This study analyses teachers’ discernment of children’s mathematical knowledge development during preschool play activities and defines how children’s pre-knowledge supports or limits their learning possibilities. In total, two preschool teachers from different preschools participated in this research study. The data consists of two documentations of children’s activities, written notes, and reflections from these two participated preschool teachers. The teachers chose the samples and defined them as important examples of how to merge mathematical knowledge development in free play. Using variation theory, the focus was to define what aspects of children’s mathematical knowledge development were offered and how teachers discern children’s pre-knowledge. The results show how the preschool teachers discern children’s pre-mathematical knowledge in relation to development of abstract abilities, and the importance of identifying their participation in free play situations for promoting mathematical understanding. 

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.

The aim of this study is to explore pre-service teachers’ procedural and conceptual understanding of pupils’ understanding of mean value, before and after an intervention. Participants are 66 pre-service teachers, and two teacher educators. The data consists of five video-recorded lessons and pre-service teachers’ pre- and post-test discussions. Based on variation theory, the analysis aimed to identify pre-service teachers’ expressed knowledge of procedural and conceptual understanding, focusing four categories of expressions: procedural (P), procedural didactics (PD), conceptual (C), and conceptual didactics (CD). The pre-service teachers’ expressed understanding of pupils’ knowledge developed differently in the five teacher student groups. A shift from procedural to conceptual focus occurred in two groups. The number of procedural related words expressed in the first group decreased from 104 (pre-test: 65 P and 39 PD) to 16 (post-test: 9 P and 7 PD), at the same time as the conceptual related words increased (pre 4 C and 0 CD and post 11 C and 16 CD). In the second group, 87 procedure-related words (36 P and 51 PD) identified during pre-test decreased after the intervention (3 P and 9 PD). Instead, concept-related words increased from 16 (12 C and 4 CD) to 29 (10 C and 19 CD). The changes occurred among participants who had got, except of theoretical instruction, challenging hands-on tasks about the Mathematical content in focus. The findings from this study contribute to understand how design of instruction for pre-service teacher contributes to enhance teacher students’ didactic perspective on pupils’ learning.

Few research studies have been conducted in a primary school in early mathematical education about the teaching of geometry. This research aims to contribute with knowledge of how teachers' awareness and understanding of necessary conditions to enhance students' abilities to discern two- and three-dimensional shapes develop. In this research, qualitative methods were used to analyse data from a lesson study in grade 4 in the subject of mathematics. Data were primarily collected through audio-recorded conversations with teachers before and after the lesson, and the results of students' pre- and post-test. The results of this research showed increased awareness of using collaboration opportunities to apply professional classroom instructions and activities to enhance students' knowledge of two- and three-dimensional shapes. This research elucidates how the practice-based professional development approach emphasised the teachers' teaching targets for understanding students' content knowledge of geometric shapes. Additionally, the result highlighted teachers' awareness and understanding of the challenges students face in learning about three-dimensional shapes from two-dimensional representations. Future research should develop a more iterative and revised research lesson design to develop more powerful content knowledge and classroom activity in this topic area.