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.

The purpose of the current empirical research, which departs from a socio-psychological perspective in terms of the concept of diagnostic teaching, is to explore and analyze mathematics teachers’ reflections about students’ learning of algebra.

The theoretical approach proposed by Bernstein and Miller, and the concept of inverse operations in algebra was applied. In this theoretical context, the purpose of developing a diagnostic framework is to include a conceptual triangulation of three domains: (1) conscious inclusion in the teaching environment of a component that addresses students’ different levels of prior knowledge and mathematical experience; (2) teacher-student content-oriented conceptual communication; and (3) a gradual sequencing of teaching content.

The methodological approach is a mathematical intervention, based on the main principles of the diagnostic framework conceptualization that is rooted in the Swedish PUMP project. The methodological tools are based on qualitative research and data collection from secondary school teachers’ group discussions, teachers’ analysis of students’ diagnostic tests, and written reflections. The participants comprise three teachers and 94 students. The crucial content of the study is a conceptual transition from whole and rational numbers to algebraic equations and problem-solving.

The findings of the study illustrate that teachers are aware of the importance of diagnostic teaching and the implementation of diagnostics in the teaching of algebra, practical awareness of students’ knowledge of algebra, and what should be taught. The results of the study show teachers’ practices and the challenges they face with handling mathematical content and content-oriented communication. The findings clearly indicate that diagnostic teaching that addresses students’ mathematical needs and teachers’ reflective teaching practice directed at student learning have not been fully developed in practice. 

The purpose of this project is to analyze how the scientific concept of inverse operations can be used as a “bottom-up” approach for teaching mathematical operations with whole and rational numbers. The primary aim of the study is to provide support for teachers in the teaching of scientific mathematical concepts to students of algebra. The theoretical approaches applied in this project are the theories of mathematical structures, especially the theory of inverse semigroups (Abelian groups), and relational thinking, in a comparison with instrumental thinking. The methodological approach uses Vygotsky’s doctrine of scientific concepts. The presentation of the analytical findings expands on previous research on additive and multiplicative inverses in teaching and conceptual learning of algebra. The findings illustrate how connections to mathematical structures and scientific concepts, such as the concept of inverse, lead to an understanding of algebraic operation rules for whole and rational numbers.

The purpose of this study is to examine how students’ repetition of arithmetic concepts is organized during the first year of upper secondary school. The theoretical approach is a triangulation of Freudenthal’s didactical phenomenology of mathematical structures, Bernstein’s repetition -without -repetition and Mezirow’s transformative learning theory. The study includes 94 upper secondary school students and their three mathematics teachers. The crucial content for repetition was whole numbers, rational numbers and potencies as prior knowledge for learning algebra-specific content, such as algebraic expressions, equations and problem solving.

The study is part of a research project about teachers’ professional development in a “research circle.”

The key components of the methodology are a qualitative analysis of the research questions and theoretical triangulation, which aim to identify significant empirical findings.

The results of the study show that limited space is given to conceptual repetition to support students’ learning of algebra. Moreover, the current textbook was not a useful source of support for the teachers. 

The objective of the present study is to analyze pre-service teachers’ mathematics knowledge base as a relationship between their substantive subject mathematics knowledge and syntactic subject mathematics knowledge. The theoretical approach refers to Shulman’s pedagogical content knowledge and Ball, Thames and Phelps’ subject matter knowledge for the teaching concept. The participating pre-service teachers (PSTs) were divided into two groups: 20 from a program course for K–3 teachers and 25 from a program course for grade 4–6 teachers. The crucial mathematical content for the study was algebraic patterns, rational numbers (fractions), proportionality and elements of combinatorics. The findings show the importance of gaining a better insight into and understanding of student teachers’ learning process and challenges related to the mathematics knowledge base (e.g., its substantive and syntactic components). The benefits of the empirical findings are that they led to new knowledge and a deeper understanding of PSTs’ learning of mathematical concepts through two learning processes (SUB and SYN), as well as the relationship and shift between these processes.

The purpose of this didactic project is to analyze a current research question, namely how student teachers’ knowledge of Abelian groups contributes to their understanding of an essential aspect of teaching algebraic concepts by extending numbers and arithmetic operations. The theoretical approaches employed are Subject Matter Knowledge, and the choice of algebraic content with focus on students’ learning of algebra. The analysis includes a broad perspective of modern algebra that leads to logical, conceptual consequences, and, for the student teachers, a course of education adapted to provide sustainable development of students’ mathematical knowledge. For example, special attention should be given to the Abelian groups as a framework for teaching and learning algebra.

The outcome of this didactic analysis is a follow-up of an earlier research project titled Student teachers’ knowledge of and perceptions of mathematics. This analytical result has been implemented and tested at the teacher training programs for courses in mathematics and mathematics didactics at the K-3 and 4-6 programs at Södertörn University in Sweden.

Current research addresses students’ arithmetic and algebraic knowledge, focusing on conceptual connections, and relationships between two aspects of knowledge. The contents in question are rational numbers and rational equations in grades 7, 8 and 9. The tools for analysis comprised an algebraic concept of rational numbers, Kieran’s theory of generalizing arithmetic into algebra, and Kaput’s  theory about the relationship between arithmetic and algebra in a conceptual context. 

 Current research shows that students’ knowledge of algebra and arithmetic has a limited conceptual connection and a weak relationship with each other. Their knowledge of arithmetic operations and solving rational equations used to be solely procedural and relied on formulas learnt in a procedural – and often mixed – manner. This caused conceptual consequences for students’ knowledge of rational numbers and their essential properties, as well as shortcomings in students’ ability to operate with rational numbers. This study highlights that conceptual transitions from rational numbers to rational equations play a crucial role in students’ learning, focusing on the conceptualization of arithmetic concepts and their ability to operate in an algebraic context.

The purpose of this study is to analyze pre-service teachers’ knowledge base in terms of their substantive subject matter knowledge and syntactic subject matter knowledge. The data collection includes pre-service teachers’ reflections upon their written analysis of teaching observations as mathematical content and written examples of how to implement a teaching moment (content) that they had created themselves. The crucial content for the study is algebraic patterns, rational numbers (fractions), proportionality, and combinatorics. The findings from the study identify and highlight areas to be developed, especially with respect to student teachers’ knowledge base and what they need to explore concerning subject-specific content for teaching mathematics, as well as crucial relevant content (substantive knowledge) and how to apply this in teaching (syntactic knowledge). Generally speaking, the study highlights the challenges student teachers face in grappling with mathematical concepts when they describe their observations, as well as in their own construction of a teaching moment. This means that in teacher education for mathematics, more attention ought to be paid to developing a knowledge base founded on the transformation of substantive knowledge to syntactical knowledge and its impact on student teachers’ learning of mathematics for teaching purposes. The study can also provide a deeper understanding of student teachers’ learning process and challenges related to the knowledge base, e.g., its substantive and syntactic components. 

The purpose of this project is to analyze how the scientific concept of inverse operations can be used as a “bottom-up” approach for teaching mathematical operations with whole and rational numbers. The primary aim of this analytical review is to provide support for student teachers in their learning of scientific mathematical concepts for the purposes of teaching. The theoretical approaches applied in this project are the theories of mathematical structures, especially the theory of inverse semigroups (Abelian groups), as well as relational thinking in comparison with instrumental thinking. The methodological approach is Vygotsky’s Doctrine of Scientific Concepts. The presentation of the analytical findings is intended to illustrate a clear connection to mathematical structures, such as the concepts of inverse, as well as how these structures can support pre-service teachers’ learning with regard to teaching mathematical operations in arithmetic and algebra. The development of a theoretical approach based on this study and its analytical findings is ongoing, but it is already being implemented in the teacher education programs for Grades K-3 and 4-6 at Södertörn University in Sweden.

This research analyzes transformative pedagogy (TP) and transformative learning theory (TL) in task-oriented teaching and presents a theoretically based model  for transformative instruction (TI model). This analytical reseach then introduces how this TI model can be specifically applied in teaching and learning of algebra-specific content.