This study investigates how preservice teachers use pictures in their written solutions to arithmetic word problems and how these visualisations reflect their reasoning processes. A total of 144 written solutions from 36 preservice teachers were analysed. The findings reveal notable differences in the types of pictures used and the reasoning processes employed by the preservice teachers, depending on whether the problems involved fractions. In fraction-related problems, preservice teachers predominantly used representational, informational, and organisational pictures, which is lacking in non-fraction problems. Reasoning processes such as selecting and exploring were common across both problem types. However, attributes such as reconfiguring, encoding, and abstracting appeared more frequently in solutions to nonfraction problems. This result suggests that, although preservice teachers more often employ visual tools when solving fraction tasks, they do not fully exploit these tools to support deeper conceptual understanding or the problem-solving process. The study highlights the need to develop preservice teachers’ competencies in using visualisations to effectively represent the problem-solving process by reasoning about the mathematical concepts involved in problems. These findings provide concrete suggestions aimed at strengthening preservice teachers’ skills in using visualisations and reasoning, which equip them to better support students’ mathematical reasoning in their future classrooms. The study also contributes to a deeper understanding of how visual tools and reasoning interact in mathematics education, thereby addressing existing gaps in the research.

While the concept of function has been extensively studied by researchers worldwide, there has been limited research conducted on this topic in Sweden. This paper aims to fill this gap by examining the significance of critical aspects and their impact on mathematical communication. The study analyzes the content presented to students regarding the concept of function, both in the classroom and through textbooks, and compares it to the actual learning outcomes. Data for the analysis was collected from two classes, involving two teachers and 45 students, including video recordings of lessons and tests. The analytical framework incorporates concepts from variation theory. The findings highlight that effective communication in the classroom hinges on considering critical aspects as the foundation of students’ learning. In this study, the argument of functions is identified as a crucial aspect in understanding functions.

In this study, we aimed to present the features of pre-service teachers’ reasoning and sense making in the process of exploring growing patterns tasks. Participants were 79 preservice primary teachers in Sweden. The analysis was grounded in the concept of critical aspects from variation theory and the concepts of assemblage, lines of flight, segmentarity, and rupture from a post-structuralist philosophical perspective. Results indicate that the concepts of assemblage, line of flight, rupture, and segmentarity can be useful tools for analyzing analysing reasoning and sense making in mathematics. The results also show that in mathematics, the line of flight can be seen in moments of creativity and innovation. Furthermore, the results show that rupture can be seen in the connection of concepts or ideas that challenge established mathematical structures and ways of thinking. Further, the results shows that segmentarity creates disciplinary boundaries and inhibits reasoning and sense making. Therefore, it is important to recognize and break down these boundaries to facilitate more creative and innovative reasoning and sense making in mathematics

While the concept of a function has been extensively researched worldwide, there has been limited investigation into how functions are taught in the classroom and the students’ opportunities to understand the uses of notation in two upper-secondary classroom settings. This paper aims to address this gap by examining the content presented in a textbook and within the teaching that occurs in two upper secondary classrooms. Data for the analysis were collected from two classes, involving two teachers and 45 students, and included video recordings of lessons and tests. The analytical framework is grounded in variation theory. The findings underscore the crucial role of teaching in providing students with the opportunity to discern the meaning of the notation related to the concept of a function.

In this study, we aimed to present the features of pre-service teachers’ reasoning and sense making in the process of exploring growing patterns tasks. Participants were 79 pre-service primary teachers in Sweden. The analysis was grounded in the concept of critical aspects from variation theory and the concepts of assemblage, lines of flight, segmentarity, and rupture from a post-structuralist philosophical perspective. Results indicate that the concepts of assemblage, line of flight, rupture, and segmentarity can be useful tools for analyzing analysing reasoning and sense making in mathematics. The results also show that in mathematics, the line of flight can be seen in moments of creativity and innovation. Furthermore, the results show that rupture can be seen in the connection of concepts or ideas that challenge established mathematical structures and ways of thinking. Further, the results shows that segmentarity creates disciplinary boundaries and inhibits reasoning and sense making. Therefore, it is important to recognize and break down these boundaries to facilitate more creative and innovative reasoning and sense making in mathematics.

Researchers note that algebraic reasoning and sense making is essential for buildingconceptual knowledge in school mathematics. Consequently, pre-service teachers’own reasoning and sense making are useful in fostering and developing students’algebraic reasoning and sense making. We report here the features of pre-serviceteachers’ reasoning and sense making in algebra, specifically in the process ofanalysing problem posing, with a focus on first-degree equations. The followingresearch questions served as a guide in the analysis of data: What are the characteristicsof the problem-posing tasks used for reasoning and sense making of first-degreeequations? What are the characteristics of pre-service teachers’ reasoning and sensemaking in problem-posing tasks?This study is part of ongoing research carried out with pre-service teachers enrolled inthe abovementioned mathematics teacher education course. All the pre-serviceteachers were older than 19, from diverse socio-economic backgrounds and attendedclass for all 10 weeks of the semester, including the seven weeks in which data wascollected. Sixty-six pre-service primary teachers participated in an anonymous writtenexam and were informed about our research (characteristics, aim, confidentialityissues, etc.). The data analysis adopted a qualitative/interpretative approach, and theunit of analysis has three dimensions: reasoning, sense making and critical aspects.According to Olteanu (2020), reasoning and sense making are closely related to eachother and to these dimensions in the manner of a ‘rhizome’ (Deleuze & Guattari, 1987).Results revealed that the pre-service teachers create a rhizomatic reasoning and sensemaking that is characterized by lines of rupture. Those lines interconnect and arise fromincorrect translations of rhizomatic problem-posing task (RPPT) into mathematicalnotations and the failure to discern the difference between variables and variables asunknown numbers; i.e., between algebraic expression and equation. The characteristicsof reasoning in RPPT and of pre-service teachers are selecting, exploring,reconfiguring, encoding, abstracting, and connecting to highlight associations andrelationships between different content, and the characteristics of sense making arerecognition, relationships, profiling, comparing, laddering, and verifying.