Background and rationale

The foundation for teachers teaching skills is made in the teaching training program. Based on this, it is important that the teaching training provides an education where the pre-service teachers (PST) have opportunities to develop their ability to teach. 

Teacher training traditionally offer PST limited opportunities to realistically practice and develop skills and strategies for teaching pupils, without the risk of adversely affecting pupils' learning (Bradley & Kendall, 2014; O’Callaghan & Piro, 2016; Piro & O’Callaghan, 2018). Experience of teaching pupils is offered to most PSTs for a limited period. Teacher education programs may not always provide PSTs with enough time, reasonably secure experiences, ordinarily acting pupils, the opportunity to repeat, or adequate feedback to develop knowledge, skills, and confidence (Ferry, Kervin, Cambourne, Turbill, Puglisi, Jonassen & Hedberg, 2004; Badiee & Kaufman, 2015). To provide more practice-oriented opportunities at the teacher preparation program a well-designed simulation, such as TeachLive, could be used to practice usually occurring situations (Mason, Jeon, Blair & Glomb, 2011).

According to Blömeke and Delaney (2012) there is a need for more knowledge about how teacher educators can support students’ development of how to teach in a professional way.  Researchers need to investigate what knowledge PST need (Blömeke & Delaney, 2012), but also how PST act when they teach a specific content (Ryberg, 2018) to fully understand their challenges. When we have knowledge about such challenges, and only then, teacher education will be able to support PSTs in an effective way. Beside or beyond traditional modes for training PSTs todays’ teacher educators can also use different kinds of simulations to practice teaching (Kaufman & Ireland, 2016). Simulation can be determined by artificial intelligence. Semi-virtual simulations, such as TeachLivE, includes virtual characters with different personal traits, but also needing a human in the loop (Ersozlu, Ledger, Ersozlu, Mayne & Wildy, 2021). Students who practice teaching with semi-virtual simulations provide the research community with unique opportunities to study how PSTs teach a specific content and interact with virtual pupils. In this paper we are focusing on PSTs enacting mathematics for teaching, explaining, or representing mathematical concepts or procedures with respect to fractions.

This papper contribute with knowledge about how PSTs teach fractions in a semi-virtual environment. The following research questions are addressed:

• How does preservice teachers explain and represent mathematics when they teach fractions? 
• How does preservice teachers interact with virtual pupils when they teach fractions?

Theoretical framework

When teachers explain and represent the subject, they handle the content. The handling of content, the enacted object of learning, is an issue about what content is present during the instruction and which aspect of the content that is brought to the foreground (Marton & Booth, 1997). The handling of content could be done in qualitative different ways with respect to how the teacher represent the subject. Venkat, Askew, Watson and Mason (2019) assert that pupils learning is connected to if the teacher can provide structural teaching. The way teachers interact with pupils affect the pupils learning opportunities (Pianta, Hamre & Allen, 2012; Hufferd­-Ackles, Fuson & Sherin, 2004; Mercer & Dawes, 2014; Park, 2011). In various interaction patterns pupils are offered different sorts of agencies. Agency include teachers 'and pupils' opportunities as individuals to act and have control over their actions, in interaction with others (Andersson, & Valero, 2016). In other words, agency are the opportunities individuals must make their voice heard and influence what happens in the classroom; it is not a specific characteristic of individuals (cf. Biesta & Tedder, 2006; Norén & Anderson, 2016). 

Methodology 102 PSTs enrolled to a teacher training programme specializing in upper elementary school attend in this study. The training to teach virtual students was done using the semi-virtual solution TeachLivE. The simulation-training was led by two instructors, one a specialist in mathematics didactics and the other was specialized in classroom management. All PSTs taught once and observed fellow PST:s teaching twice. All groups visited the simulation three times. Being a semi-virtual simulation meant that some traits for each virtual pupil were programmed while others had to be created by a simulation-specialist that orchestrated the virtual pupils in relation to a content, fractions (Chini, Straub & Thomas, 2016). The simulation specialist had in advance, together with the teacher in the course and researchers in the project planned how each virtual pupil could represent different possible misunderstandings about fractions (Reynolds & Muijs, 1999). Each session was ended with reflections and feed-back. Analyze procedures The qualitative explorative analyse started as the researchers on their own, one by one, looked at some randomly chosen recordings among the amount of 102 recordings. The analyse was based on an impression, or a foreshadowed problem (Malinowski, 1922) that there was something interesting with the variation of how the PSTs interacted with the virtual pupils while explaining and representing mathematics. The first inductive phase of watching recordings was followed by a second collective phase step when the researchers came together and looked at the same recordings. Doing that also included reasoning and arguing about what had been seen in the recordings. In this phase several ideas were discussed, for example how questions where posed and how PSTs interacted with pupils. The second phase resulted in a preliminary model illustrating that some PSTs directed the interaction with questions to the virtual pupils while other followed the interaction while answering questions from the virtual pupils. That led to a third phase were the researchers once again on their own searched for patterns among some specifically chosen recordings that illustrated a variety of ways of teaching. In this phase two focus-areas where chosen, (a) different aspects of structure and handling the content, and (b) different patterns of interaction during the lessons. From that followed the fourth phase were the researchers distinguished four prototypical categories (Lakoff, 1987; Maxwell, 2002) of different ways that the PSTs taught the virtual pupils.

This paper contributes with important insights about how PSTs teach and interact with virtual pupils and what challenges they are struggling with enacting mathematics while teaching about fractions. The four teaching styles founded during the analyze (a) a confusing way, (b) an improvising way, (c) a lecturing way and (d) an orchestrating way, shows that there is a big variety among PSTs with respect to how to handle the content and interact with pupils. It also identifies areas where the PSTs need further support in their development to become teachers. Teacher educators can use the results as a starting point in a nuanced and qualified discussion about how to arrange a teacher education program. The results imply that the following aspects can be crucial to focus on in a teacher education program: (a) PSTs’ content knowledge,( b) PSTs’ ability to choose examples that highlight the content (c) PSTs’ ability when creating an internal logic in the lesson and (d) PSTs’ ability elaborating the content with pupils. Teaching is a highly qualified task, and it takes a lot of practice to become a good teacher. Therefore, PSTs' need to be given lots of opportunities to practice (McDonald, M., Kazemi, E., & Schneider Kavanagh, S., 2013). It is not enough to practice during the field studies. The semi-virtual simulation used in this study has made it possible to see details in how PSTs' teach, making it possible to identify their challenges, and in the next step provide a better teacher education. For PSTs' teaching virtual pupils can provide an opportunity to practice on specific teaching skills in a safe environment (Dawson & Lignugaris/Kraft, 2017).

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