Using video-recordings to enhance pre-service teachers’ abilities to identify mathematical activities in pre-school

Abstract Previous studies highlight the importance of investigate the construction in videorecorded learning situations. The aim of this study is to investigate how video-recordings can be used to enhance pre-service teachers’ abilities to identify children’s exploration of mathematics in preschools. Even if some previous studies have focused the use of video-recordings of learning situations to improve teaching, only a few of them focused how constructed videos enhance pre-service teachers’ abilities to identify when children explore mathematics through learning-situations in preschool. The participants were pre-service teachers in a higher education course. Data consisted of pre-service teachers’ written responses after viewing eleven edited video clips before, during, and after a compulsory mathematics teacher education course. The analysis is based on variation theory and show the impact of the construction of the video clips and how to design the clips to enhance the pre-service teachers’ abilities to define mathematical components during the children’s activities.


Introduction
Pre-service education for teachers strives to develop knowledge needed for teachers' professional work, in another institutional environment, the practice. There is a challenge for teacher educators to make the university courses developing a knowledge base for how to handle future teaching design. Tirosh et al. (2014) explore whether videos can be used to offer learning situations for teachers to study their practice. They found that teachers in a professional development program used the theoretical tools provided to identify pupils' conceptualisation without having any defining theories for their analyses. The teachers discussed the pupils' knowledge expressed during the video-recorded lessons, and formulated strategies of how to evaluate pupils' knowledge. However, if it is enough to leave the teachers to discuss by themselves, and by that develop their own knowledge, are questioned (Tirosh et al., 2014).
Previous studies have identified the benefits of using video-recorded learning situations for knowledge development in professional development efforts (Santagata & Guarino, 2011;Sherin & Han, 2004). The main benefits found are teachers' increased ability to observe pupils' learning, their increased awareness of pupils' mathematical abilities, and the development of teachers' ability to analyse the pupils' thinking. Sherin and Han (2004) argued that viewing videos from the classroom, showing interactions between the teacher and the pupils and discussing these videos with other teachers, promoted an attitude whereby teachers were more anxious to ask questions, reflect on their work and learn more about teaching effectively. This process of reflection and thinking can become a resource for teachers and contribute to their continuing exploratory learning about pupils' learning and about teaching and learning.
Similarly, Tirosh et al. (2014) emphasised that videos can be used as a tool in continuing professional development whereby preschool teachers continue to challenge their own practice through discussing common issues in relation to their teaching activities, even after an intervention. These common issues are based on theories that generate conceptual tools for perceiving both mathematical concepts and theoretical constructions in situations where mathematics is taught. These conceptual tools enable members of the teaching community to reflect on their practice and interact with colleagues (Coles, 2019;Garcia et al., 2006;Helenius et al., 2015Helenius et al., , 2017Tirosh et al., 2014). Although teachers do not explicitly describe the theoretical concepts involved, previous studies have shown that theories and concepts can contribute to the development of conceptual tools through watching pictures, and thus improve both teachers' analytical ability and their mathematical knowledge (Helenius et al., 2017;Tall & Vinner, 1981). According to Coles (2019) the facilitator during the conversation about the teaching design in the videos has an important role for the learner's development understanding, to highlight the aspects which the learners are supposed to identify. The main part of the studies is done in a context where the video-recordings are discussed collaboratively in teams of teachers, few studies have investigated professional development based on online videorecorded mathematical learning situations without collaboratively discussions between teachers (Helenius et al., 2017). Güler and Çelik (2022) study focuses on providing professional development for novice mathematics teachers and aiming to improve their lesson analysis skills which are accepted as an element of teacher competencies (c.f Ball et al., 2008). To capture the novice teacher's development skills during the online intervention, they offered a well-structured framework (which, for example, caught students thinking) together with video recorded classroom in the online program. In their result, it was clear how the video-recorded situations contributed to changes in what the novice teachers recognised as necessary for student learning. In the beginning, the novice teacher focused on constructing the learning environment. In the end, they took more notice of students' misconceptions and how their instruction could be more prepared to highlight different students' misconceptions and thinking in the learning situation.
Videos have also been used in teacher education and professional development to illustrate children's mathematical activities in free play-oriented situations (Ginsburg, 2014) and in professional development to improve instruction (Güler & Çelik, 2022;Holmqvist, 2011). However, teacher education programs often do not support trainees in analysing and interpreting classroom situations which can affect student thinking (Güler et al., 2020;Svensson, 2022). In addition, Santagata et al. (2021) found in their review article that combining video with well-structured frameworks supported by well-prepared facilitators optimises video-supported programs. Both discussion sections of reviewed publications and their evaluation of studies led them to the conclusion that although the evidence is clear that engaging teachers in video-supported activities lead to meaningful changes and improvement in their noticing competencies concerning students thinking and learning, there is a need for more investigation of specific elements that make programs, videorecorded learning situations successful (Santagata et al., 2021).
Furthermore, few previous studies have examined the use of constructed video-recorded to analyse pre-service teachers' ability to identify children's expressions of mathematical knowledge based on mathematical activities in preschool, which this study addresses.

Study aims and research questions
The aim of this study is to investigate how video-recordings can be used to enhance pre-service teachers' abilities to identify children's exploration of mathematics in preschools. The mathematic content captured at the videorecording and made discernible for the pre-service teachers is based on Bishop's six mathematical activities (Bishop, 1988b). The analyses, based on variation theory, aims to identify aspects for constructing the video-clips regarding if and how the content is made discernible for the pre-service teachers. The video-recorded activities are used to identify preservice teachers' ability to discern mathematical content in the video-clips before, during, and two weeks after a compulsory mathematical course in higher education. This means three measurements made to follow and analyse the pre-service teachers' process of knowledge development.
The following research questions (RQs) were formulated: RQ 1. In what ways do the design of the video-recordings enable pre-service teachers to express their knowledge before, during, and two weeks after the course? RQ 2. What aspects of the children's knowledge were identified by the pre-service teachers before, during, and two weeks after the course? Bishop (1988a) argued that the curriculum should be developed in terms of mathematical enculturation and consider mathematics in its cultural context. Furthermore, Bishop (1988a) argues that the education curriculum should consider whether mathematics is only related to numbers and counting. Mathematics takes many forms in relation to its cultural context, but there are six general activities that tend to be common to all cultures when it comes to the production and use of mathematics: counting, measuring, locating, designing, playing, and explaining (Bishop, 1988b). Consideration of these six mathematical activities has proven to be useful for preschool teachers aiming to understand the teaching of mathematics in Sweden (Helenius et al., 2017;Johansson, 2015). Reikerås et al. (2020) visualised in their study how children explored mathematics based on all Bishops six mathematical activities in a play-based outdoor activity. However, the most notable result was the teacher's role for promote children explore mathematics in this outdoor learning situation. Furthermore, mathematics for young children has been introduced by these mathematical activities in teacher education in Sweden (Helenius et al., 2015). Johansson (2015) found that Swedish preschool teachers tend to easily adopt and create a terminology for the six common cultural activities for mathematical situations, which often occurs in the preschool context. But some mathematical activity, for example playing and explaining seemed to be more difficult for pre-service teacher to explore understanding of and connect them to children knowledge development in free learning situations (Svensson, 2015(Svensson, , 2017 By that, the mathematical course used in this study is based on these activities to promote a wider understanding of mathematics learning situations in preschool.

Theoretical framework
To analyse if and how pre-service teachers through videorecording enhance abilities to identify children's exploration of mathematics in preschools, this study uses variation theory as an analytic lens. Variation theory (Marton, 2015), which provides analytical tools for exploring the necessary conditions for learning, is used in this study to investigate pre-service teachers' knowledge of children's exploration of mathematical understanding. It entails assumptions about learning that can be applied to enhance learners' ability to discern aspects of the object of learning that have not previously been identified (Holmqvist & Selin, 2019;Marton, 2015). Previous studies of how preservice teachers develop their understandings of students' learning, based on variation theory, have shown how the pre-service teachers' skills develops (Brante et al., 2015). In variation theory, the purpose of learning is to gain the ability to do something with something (Marton, 2015). To understand learning, especially what is crucial for learning, it is necessary to consider what is learned in different learning activities (Marton, 2015). Further, in variation theory, the different ways in which various aspects are simultaneously perceived are significant for learning. However, seeing the differences in learning as a difference in the ability to discern aspects do not simply add to the learning of the discerned the aspects. From a variation theory perspective, learning is a question about being able to discern and differentiate the aspects that constitute a chosen of an object of learning. It is thus through expanded understanding of what is critical to distinguish which increases knowledge and conditions for learning in relation to the goals. Gibson and Gibson (1955) point precisely to the importance of differentiate to highlight the learning, based on the perception, becomes richer through different answers. Thus, learning becomes a matter of being able to distinguish the differences and find connections between what is essential to understand something specific. To experience or understand something in a certain way, certain (critical) aspects must be discerned, and it is the discernment of these aspects that is crucial for learning (Marton, 2015). From an educational point of view, aspects that are considered important are based on the learning in relation to the chosen object of learning. Marton (2015) describes how differences in how the same phenomenon is experienced have to do with differences in discernment of the learning activity. Marton and Pang (2006) show that how teachers organise learning does not make a difference to the learning, for example group work or whole class teaching. It's in the how the teacher chooses to treat or direct the attention of learning towards specific aspects of selected object of learning that make a difference, for what becomes possible for the student to learn. Furthermore, learning is also a question of being able to determine those aspects that are considered critical for a specific group in a specific situation based on the object of learning (Marton, 2015). Thus, it is an increased understanding of what it is critical to distinguish that increases knowledge and the conditions for learning in relation to the learning goals. The intended object of learning, the content the teachers plan for to be learnt by the learner, refers to the knowledge, ability, or skill that the learning situation intends to develop (Marton et al., 2004;Runesson, 2005). Holmqvist Olander and Sandberg (2013, p. 284) state that what is planned for and designed by the teachers, the intended learning, becomes the enacted object of learning in the classroom meeting the students' experiences of the content. This process refers to the way in which the object of learning is handled by the learners and the teacher together. The object of learning is also possible to be analysed by researchers or other teachers, as the lived object of learning defined as the actually learning expressed by the learners (Holmqvist Olander & Sandberg, 2013;Svensson, 2022).
In this study, the activities (Bishop, 1988b) represented six phenomena as intended objects of learning. However, within the six phenomena, there are features with regard to aspects in relation to the learning situation (c.f. Bishop, 1988b). According to Marton (2015) learning is to be found in the pattern of variation and invariance., there invariance means to keep something constant at the same time you vary something else. In this study the mathematical activity counting is kept invariance and the other five mathematical activities are varied as intended object of learning in the eleven videorecorded learning situations.

Method
The course, which the participating pre-service teacher take part of intends to develop, through lectures and workshops in a seminar format, their knowledge about the subject of mathematics with relevance to preschool children's learning opportunities based on Bishop's six mathematical activities (Bishop, 1988b).

Participants
In total, pre (143), during (96), and two weeks after the course (119) pre-service teachers participated in the data-collection, were analysed. To participate in seminars are voluntary for the preservice teachers. Therefore, the number of pre-service teachers participating in the data-collection are different on each of the three measurement occasions. The analysis is made at a group level and increase the possibility to handle the dropout during the three measurement occasions. The pre-service teachers are divided into five groups for different workshops and seminars with different teacher educator for respective group and because of that they are also divided into five groups in the analysis in this study. Participants were informed of their right to refuse to participate, and their verbal and written consent was obtained prior to their participation. The collected data was anonymised and coded to maintain confidentiality. Further, the data was kept in a locked safe to prevent unauthorised access. The names of the participants and other sensitive data was removed prior to the data being stored.

Data collection
The collected data included the pre-services teachers' written responses to the video clips in the test. The written responses are documentations of the pre-service teachers' observations regarding the children's reasoning and exploration in relation to mathematics in preschool. The study commenced in November 2017 and ended in January 2018. All activities took place during the regular teacher education program. The pre-service teachers attended in same room during the data collection, all three times. They answered the pre-, post-and delayed post-tests before the ordinary lecture in the course. The delayed post-test was taken two weeks after the course ended, answered right before the introduction of the next ordinary course. The video-recordings which the pre-service teachers reflected upon at the test-situations were presented as a movie setting, placed in front of the pre-service teachers in a large lecture room.

Data analysis
The analysis of the lessons was carried out with the support of variation theory (Holmqvist, 2006;Lo, 2012) and determined what aspects could be discerned by pre-service teachers to explore the children' knowledge of mathematic during the learning situation. The analysis is based on written answers from the pre-service teachers. In total 353 pages have been verbatim transcribed and analysed. There are five pages from the latest measurement with no comments from participant student. Therefore, the total number of pages are 353 instead of 358 (143, 96, 119). Variation theory was applied in the design of the activities in the eleven video-recorded by offering opportunities for discernment through variation and invariance two mathematical activities (Bishop, 1988b) simultaneity, following the procedure of Brante et al. (2015). This enabled the study to identify which aspects of the enacted object of learning (Holmqvist Olander & Sandberg, 2013) the pre-service teachers discerned during their analysis of the children's mathematical knowledge before, during, and two weeks after the course. Further, the analysis aimed to identify what in the observed learning situation in each video were discerned and expressed by the pre-service teachers as identification of learning (experienced by the pre-service teachers) in relation to the enacted object of learning (which a focus on the content discerned) and the lived object of learning (what knowledge the pre-service teachers developed). This will, in each of the eleven videorecordings' design, capture what the pre-service teachers discerned to identify as mathematical activities (Bishop, 1988b). The unit of analysis was pre-service teachers' expressed discernment of children's activities that was identified as mathematics, explored in a series of eleven videorecordings (with a total time of 15 m 25 s), in structured and unstructured learning situations, capturing Bishop's six mathematical activities (Bishop, 1988b). To capture changes of the preservice teachers' skills, they were required to write their observations on paper before, during, and two weeks after the course in relation to the eleven videorecording learning situations. The same eleven learning situations were used in all three measurements occasions. This research search for connection between differences and similarities of the pre-service teachers expressed discernment of children's activities in relation to mathematic. Therefore, through these different answers. This study tries to distinguish the differences and find connections between what is essential to understand in the constructed videos by using five groups of pre-service teachers expressed skills in same higher education course (c.f. Gibson & Gibson, 1955).
In the analysis process, each video clip was first analysed separately to the intentional learning object based on Bishop's (1988b) mathematical activities. In the next phase, colour coding used to mark how the pre-service teachers expressed the intentional object of learning in their written notes. This was done first by looking over 20 written notes to find connections between how, for example, numbers were expressed. The most frequently expressions formed a unit of "everyday expressions" for mathematical numbers. Secondly, this unit of analyses was divided with the total of words expressed by the pre-service teachers. By starting from the pre-service teachers' expression of the mathematical concepts, the analysis contributes to how and if the video-recorded mathematical situations were offered discernment of the intentional learning object to the enacted learning object. Finally, the lived object, the enacted learning object in each video clip, was analysed and contributed to how the pre-service teachers expressed the object of learning. The latter phase of the analysis answers the purpose of this research article: how video recordings can enhance pre-service teachers' abilities to identify children's exploration of mathematics in preschool.

Design of pre-, post-and delayed post-test
Eleven video-recordings were used in the tests, exactly the same recordings in all three tests occasions. Initially, the pre-service teachers were asked about which mathematical activities of Bishop's six mathematical activities, were possible or could be possible for the children to develop their mathematical knowledge of, in the different learning situations. These eleven situations were based on previous studies of how preschool teachers experience mathematics in preschool (Helenius et al., 2017). Each mathematical activity represented one or two phenomenon/a with features based on Bishop's six mathematical activities (Bishop, 1988b) related to the recorded learning situation. The author constructed the videos based on eleven selected video-recordings from preschool teachers, which they have captured in their own preschool context. Inspired by previous studies (Helenius et al., 2017) on how preschool teachers used to experience mathematics in preschool through mathematical activities (Bishop, 1988b), all eleven mathematical situations in the recording involve aspects of counting (invariant content) with one or two of the other five mathematical activities (varied content). Thus, aspects of counting simultaneity with other mathematical activities enabled discernment of the intended object of learning, as well as the discernment of aspects in relation to the enacted objects of learning (Holmqvist Olander & Sandberg, 2013). The intended object of learning is marked in italics (see appendix), where the video-recordings are presented, and the talk between teacher/s and child/ren are verbatim transcribed.
In line with the aim of this study, the analysis is based on the pre-service teachers' written expression about the video-recorded eleven mathematics situations made three times during a course. The analysis revealed the aspects that pre-service teachers discerned regarding the children's mathematical knowledge as expressed in the video clips. Firstly, the discerned object of learning in each of the eleven video clips is presented. Secondly, the pre-service teachers' statements regarding what knowledge the children have are analysed. Table 1 presents a compilation of the words used by the pre-service teachers to express aspects of mathematical knowledge they observed based on the mathematical activity shown in the videorecordings (see Table 1). However, in relation to video-recordings 10 and 11, these aspects changed based on the pre-service teachers' stated knowledge in relation to the mathematical activities. In video 10, the aspects associated with locating were instead interpreted as playing. For example, when child 1 pushes child 2 gently and says, "You should stand there" while pointing to a position in front of the cash register, this situation was discerned as modelling and abstracting to enable the play to continue. In another setting, the situation could be discerned based on aspects associated with locating (direction, distance, spatial environment) shown in Table 1. In video 11, the aspects associated with locating and designing were discerned differently. This was based on the specific situation of the child having a template rather than free drawing of the different shapes. This means that in Table 2, the aspects were associated with locating in playing or designing in videos 10 and 11 depending on the situations. Table 2 presents the results of the pre-service teachers' expressed knowledge of the intended object of learning, regarding the aspects presented in Table 1. The six activities are presented in the following order: playing, explaining, designing, locating, measuring, and counting. For example, in relation to the first video clip, the documentation was based on the mathematical activities of playing (1a) and counting (1b). Word counts of expressions are relating to different types of mathematical activity. The analysis is based on the number of words expressed of the aspect/s discerned, the lived object of learning, (see Table 1) divided by the total number of words expressed by the group. The results are shown as percentages, to show the differences in expressed discernment, following

The pre-service teachers' expressed knowledge
To enable the video clips, making the aspects intended to be discerned also ending up as the aspects discerned, reflecting the enacted object of learning, the following variables were found   Note: *n/a: not applicable -no words expressed by the pre-service teachers. critical. Discernment of the intended object of learning increased when the video-recordings used a pattern which was structured through invariance and variation simultaneously, where the invariant aspect was related to the intended object of learning. This was identified in the analyses by the pre-service teachers' expressions of aspects in the second video clip (e.g., group B from 7-46%), there it was notable that the discernment of numbers of words increased in all groups. There was less mathematical content expressed by words in all groups initially, to describe the situation. The child is given the opportunity to view the same animal in different ways depending on what category the preschool teacher uses to challenge the child. In the video-recorded learning situation, the preschool teacher targets the enacted object of learning by retaining the animal's invariance (constant) while varying the categories (variation).
In the fourth video clip featuring a collection of stones, there was an increase in the numbers of words expressed (e.g., in group E from 11-25 %) indicated discernment of aspects related to the intended object of learning, between the pre-and post-tests for all groups. However, only group B showed a further increase between the post-test and delayed post-test (from 14-20 %), with the other groups either remaining unchanged or regressing slightly (e.g. group A from 21-19 %). In the situation, the preschool teacher starts with the children's placement of the stones. Unlike the second video-clip, which also deals with explaining, both categories and objects are varied in this situation. Based on variation theory, this makes it more difficult to discern what was intended to be discerned, as all aspects vary. However, similar to the second video-clip, the preschool teacher in the video-clip directs her questions towards the categorisation of the objects that, in this situation, the stones represent. Both the objects and the questions from the preschool teacher are focused on the intended object of learning.
In the fifth video-clip, the numbers of words expressing the discerned intentional aspect of counting showed an increase between the post-and delayed post-test for groups A and C (group A: 8-14%, C:6-15%), while the other groups showed a minor decrease. The video-clip was constructed in relation to locating, but the discerned aspects did not include related words for the feature of the aspect, for example depth, location, and distance. Instead, the discerned aspect seemed to be counting as the number of expressed words related to counting increased. The verbal interaction between the preschool teacher and the children is based on words related to counting such as counting rhymes and pair formation. There tended to be a gap between the intended object of learning (counting) and the choice of materials and learning environment (locating) in this video-clip, making other aspects discernible of the enacted object of learning, than the intended.
In the seventh video-clip, there was an increase in the number of words expressed in all group from pre-to post-test (e.g. in group D: 7-24%) discernment of the aspect locating, which also was the intended object of learning. However, it is noteworthy that initially, no words were expressed that could be associated explicit with locating. The preschool teacher in the videorecorded learning situation focused on the aspect locating by asking questions about where the child lived, at a map. The route that the child takes to preschool tends to increase the possibilities of distinguishing the intended object of learning with regards to the geographical aspect of mathematics. This is in comparison with the fifth video (which also features locating as the intended object of learning), wherein the preschool teacher focuses more on words in relation to counting than on words related to geographical aspects such as the features placing, depth, and distance. Further, this video-clip is seen as more in line with the geographical aspects that are perceived in everyday life in a spatial environment.
In the ninth videorecording, the numbers of words expressed in relation to the intended object of learning (measuring) increased in most groups between the pre-and delayed post-test (e.g. group A:16-42%), while those in relation to designing decreased. The discerned aspect is by that measuring. In the pre-test, more words were expressed showing the discernment of the aspect designing, than discernment of the aspect counting. In the videorecording, the child uses its finger as a measuring tool to locate the centre of the object, a kitchen roll. The preschool teacher in the learning situation challenges the child with questions such as, "How do you know it is the middle?" The teacher challenges the child with reasoning questions, and thus gives the child the opportunity to apply reason and then explain its reasoning. The child does this through gestures, an aspect the situation is focused on, showing how the child quietly counts from the middle to the edge of the roll using a finger as a measuring tool. However, it is interesting to note that the numbers of words related to the discernment of the aspect designing increased between the post-and delayed posttests, while the discerned aspects related to measuring decreased. The description of the construction of a rocket was seen as contributing to the increased emphasis on designing.

The aspects of the children's knowledge the pre-service teachers discerned
In relation to all video-recordings, the pre-service teachers were required to explain what happens in the video clip prior to the pre-test. For example, regarding the video clip Numbers, they might explain that the aim is to "build numbers and count numbers", and in relation to the video clip Animals, they might explain that the focus is on "sizes, measuring, and counting tails/legs". There were few statements regarding Bishop's mathematical activities involved. Further, in the post-test two categories of responses were identified. Firstly, the pre-service teachers identified several aspects of mathematical activities without regarding them as mathematical content. For example, in relation to the video clip Map, they identified "locating, measuring, counting, and explaining". Secondly, they were able to discern aspects of the mathematical content but were unable to relate it to one of Bishop's mathematical activities, except in relation to the final videorecording, Drawing shapes. In this, shapes appear with regards to designing already at the post-test. In the delayed post-test, two weeks after, the discernment is concerning aspects how the mathematical content, and the aspects are in relation to the mathematical activities, for example regarding the aspect in video clip Numbers, features as counting, counting rhyme, cardinal numbers, were discerned. This might be explained by fewer mathematical activities, which are more in line with the intended object of learning in the videorecording. If there are two aspects of mathematical activities, the pre-service teacher expresses discernment of both (locating and counting) as regarding the example Map, the aspect location, where features as direction, topological map, identification and designation were discerned. It is also notable that the pre-service teachers were able to mention other aspects, such as measuring and counting activities, that could have been undertaken instead of those displayed in the videos. For example, in relation to the video-clip Macaroni, they suggested that the children could have "compared the weights of the different objects" and could have "counted and understood cardinal numbers". This could be connected to the design of the video-clip. The aspects that varied in relation to the intended object of learning, for example in the video-clips Macaroni and Stone, as mentioned in the first section, the pre-service teacher discern the mathematical aspects where the variation of features is implemented. In the two videos Kiosk and Animals it is notable that the pre-service teachers recognize counting, measurement, and animals at the pre-test, but later on they express their discernment of aspects of the mathematical activities playing and explaining, which offered discernment by varying their features. Table 3 presents examples of the pre-service teachers' expressed identification of aspects of mathematical knowledge of the enacted object of learning, in relation to the intended object of learning.

Discussion
In this study, critical aspects intentionally planned to be discerned, were also found to be discerned when they varied in the videorecorded teaching design, based on an invariant background. If the aspects that were intended to be made discernible when planning the instruction (in this case the video-recordings) also where those that actually varied in the enactment, against an unvaried (invariance) background, the pre-service teachers discerned them. By that, the results show that designing video-recordings to be used as learning material in teacher education, requires an accurate identification of what aspects are to be made discernible by variation against Table 3. Discerned aspects in relation to the intended object of learning (see Table 1 an unvaried background. The analyses presented in Table 2 showed an increase in the pre-service teachers' discernments of children expressed mathematical knowledge, in relation to the intended object of learning, what was meant to be learned. This was observed in relation to all five groups of pre-service teachers. The result could be interpreted as confirming that the aspects in the preschool teacher's teaching design in the videorecording were structured in relation to what aspects have to vary of the intended object of learning to make discernment possible. This was indicated in the changes of expressed aspects and features of the object of learning while analysing words in the pre-service teachers' written expressions. The quality of the design of the video-clips, regarding aspects possible to discern, also had an impact on changes of expressed discernment of the aspects for the pre-service teachers. Regarding variation theory (Holmqvist Olander & Sandberg, 2013), the variation of aspects and features of the phenomenon (Holmqvist & Selin, 2019) was discernible in terms of an awareness of the way in which the video-recordings contributed to improve the pre-service teachers' abilities to identify the children's expressions of mathematical content knowledge (see Table 2). Further, the ways in which aspects and features in relation to the phenomenon were used to make the intended object of learning the same as the enacted object of learning (see Tables 2 and 3), was also related to what mathematical activities were focused. For example, depending on the situation, the aspect direction and distance could have features such as locating (spatial environment) and playing (modelling, abstraction, hypothetical thinking). This can be seen in relation to the video-clip Kiosk with a shift from expressing the aspect "locate" (post-test) to "think abstractly" and "problem solving" at the post-test (see Table 1). Further, the results show the ways in which the pre-service teachers' discernment develops and change between the test-situations (pre-, post-and delayed post-test) (see Table 2). The construction of the video-recordings, by using two mathematical activities simultaneously (Marton, 2015) made it possible to enhance the discernment of aspects and features not yet discerned by the pre-service teachers. Furthermore, through the repeated measurement (three times) during a course which contribute to developing pre-service teachers' ability to enhance children's mathematical learning opportunity in preschool, the analysis was able to contribute to distinguishing in relation to the intentional, enacted and lived object of learning critical aspects in the construction of the video clips. Notable is that through using five groups of pre-service teachers with different teacher educator teaching the same mathematical content different and similarities could be interpreted and distinguished, and furthermore connections between what was essential and critical to understand in the constructed videos. Ginsburg with colleagues (2014) used video-recordings to illuminate children's exploration of mathematics. Santagata et al. (2021) argues for more investigation of specific elements that make program through video-recorded learning situations successful. In this research study, it is not just the video-recordings that are important, the design of them is crucial. Teachers must be aware of which aspect of learning is to be elucidated and which aspect of learning is intended to be discernible during the learning situation (Marton, 2015;Svensson & Holmqvist, 2021). (Güler & Çelik, 2022) mentioned from their result from an online program the important of offered a wellstructured framework to the novice teacher. This for contributing the teachers in the intervention to capture students thinking and learning in the videorecorded classroom situations. In this study, pre-service teachers took part in a course to develop understanding of mathematical theories based on Bishop six mathematical activities (Bishop, 1988b) as a theoretical tool to develop knowledge of mathematical learning in preschool. The results show how the pre-service teachers expressed a developed understanding visualised in changes in word used during identification of mathematics in the preschool setting in relation to mathematical concepts. Though the concepts underlying Bishop's six mathematical activities (Bishop, 1988b) were not identified in the preservice teachers' pre-test results, it can be interpreted that they played an important role, as a framework, in enabling them to identify the mathematical content and learning displayed by the children (c.f. Garcia et al., 2006;Güler & Çelik, 2022;Helenius et al., 2017;Tirosh et al., 2014). However, the result in this study, by using variation theory, also show how the design of the teaching situations in the video-recorded learning situation played an important role for enable the pre-service teacher to discern children's mathematical knowledge. In previous research studies (Svensson, 2015(Svensson, , 2017 some of the mathematical activities (Bishop, 1988b), for example playing and explaining, have seemed to be more difficult for the preschool teachers to be aware of in learning situations. In this study all of the six mathematical activities (Bishop, 1988b) were explored by the participants (see Table 1) even playing and explaining. It was not depending on which mathematical activity the video-recordings focused on. Instead, it was how the aspects of the intended mathematical content were visualised in the construction of the teaching activity between the child and the preschool teacher that make difference for how the learning object was distinguished. Therefore, from this point of view it is not only important to use a structured framework for the mathematical content it is also necessary to use a structured framework for the teaching and learning activity for offering distinguish the learning object in the videorecordings. However, this is a small study and there are needs for more research in this research area-constructing videos for test and learning situations in teacher education programs.

Disclosure statement
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Citation information
Cite this article as: Using video-recordings to enhance preservice teachers' abilities to identify mathematical activities in pre-school, Christina Elisabet Svensson, Cogent Education (2023), 10: 2194778.