This study is analytical research that examines a theoretical approach to transformative pedagogy (TP) and transformative learning (TL) in order to conduct a logical deductive construction of the transformative instruction model (TI model). The study aims to enhance the understanding and effectiveness of the interplay between teaching content and learning in the form of requires for the high-quality teaching of students in the cognitive development of mathematics. It also analyzes the conditions for transformative teaching and learning in a task-oriented context. The crucial psychological outcomes focus on the domains of TP and their interpretation within the context of Mezirow’s TL theory, emphasizing an instrumental, dialogic and self-reflective learning discourse, as well as meaning schemes in learning. The methodological approach is based on a systematic deductive analysis. The analytical results as a conceptualization and an analytical construction of a TI model related to transformative teaching and learning algebraic concepts. The TI model is the result of extending theoretical approaches to TP and TL, as well as logical connections to research about the formation of algebraic concepts. Benefits of the TI model is a new knowledge about an effective teaching of algebra, where students’ algebra learning, and knowledge development is crucial.

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.

This study examines two students’ reasoning processes when solving a Calculus problem, focusingon their discourse characteristics and factors influencing narrative endorsement. Using videorecordings of paired discussions, the results show rituals, social acceptance, ambiguity,inconsistency, and the impact of mathematical representations as characteristics of the narratives.The results also indicate that rituals and ambiguities are key factors in determining whether thenarrative becomes endorsed. Additionally, ambiguity provides opportunities for the two students tothink critically and creatively, which can support their reasoning process. 

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.

Researchers highlight the critical role of algebraic reasoning and sense-making in developing students’ conceptual understanding of mathematics. Consequently understanding and developing pre-service teachers’ own reasoning and sense-making abilities are essential for developing students’ algebraic skills. In this presentation, we present the results of a study in which we explore pre-service teachers’ reasoning and sense-making in algebra across two distinct types of tasks: problem-solving concerning growing patterns and problem-posing tasks involving first-degree equations. The research question guiding the analysis is: Which opportunities do different tasks provide for pre-service teachers’ reasoning and sense-making, and how do pre-service teachers utilize these opportunities in their reasoning and sense-making? The studyinvolved 79 pre-service primary teachers taking a mandatory written exam as part of amathematics teacher education course. The analysis applied a qualitative/interpretative methodology, utilizing variation theory and assemblage theory as analytical frameworks (Deleuze & Guattari, 1987; Marton, 2015; Olteanu, 2022). Key analytical tools - lines of flight, segmentarity, and rupture - inform diverse connections. The study results indicate that problem-solving and problem-posing tasks offer dynamic opportunities for connecting reasoning and sense-making among pre-service teachers. This connection is characterized by essential actions in reasoning like selecting, exploring, abstracting, encoding, and connecting information, while sense-making involves recognition, relationship identification, comparison, explanation, and verification. Pre-service teachers utilize these opportunities through the interplay of lines of flight, which provide new ways for reasoning and sense-making that enable the distinction of critical aspects. While pre-service teachers connect reasoning andsense-making by segmentarity and rupture in problem-solving tasks, they do this by employing segmentarity in problem-posing tasks. Segmentarity restricts changes i npre-service teachers’ reasoning and sense-making, thus limiting the distinction of critical aspects, while rupture leads to transformation or introduces new opportunities for distinguishing critical aspects.

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

In recent decades, approaches have evolved to define the subject matter knowledge essential for teaching, based on Shulman’s framework (1986) and additional insights provided by Ball et al. (2005). At the same time, scholarly researchers agree that teachers’ subject matter knowledge has a strong impact on the quality of instruction, something that plays a central role in the students’ subsequent ability to learn mathematical concepts and acquire mathematical skills (Adler & Ronda, 2015). In line with this, it is relevant to extend research and further investigate by addressing the question of teaching that is linked clearly to students’ solid and deep learning of mathematics, both by theoretical and empirical studies. This presentation introduces analytical findings from our research that examined a transformative teaching approach as an extension of teachers’ reflective practice with the starting point being different learning processes in the theoretical context of transformative pedagogy (TP) and transformative learning (TL) (Mezirow, 2003). The TP and TL theoretical framework proposed in this study prompts discussions on the structural aspects of task-oriented teaching, offering a conceptually grounded model for transformative instruction (TI).The focus in the TI- model is on various types of task-oriented learning, encompassing instrumental, dialogic, and self-reflective learning and conceptual overlap from prior knowledge to profound knowledge of mathematical concepts. The analytical result is also illustrated in the analytical application of the TI model by a practical example involving the conceptual overlap in students’ learning, from rational numbers to rational equations. The evaluation of these analytical findings will be improved by empirical implementation related to transformative teaching and the students learning specific-algebra concepts.