The article contributes to the ongoing discussion on ways to deal with the diversity of theories in mathematics education research. It introduces and systematizes a collection of case studies using different strategies and methods for networking theoretical approaches which all frame (qualitative) empirical research. The term 'networking strategies' is used to conceptualize those connecting strategies, which aim at reducing the number of unconnected theoretical approaches while respecting their specificity. The article starts with some clarifications on the character and role of theories in general, before proposing first steps towards a conceptual framework for networking strategies. Their application by different methods as well as their contribution to the development of theories in mathematics education are discussed with respect to the case studies in the ZDM-issue.

Learning situations that concentrate on conceptual understanding are particularly challenging for learners with limited proficiency in the language of instruction. This article presents an intervention on fractions for Grade 7 in which linguistic challenges and conceptual mathematical challenges were treated in an integrated way. The quantitative evaluation in a pre-test post-test control-group design shows high effect sizes for the growth of conceptual understanding of fractions. The qualitative in-depth analysis of the initiated learning processes contributes to understanding the complex interplay between the construction of meaning and activating linguistic means in school and technical registers.

No abstract

In spite of the widely accepted need for language-responsive subject-matter teaching, few teachers are prepared for this challenge due to the lack of empirically founded subject-specific professional development (PD) programs for language-responsive classrooms. The design research study presented in this article pursues the dual aim of (a) promoting teachers' expertise in language-responsive mathematics teaching using PD courses and (b) investigating teachers' developing expertise in qualitative case studies. Both aims are pursued based on a conceptual framework for teacher expertise in language-responsive mathematics teaching, starting from typical situational demands that teachers face in language-responsive mathematics teaching and the orientations, categories, and pedagogical tools they need to cope with these situational demands, especially the demand to identify mathematically relevant language demands. For language-responsive teaching, the interplay of categories for mathematical goals and language goals turns out to be of high relevance. Keywords Teacher expertise. Language-responsive mathematics. Individual use of categories. Design research for teachers For nearly 40 years, research has pointed to challenges in linguistically diverse classrooms (Austin and Howson 1979; Ellerton and Clarkson 1996). Until now, language proficiency in school academic language has been a crucial factor with a high impact on mathematics achievement (Secada 1992; Haag et al. 2013; Marshman

research (in Sect. 2.1) and identify dimensions of variability (Sect. 2.2). Distinguishing two archetypes, 'design research with a focus on curriculum products', and 'design research with a focus on learning processes' (in Sect. 2.3), this thematic issue and the overview on achievements and challenges mainly concerns the elaboration of the latter archetype for mathematics education, where it is the dominant approach.We first characterize this approach by grounding it in background theories that require an active role of students in constructing their own knowledge (Sect. 3). This is followed by a methodological reflection on this type of design research (Sect. 4), which takes its starting point in some of the critique on design research. As generating theories is crucial for design research with a focus on learning processes, we discuss the variety of theories that play a role, and the process of developing theories in design research. We close with an overview of how those considerations play out, elucidated with reference to the contributions to this thematic issue (Sect. 5). Design research as a research program of increasing importance in general education and mathematics education 2.1 Different origins of design researchDesign research emerged in different places, under different names (e.g., "design research", "design-based research", "design experiments", "design theories", "educational design research", and "developmental research"), and-what is more important-to serve different needs, resulting in different characteristics. In this article we use the term "design research" as a generic, symmetric term, Abstract Design research continues to gain prominence as a significant methodology in the mathematics education research community. This overview summarizes the origins and the current state of design research practices focusing on methodological requirements and processes of theorizing. While recognizing the rich variations in the foci and scale of design research, it also emphasizes the fundamental core of understanding and investigating learning processes. That is why the article distinguishes two archetypes of design research, one being focused on curriculum innovations, one being focused on developing theories on the learning processes, which is the main focus of the thematic issue. For deepening the methodological discussion on design research, it is worth to distinguish aims and quality criteria along the archetypes and elaborate achievement and challenges for the future.

Given the challenges of scaling up content-related professional development (PD), the PD facilitators have gained increasing attention in PD research. In this mainly programmatic and structural article, we try to systematize existing research strategies which take into account the multifaceted and multi-level structure of PD research. We discuss the lifting strategy which draws upon structural analogies between the classroom and the PD level, the nesting strategy for research-based design, and the unpacking strategy for design-based PD research and locate these strategies within a Three-Tetrahedron Model for PD research. Thereby, we present a framework to systematize and explain existing research approaches and to identify necessary but missing research contributions. The article provides a language for initiating a discourse on research agendas which can support PD facilitators: The Three-Tetrahedron Model for PD research and design captures the complexity of learning and teaching at the classroom, teacher, and facilitator level that is needed to inform design and research into PD as well as to uncover gaps in the literature.

Subtraction can be understood by two basic models-taking away (ta) and determining the difference (dd)-and by its inverse relation to addition. Epistemological analyses and empirical examples show that the two models are not relevant only in single-digit arithmetic. As curricula should be developed in a longitudinal perspective on mathematics learning processes, the article highlights some exemplary steps in which the inverse relation is discussed in light of the two models, namely mental subtraction, the standard algorithms for subtraction, negative numbers and manipulations for solving algebraic equations. For each step, the article presents educational considerations for fostering a flexible use of the two models as well as of the inverse relation between subtraction and addition. In each section, a mathedidactical analysis is conducted, empirical results from literature as well as from our own case studies are presented and consequences for teaching are sketched. Two models of subtractionMany adults as well as school children understand subtraction solely as taking away. In this paper, we shall show the importance of the second model of subtraction (determining the difference) and the relevance of the inverse relation between addition and subtraction by adopting a longitudinal perspective. 1 Beforehand, some remarks are necessary with respect to the notions that we use. 1 Note that we do not present an empirical longitudinal study where we followed a set of students or a programme over a longer period of time. We adopt a longitudinal perspective for conducting a mathedidactical analysis (van den Heuvel-Panhuizen and Treffers, 2009).C. Selter (*) : S. Prediger : M. Nührenbörger : S. Hußmann

No abstract