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 deals with students’ arithmetical and algebraical knowledge, with focus on a conceptual connection, and the relationship between two aspects of knowledge. The content in question is rational numbers, rational equations and problem-solving about proportion and ratio in grades 7, 8 and 9. The method contains three tests given to 400 students in grades 7–9. Tools for analysis were theories of generalizing arithmetic into algebra (Kieran, 2004) and the relationship between arithmetic and algebra in a conceptual context (Kaput, 2008). 

Current research shows that students’ knowledge of algebra and arithmetic often has a limited conceptual connection and a weak relationship. Their knowledge of arithmetic operations and solving rational equations use to be just procedural and rely on formulas learnt in a procedural way and often mixed up. The study also shows that students’ procedural strategies in finding formulas suitable for solving the equations, as well as carrying out the corresponding calculations, were often insufficient.  

The study pays attention to shortcomings in students' conceptualization of arithmetic operations with rational numbers and how to apply them solving rational equations. One reason for this is lack of continuity in instruction and learning.  

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.

Den första delen av boken handlar om matematikens didaktik innebörd som är ett tvärvetenskapligt forskningsfält, där möts olika vetenskaper. I boken reder författarna ut  individualiserings-och konkretiseringsprocesser samt vad diagnostisering och kontinuiteten i undervisningen kan innebära. I andra delen av boken beskrivs om hur matematiskt innehåll inom olika områden i skolans matematik för årskurs 1-6 kan utformas och hur undervisningen inom dessa områden kan gå i klassrumspraktiken. 

The purpose of this article is to draw attention to, analyze and discuss the following issues: (1) What mathematics should teacher education include, in order for student teachers to gain knowledge of a teaching practice that ensures the progression in students' mathematics development, and (2) How can the subject-specific content in an algebra course for student teachers be designed through an interaction between formal concepts in mathematics and the content of practical mathematics teaching with focus on algebra. An analysis of these issues is carried out within a theoretical framework of didactics of mathematics, related to a research context.

This article is based on two research projects, MIL (Mathematics in teacher education) and SKUM (Student teachers’ knowledge and perceptions of mathematics) as well as ongoing research work with a focus on the quality of student teacher education in mathematics and the didactics of mathematics in the K–3 and 4–6 programs at Södertörn University. 

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. 

Proportional reasoning causes considerable difficulty for students. One reason for this is, that a lack of basic understanding of fractions in earlier years, causes difficulties in the middle years. Moreover, learning of fractions and proportion is a long-term process and students encounter it continuously from grade 1 to 9. It is also difficult to teach, and teachers’ mathematical content knowledge plays a crucial role in students’ learning of mathematics. The purpose of this study is to analyse the effects of teaching, and the influence of teachers’ mathematical content knowledge about fractions and proportion on students learning in progression from grade 4 and 5 to grade 8. The method contains pre- and post-tests for 86 students, classroom observations, checking students’ written solutions and interviews of 35 selected students after the post-test. The intervention includes construction of educational materials (EM), teachers participating in seminars-training related to EM and teacher’s implementation of EM. The EM contains tasks for students and a teachers’ guide with aims and goals for teaching and a theoretical background. Tools for analysis were a methodological design, Variation theory, and a theoretical approach, Mathematical Content Knowledge for Teaching. In focus of the analysis was students learning in progression, related to variation and crucial aspects of learning. Findings from this study shows that teachers’ mathematical content knowledge and their ability to identify the objects of learning and apply this in teaching is very important for students’ learning and progression in their learning. Most students showed an ability to learn, but their performance was intimately linked to teachers’ perception of the crucial aspects in teaching, and variation. Moreover, anomalies in students’ perceptions of basic concepts caused obstacles in their learning. Some anomalies seem to have followed students from middle School to grade 8. Finally, the study illustrated how anomalies arise if misconceptions are not noticed by teachers. The outcome of the study can explain more about crucial steps in teaching and learning of fraction and proportion. The study pay attention to challenges in mathematics teaching.

This paper examines and analyzes how students learn multiplication tables, specifically the role of multiplicative structures and how these are used as students learn to master the tables. The analysis is performed in the context of the generalization process related to the teaching activity focusing students’ perception of concepts. The theoretical approach applies Davydov’s concept of theoretical generalization as perception-conception-elementary concept (PCE model) and Vergnaud’s theory of multiplicative structures in three classes: mapping rule (MR), multiplicative comparison (MC), and Cartesian product (CP). For the methodological design, Marton’s variation theory has been chosen.

This study includes two teachers and 40 students in two Year 3 classes, followed two years later by one teacher and 25 students in one Year 5 class. The analysis of the outcome is based on documented classroom observations, one-on-one interviews with students and teachers’ reflections on students’ learning outcomes. The conclusion of the study is that the generalization of multiplication is a difficult process for students, especially in the classes MC and PC, and one that sometimes results in challenges to identifying multiplicative situations and relating these to the multiplication tables. This illustrates that teaching activities and teachers’ support are necessary conditions for students’ learning. The study also shows that multiplicative structures can help students to find and systematize crucial patterns in the multiplication table, allowing them to learn the multiplication table in a more efficient and structured manner. During the one-on-one interviews, students actively searched for and found structures and solutions that did not come up during lessons. This shows that multiplicative structures are a suitable didactic tool for identifying patterns in multiplication tables, thereby facilitating learning other than by rote.

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 study examines the impact of teachers’ subject matter knowledge on students’learning. The mathematical content deals with rational number as fractions andproportion. The study includes pre- and post-tests from 99 students, classroomobservations, students’ written solutions and interviews with 48 selected studentsafter the post-tests. Findings from this study show that the impact of teachers’ subjectmatter knowledge and ability to identify the objects of learning, and apply this inteaching, strongly influenced the development of students’ conceptual learning aboutfractions and proportion.

This chapter examines and analyzes students’ learning about aspects of the concept of multiplication with a focus on perceptions and representations, and how they apply this to handling multiplicative situations and patterns in the multiplication tables. The analysis has been performed in the context of the generalization process related to teaching activities, with a focus on students’ perception of multiplication. The theoretical approach is based on Davydov’s (1990) view of theoretical generalization as a perception-conception-elementary concept (PCE model). The current mathematical content was classified according to: (1) multiplicative structures (Vergnaud, 1983);and (2) basic laws of algebra (van der Waerden, 1971). The relationship between students’ learning and the teaching process was studied in order to identify students’ learning in action. The study comprises two teachers and 40 students in two classes in grade 3 and was followed up two years later in grade 5 with one teacher and 25 students. The findings of this study can provide knowledge about students’ learning about multiplication using structures and multiplication tables in a conceptual context.

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.