢哋唔係呢個材料，但係都肯讀囉 ((tr. Even though they are not good at mathematics and they don't have the talent, at least they will put in effort in learning mathematics.)) T aims to offer an example that has no direct relevance to mathematics in order to arouse students' interest and deepen their understanding of the purpose of doing maths.
T aims to lighten the mood in the classroom and motivate students' interest in learning mathematics.
Acknowledging the students' weak ability in learning mathematics.
T believes that the students need more support and encouragement in learning mathematics. Hence, T's strategy is to create a humorous context in order to bring mathematics to life.

(Continued)
Classroom-interaction transcript Haha. That can allow students to notice that: 'hey the guy out there is from the department of national security. He is also doing something similar like us.' Haha. So that they will be immersed into the whole situation. If the whole drama is not well-performed, the students won't be engaged. So yeah)).
T attributes his skills of adopting various character views to his experience of engaging in drama activities.
T aims to encourage students to imagine himself as the Secretary for National Security so that they can be immersed in the hypothetical context.
The researcher acknowledges the role of switching between character views in order to construct various discursive identities.
T's prior experience in drama activities enables him to adopt different discursive identities and construct the hypothetical scenario.
T shifts his footing by imagining his students' reactions when they are looking at T's dramatic performance.
T argues that creating a situation related to national security can arouse students' interest and deepen their understanding of the purpose of doing mathematics. It encourages students to think about the significance of mathematics in real-life situations. He quotes an English phrase from the classroom interaction, 'it's all about math', in line 2 in order to reiterate his pedagogical goal for creating such a hypothetical context in the classroom. Moreover, T also aims to lighten students' moods and motivate students' interest in learning mathematics. In the interview, T acknowledges that the students in this enhancement class need further support and encouragement for learning mathematics. In order to achieve his goal of developing students' interest in learning mathematics, T's strategy is to create a humorous context in order to bring mathematics to life and motivate students to immerse themselves in the hypothetical context (lines 4-20). In order to successfully create the hypothetical context dramatically, it is noticeable in the MCA analysis that T enacts two characters simultaneously by using his Cantonese and English utterances, eye gazes and gestures in order to vividly depict the discursive identities of the secretary and the assistant. T attributes his skills of adopting various character views to his experience of engaging in drama activities when he was a secondary and university student. Particularly, T shifts his footing by imagining himself as his students' reactions when they are looking at T's dramatic performance, '啊，出邊望緊嗰個人係 國防部 . . . 佢都咁樣做咁樣囉' (hey the guy out there is from the department of national security. He is also doing something similar like us), in order to display his goal of enabling students to immerse themselves in the hypothetical scenario. Therefore, it can be argued that T's pedagogical goal of bringing mathematics to life and also his prior experience in acting shape his embodied enactment in adopting different discursive identities and constructing the scenario accordingly.

Extract 3: Enacting a moment of fighting
In Extract 3, students need to find the coordinates of a point through the condition for parallel lines. Specifically, students are asked to draw on the equation for calculating the length of a slope (i.e. m = (Y2−Y1) ÷ (X2−X1)). Students are expected to first substitute the coordinates of a point into the equation and subsequently follow the operational process of subtraction and division in order to generate the value of the slope.
The mathematical question provides three points (A, B and C) to students. Each point involves the values of x-axis and y-axis. The three points A (1, 7), B (4, 5) and C (3, 2) are provided. The first question requires students to find the slope of BC. The second question involves students in searching for the coordinates of D if the straight line passing through point A and parallel to BC intersects the y-axis at point D.
Prior to the extract, student 4 (S4) pointed out that inserting a formula into the calculator could allow her to obtain the same result and it was unnecessary to understand the equation. T explained that students need to show the mathematical steps when answering extended questions, similar to the mathematical question that they had to solve in class. S4 continued the debate with T, and she noted that students could use the calculator to check their answers. T agreed with S4's suggestion, and he reminded students not to rely on the calculator in case they forgot the formula when answering extended questions. In Extract 3, T is enacting a hypothetical context and pretending he is a fighter who is trying to solve a mathematical equation.
Before T offers advice to students, T is going through a question as he utters the English letters and writes down the corresponding mathematical value on the blackboard. T then initiates a side sequence in lines 104 and 110 and switches to Cantonese in order to explain that setting a formula in a calculator is like a heterodox manner, '好 多旁門左道嘅方法 (the outside world includes lots of heterodox ways)'. In line 112, when T explains the need for students to establish a strong basis for doing mathematics, '打好過根基', T slightly kneels down and places his right hand at his waist, which potentially indicates to students that what is going to come is performative. After a 0.5-second pause, T first holds his fist and enacts a fighting gesture ( Figure  10) as he explicitly mentions the need to deal with the questions, '對付嗰啲題目' (line 114). Such an enactment allows students to imagine T as a fighter who has solid training and the ability to solve mathematical questions. In line 116, T reiterates the need for a soldier to build up a solid basis for fighting in Cantonese, which signals the end of the enactment.
In Extract 3, T's embodied enactment of a fighter is represented through his use of body movements, gestures and his Cantonese utterances. By doing so, T aims to emphasize the need for students to develop a strong mathematical knowledge basis in order to address the mathematical questions easily in the examination context. In the MCA analysis, T and students portray the fact that the goal of learning mathematics is to manage school examinations through memorizing formulas. It is evident that school examination culture is apparent in the extract since students in the remedial class are expected to enhance their school mathematics examination results (fieldnotes, ethnographic interview with T), and T in this extract encourages students to understand the purpose of applying the memorized equation when calculating the value of a slope, instead of merely using a calculator to obtain the value. It can be suggested that such a vision goes against the perspective of a socially oriented mathematics classroom that prioritizes students' development of conceptual understanding. In the video-stimulated-recall interview, T is invited to comment on how the embodied enactment enables him to achieve his pedagogical goal of preparing students for the school examination (Table 3). in order to find out the length of the slope, the calculator will not display the whole formula on the screen for students. So, it's easy for students to forget about the actual formula. Creating such an imaginary context can alert students so that they are aware of the possible situation that may happen to them. Therefore, students can prevent making such a mistake.)) T explains the rationale of creating an imaginary context of a soldier fighting. T aims to make it more interesting for students to understand his advice.
T is explaining how entering the 4 values (i.e. the values of x-axis and y-axis) can allow students to find out the length of the slope. However, T notes that it does not allow students to see the actual formula on the calculator's screen.
Students may easily forget the formula and the goal of creating the hypothetical scenario is to draw students' attention to the importance of memorizing the formula. The researcher is interested to know how the embodied enactment enables T to achieve his pedagogical goal.
Such an imaginary context serves as a metaphor to students as they are the soldiers who will need to develop a solid knowledge base to solve mathematical equations.
The researcher acknowledges the playfulness of T's creation of the hypothetical context. T is drawing on his prior experience of doing mathematics. He recalls a moment when he forgot the formula when he was working on an extended question. Such an experience motivates T to create a hypothetical context in order to give a heads-up to his students.
In the interview, T first explains the rationale of creating a hypothetical scenario of a soldier fighting against enemies. T aims to make it more interesting for students to understand his advice. Metaphorically speaking, the students are the 'soldiers' who will need to develop a solid knowledge base and solve the mathematical equations accordingly. T further emphasizes the need for students to memorize the mathematical formula so that they are equipped to deal with extended questions that are related to calculating the value of a slope. Particularly, T draws on his prior experience of doing mathematics. He recalls a moment when he forgot the formula when he was working on an extended question. Such an experience motivates T to create a hypothetical scenario to give a heads-up to his students. Although T's motive to ask students to memorize the formula goes against the calls for prioritizing students' development of conceptual understanding, it can be argued that the teacher's creation of this hypothetical scenario is shaped by his prior experience of forgetting a formula when working out a mathematical question. This serves as T's motivation to remind students not to solely rely on the calculator for solving any mathematical equations.

IV Discussion and implications
The aim of this article is to examine how the teacher's embodied enactment extends his multilingual and multimodal repertoires to allow him to translanguage fluidly in order to construct hypothetical scenarios in EMI mathematics classrooms. This, in turn, facilitates students' learning of mathematical concepts (Extract 1) and develops students' interest in learning mathematics (Extracts 2 and 3). In Extract 1, it reveals how the teacher employs multilingual, multimodal and semiotic resources to create the hypothetical context of a pilot escaping from the tower in order to assist students in understanding the complex mathematical question. It is argued that the use of gestures and drawings on the blackboard plays a role in shaping the construction of the scenario of a pilot flying an aircraft. In Extract 2, the teacher uses diverse linguistic and non-linguistic cues, particularly eye gaze, pauses and gestures, to adopt two different character perspectives (i.e. secretary for national security and the assistant) for creating a hypothetical context of a secretary making an informed decision about attacking an aircraft. This allows students to understand the purpose of employing bearings in trigonometry for solving urgent issues, such as protecting the nation's security. Extract 3 illustrates how the teacher uses his bodily actions to enact a hypothetical context of himself as a fighter solving mathematical equations. The teacher's pedagogical goal here is to encourage students to apply the specific mathematical formula that they have learnt in class for solving unknown mathematical equations, rather than relying solely on the calculator. By doing so, students will draw on their knowledge base when they are dealing with difficult mathematical questions. In all extracts, the MCA analysis reveals that the teacher is engaging in embodied enactment to creatively create different hypothetical scenarios in order to achieve his pedagogical goals at specific moments of the classroom interactions.
This article demonstrates that the creation of different hypothetical scenarios through embodied enactment allows the EMI teacher to extend his multilingual and semiotic repertoires, which enables him to create a translanguaging space (Li, 2011(Li, , 2018Tai & Li, 2020, 2021a, 2021b, 2021cTai & Wong, 2022;Tai, 2022a;Tai, 2022b;Tai, 2023a;Tai, 2023b) to bridge the different language use situations in the classroom and beyond (Tai & Brandt, 2018;Tai & Khabbazbashi, 2019a;Tai & Khabbazbashi, 2019b). It is evident that the teacher puts emphasis on developing students' understanding of the procedural aspects of a mathematical question (Extract 1) and encouraging students to memorize mathematical formulas (Extract 3). Nevertheless, the teacher's translanguaging practices in Extracts 1, 2 and 3 can be viewed as creative because the teacher's translanguaging practices involve more than simply using one's body to add a visual description of a concept. Instead, the teacher physically and visually illustrates how the mathematical concepts (e.g. bearings in trigonometry in Extracts 1 and 2 and the concept of slope in Extract 3) can be related to students' familiar situations. The findings echo Tran's et al. (2017, p.3) view that 'integrating the body into the [teaching and] learning experience can, therefore, improve mathematical understanding by providing a connection between concrete referents and abstract concepts'. The MCA analysis presented in this article has demonstrated how the teacher's use of embodied enactment connects various mathematical ideas to the relevant out-of-class knowledge, which is a significant first step towards reconceptualizing mathematics teaching as creative and prioritizing in motivating students' in learning mathematics (Verschaffel et al., 2012). Specifically, the classroom extracts have revealed how the teacher's embodied enactment can function as a translanguaging practice which potentially facilitates students' learning of abstract mathematical knowledge and advice and motivates students' interest in learning mathematics. This can also potentially broaden the students' perspective as they recognize the meaning and value of academic knowledge beyond the instructional context.
In addition to drawing on diverse multilingual and embodied resources in the process of teaching mathematics, the teacher also brings diverse funds of knowledge into the creation of various hypothetical scenarios. For example, the teacher's creative use of gestures in depicting the shape of an aircraft's fuselage is influenced by his personal interests and childhood experience (Extract 1). Moreover, the teacher's prior experience with acting (Extract 2) and his experience of failing to remember the correct mathematical formula (Extract 3) also contribute to the creation of various hypothetical scenarios in the classrooms. These are the sociocultural factors that the teacher draws on which shape his translanguaging practices in the production of embodied enactments.

V Conclusion
The findings contribute to the current literature on translanguaging and EMI mathematics teaching and learning in a number of ways. Methodologically, this study demonstrates how combining MCA and IPA can help researchers to analyse how the EMI mathematics teacher creates embodied enactments through using different multilingual and multimodal resources for scaffolding abstract mathematical concepts to all students in the classroom (Tai & Li, 2020, 2021a, 2021b, 2021c. Particularly, adopting MCA enables researchers to describe how mathematics teaching and learning are achieved from the situated actions of classroom participants interacting in the classroom (Ingram, 2018). Although the study does not illustrate how the teacher's creation of embodied enactment through translanguaging can lead to positive outcomes on students' mathematical learning and L2 English development, this study aims to reveal how teachers can bring diverse resources and different kinds of knowledge into the classroom with the aim of promoting students' understanding of mathematical concepts and developing their motivation about mathematics learning. Future research can carry out a longitudinal case study in order to examine how the teacher's use of gestural resources can extend the students' cognitive abilities in grasping abstract mathematical ideas. Pedagogically, the findings draw attention to the importance of raising EMI mathematics teachers' understanding of translanguaging practices in supporting mathematics teaching and learning. The EMI policy restricts teachers and students to use English only in the classroom, and it is not explicit on the use of semiotic tools (Li, 2020). In the study, it is revealed that the teacher uses diverse multilingual and multimodal resources to enhance students' learning even though he has no training in how and when to use them strategically. The findings offer illustrative examples of an EMI teacher's translanguaging strategies and invite preservice and in-service teachers to think about using translanguaging for creating hypothetical scenarios through embodied enactments in their own professional contexts.

(word)
Approximations of what is heard ((comment)) Analyst's notes # Indicating the exact locations of the figures in the transcripts + Marks the onset of a non-verbal action (e.g. shift of gaze, pointing) XX

Multimodal conversation analysis (MCA) transcription conventions Sequential and timing elements of the interaction
[ Beginning point of simultaneous speaking (of two of more people) ] End point of simultaneous speaking = Talk by two speakers which is contiguous OR (i.e. not overlapping, but with no hearable pause in between) continuation of the same turn by the same speaker even though the turn is separated in the transcript (0.2) The time (in tenths of a second) between utterances (.) A micro-pause (one tenth of a second or less)

Paralinguistic elements of interaction
wo:rd Sound extension of a word (more colons: longer stretches) word.
Fall in tone (not necessarily the end of a sentence) word, Continuing intonation (not necessarily between clauses) wor-An abrupt stop in articulation word?
Rising inflection (not necessarily a question) word (underline) Emphasized word, part of word or sound word↑ Rising intonation word↓ Falling intonation°w ord° Talk that is quieter than surrounding talk hh Audible out-breaths .hh Audible in-breaths w(hh)ord Laughter within a word >word< Talk that is spoken faster than surrounding talk <word> Talk that is spoken slower than surrounding talk $word$ Talk uttered in a 'smile voice' Other conventions