Theories of and in Mathematics Education
Abstract
How far has the didactics of mathematics developed as a scientific discipline? This question was discussed intensively in Germany during the 1980s, with both affirmative and critical reference to Kuhn and Masterman. In 1984, HansGeorg Steiner inaugurated a series of international conferences on ‘Theories of Mathematics Education’ (TME), pursuing a scientific program that aimed at founding and developing didactics of mathematics as a scientific discipline. Today, a more bottomup metatheoretical approach, the networking of theories, has emerged which has roots in the early days of discussing the developmental of mathematics education as a scientific discipline. This article presents an overview of this thread of development and a brief description of the TME program. Two theories from Germanspeaking countries are outlined and networked in the analysis of an empirical example that shows their complementary nature traced back to the TME program.
Keywords
Theories Theory of mathematics education (TME) Networking theories Mathematics education as a scientific discipline Learning activity Sign use Semiotic game7.1 Introduction
This chapter^{1} begins with a description of the historical situation of the community of mathematics education in Germanspeaking countries. The historical development and discussions surrounding the concept of theory related to mathematics education as a scientific discipline are traced from the 1970s up to the beginning of the twentyfirst century, in the Germanspeaking countries as well as internationally. We will describe the main points of the Theory of Mathematics Education (TME) program as introduced by HansGeorg Steiner. Two theoretical approaches, the theory of Learning Activity developed by Joachim Lompscher and Willi Dörfler’s semiotic view of doing mathematics related to diagrammatic reasoning and its semiotic game, are summarized and concretized through the application of them to the analysis of an empirical example, a students’ group solution of a mathematical task. Based on this example, we depict the networking of theories and the subsequent contribution to the TME program.
7.2 The Role of Theories in Relation to Mathematics Education as a Scientific Discipline: A Discussion in the 1980s
On an institutional and organizational level, the time span from the 1970s until the early 1980s had been a period of considerable change for mathematics education in former West Germany^{2}—both in school and as a research domain. The Institute for Didactics of Mathematics (Institut für Didaktik der Mathematik, IDM) was founded in 1973 in Bielefeld as the first research institute in the Germanspeaking countries specifically dedicated to mathematics education research; 1975 saw the inception of the Society of Didactics of Mathematics (Gesellschaft für Didaktik der Mathematik, GDM) as the scientific society of mathematics educators in the Germanspeaking countries (cf. Bauersfeld et al. 1984, pp. 169–197; Toepell 2004).
The teacher colleges (‘Pädagogische Hochschulen’), being the home of many mathematics educators at that time, were either integrated into full universities or developed into universities of education entitled to award doctorates. The Hamburg Treaty (‘Hamburger Abkommen’, KMK 1964/71), adopted in 1964 by the Standing Conference of Ministers of Education and Cultural Affairs (KMK), led to considerable organizational changes within the German school system. Although the traditional, more vocationally oriented ‘Volksschule’ (a common school covering both primary and secondary education, grades 1–8) was abolished, a new track called ‘Hauptschule’ was instituted for grades 5–8, reinstating a third track besides ‘Realschule’ and ‘Gymnasium’ for the years to come. In 1968, the Standing Conference’s ‘Recommendations and Guidelines for the Modernization of Mathematics Teaching’ introduced profound changes to the content of mathematics education at all ages. Along with this, the traditional designation of the school subject as ‘Rechnen’ (translated as ‘practical arithmetic’) was also abandoned for primary school education in favour of the subject designation ‘Mathematik’ (cf. Griesel 2001; Müller and Wittmann 1984, pp. 146–170).
Likewise, there was a vivid interest in discussing how far mathematics education had developed as a scientific discipline, as documented in both of the German language journals on mathematics education founded at that time: Zentralblatt für Didaktik der Mathematik (ZDM, founded in 1969) and Journal für MathematikDidaktik (JMD, founded in 1980). These discussions mainly addressed two aspects: the role and suitable concept of theories for mathematics education, and the question of how mathematics education was to be founded as a scientific discipline and how it could be further developed. Of course, both aspects are deeply intertwined.
Issue 6 (1974) of ZDM was dedicated to a broad discussion about the current state of the field of ‘Didactics of Mathematics’/mathematics education. The issue was edited by HansGeorg Steiner. It comprised contributions from Bigalke (1974), Freudenthal (1974), Griesel (1974), Otte (1974) and Wittmann (1974), among others. These articles were focused around the questions of (1) how to conceptualize the subject area or domain of discourse of mathematics education as a scientific discipline; (2) how mathematics education may substantiate its scientific character; and (3) how to frame its relation to reference disciplines, especially mathematics, psychology and educational science. While there was a rich variety in the approaches to these questions, and, likewise, to the definitions of ‘Didactics of Mathematics’ given by the various authors, cautioning against reductionist approaches seemed to be a common topic of these papers. That is, the authors agreed upon the view that mathematics education cannot be meaningfully conceptualized as a subdomain of either mathematics, psychology, or educational science alone.
The role of theory was more explicitly discussed about 10 years later in two papers (Burscheid 1983; Bigalke 1984) and in two comments (Fischer 1983; Steiner 1983) published in the ‘Journal für MathematikDidaktik’ (JMD). As an example of the discussion about theory of that time, we will convey the different positions in these papers in more detail.
 (1)
symbolic generalizations as “expressions, deployed without question or dissent […], which can readily be cast in a logical form” (Kuhn 1970, p. 182) or a mathematical model—in other words: scientific laws, e.g. Newton’s law of motion;
 (2)
metaphysical presumptions as faith in specific models of thought “shared commitment to beliefs”, such as “heat is the kinetic energy of the constituent parts of bodies” (p. 184);
 (3)
values and attitudes “more widely shared among different communities” (p. 184) than the first two components;
 (4)
exemplars, such as “concrete problemsolutions that students encounter from the start of their scientific education” (p. 187)—in other words: textbook or laboratory examples.
 (a)
metaphysical or metaparadigms (refers to 2);
 (b)
sociological paradigms (refers to 3);
 (c)
artifact or constructed paradigms (refers to 1 and 4).
Each paradigm shapes a disciplinary matrix according to which new knowledge can be structured, legitimized, and embedded into the discipline’s body of knowledge. Referring to Masterman, Burscheid used these types of paradigms to identify the scientific state of mathematics education with respect to four development stages of a scientific discipline (see Burscheid pp. 224–227):
Stagemodel of the development of a scientific discipline (p. 224, translated)^{a}
Stage  Paradigm characteristics  Core description 

1  Nonparadigmatic  Founding phase of the scientific discipline 
2  Multiparadigmatic science  Scientific schools based on paradigms emerge 
3  Dualparadigmatic science  Mature paradigms compete 
4  Monoparadigmatic science  A mature paradigm determines the scientific discipline 
Following the disciplinary matrix, Burscheid (pp. 226–236) identified paradigms in mathematics education and features at that time, according to which different scientific schools emerged and could be distinguished from one another, e.g. according to forms, levels and types of schools, or according to reference disciplines such as mathematics, psychology, pedagogy, and sociology. The constructed paradigms dealt in principle with establishing adequate theories in a discipline. Concerning building theories, however, the transfer of the model of Masterman and Kuhn was difficult to achieve because symbolic generalizations and/or scientific laws can be built more easily in the natural sciences than in mathematics education. This is because mathematics education is concerned with human beings who are able to creatively decide and act in the teaching and learning processes. Burscheid doubted that a general theory such as those in physics could ever be developed in mathematics education (p. 233). However, his considerations led to the conclusion that “there are single groups in the scientific community of mathematics education which are determined by a disciplinary matrix. […] That means that mathematics education is [still] heading to a multiparadigm science” (translated, p. 234).
Burscheid’s analysis was immediately criticized from two perspectives. Fischer (1983)^{3} claimed that pitting mathematics education against the scientific development of natural science is almost absurd because mathematics education has to do with human beings (p. 241). In his view “theory deficit” (translated, p. 242) should not be regarded as a shortcoming but as a chance for all people involved in education to emancipate themselves. The lack of impact to practice should not be overcome by topdown measures from the outside but by involving mathematics teachers from the bottomup to develop their lessons linked to the development of their personality and their schools (p. 242). Fischer did not criticize Burscheid’s analysis per se, but rather the application of a model postulating that all sciences must develop in the same way as the natural sciences towards a unifying paradigm (Fischer 1983).
Views on mathematics education as a scientific discipline
Discussant  View on mathematics education as a scientific discipline and its development 

Burscheid  Theories and theorizing are in the core of mathematics education as a scientific discipline. Taking the development of natural science as a role model, Burscheid assumes that the development of mathematics education advances by a process of maturing and competing paradigms 
Fischer  Fischer dismisses to take natural science as role model for scientific development since mathematics education has to do with human beings and it is practice based. It develops from practice bottomup by the development and emancipation of teachers 
Steiner  Steiner dismisses to take natural science as role model for scientific development of mathematics education because not even physics fits this model in all respects. Mathematics education as a scientific discipline is systemic and interdisciplinary at its core. It develops from the inside as a system of interrelations among mathematics, further disciplines and through the relation of theory and practice 
Bigalke  The nature of mathematics education as a scientific discipline follows scientific principles. Its theory concept consists of an unimpeachable kernel and an empirical surrounding. From the contextual nature of the scientific knowledge of mathematics education Bigalke infers the necessity to accept multiple principles and theories. This knowledge develops from the inside while theories are inspired by practice and have to prove being successful in research and practice 
One year later, Bigalke (1984) proposed exactly such an analysis from the inside. He analyzed the development of mathematics education as a scientific discipline as well, but this time without using an external developmental model. He proposed a “suitable theory concept” (translated, p. 133) for mathematics education on the basis of nine theses. Bigalke urged a theoretical discussion, and reflection on epistemological issues of theory development. Mathematics education should establish the principles and heuristics of its practice, specifically of its research practice and theory development on its own terms. Bigalke specifically regarded it as a science that is committed to mathematics as a core area with relations to other disciplines. He claimed that its scientific principles should be created by “philosophical and theoretical reflections from tacit agreements about the purpose, aims, and the style of learning mathematics as well as the problematization of its prerequisites” (translated; p. 142), and he emphasized that such principles are deeply intertwined with research programs and their theorizing processes.
A theory in mathematics education is a structured entity shaped by propositions, values and norms about learning mathematics. It consists of a kernel, that encompasses the unimpeachable foundations and norms of the theory, and an empirical component which contains all possible expansions of the kernel and all intended applications that arise from the kernel and its expansions. This understanding of theory fosters scientific insight and scientific practice in the area of mathematics education. (translated, p. 152)
Bigalke (1984) himself pointed out that this understanding of theory allows many theories to exist side by side providing a frame for a diversity of theories. It was clear to him that no collection of scientific principles for mathematics education would result in a ‘canon’ agreed across the whole scientific community. On the contrary, he considered a certain degree of pluralism and diversity of principles and theories to be desirable or even necessary (p. 142). Bigalke regarded theories as being inspired by the practice of teaching and learning of mathematics thus providing the link to this practice, founding mathematics education as a scientific discipline in which theories may prove themselves successful in research as well as in practice (Bigalke 1984). Progress of the scientific discipline results from the challenge to overcome the tension between the scientific principles and the values and norms in the practice of teaching and learning mathematics. Theories are the tools to overcome this challenge (p. 159), hence, allowing various forms of theories to be developed.
7.3 Theories of Mathematics Education (TME): A Program for Developing Mathematics Education as a Scientific Discipline

“Development of the dynamic regulating role of mathematics education as a discipline with respect to the theorypractice interplay and interdisciplinary cooperation.

Development of a comprehensive view of mathematics education comprising research, development, and practice by means of a systems approach.

Metaresearch and development of metaknowledge with respect to mathematics education as a discipline” (emphasis in the original; Steiner 1985, p. 16).
Steiner characterized mathematics education as a complex referential system in relation to the aim of implementing and optimizing teaching and learning of mathematics in different social contexts (p. 11). He proposed taking this view as a metaparadigm for the field (Steiner 1985, p. 11; Steiner 1987a, p. 46), addressing the necessity of metaresearch in the field. According to Steiner, the field’s inherent complexity evokes reduction of its complexity in favor of focusing on specific aspects, such as curriculum development, classroom interaction, or content analysis. According to Steiner, this complexity also creates a differential classification of mathematics education as a “field of mathematics, as a special branch of epistemology, as an engineering science, as a subdomain of pedagogy or general didactics, as a social science, as a borderline science, as an applied science, as a foundational science, etc.” (Steiner 1985, p. 11). Steiner required clarification of the relations among all these views, including the principle of complementarity on all layers, which means considering research and metaresearch, concepts as objects and concepts as tools (Steiner 1987a, p. 48, 1985, p. 15). He proposed understanding mathematics education as a human activity, hence, he added an activity theory view to organize and order the field (Steiner 1985, p. 15). The interesting point here is that Steiner implicitly adopted a specific theoretical view of the field but points to the multiple perspectives in the field which should be acknowledged as its interdisciplinary core.
Steiner (1985) emphasized the need for the field to become aware of its own processes of development of theories and models and investigate its means, representations and instruments. Epistemological considerations seemed important for him, specifically concerning the role of theory and its application. In line with Bigalke, he proposed considering Sneed’s and Stegmüller’s view on theory as suitable for mathematics education, since it encompasses a kernel of theory and an area of intended applications to conceptualize applicability being a part of the very nature of theories in mathematics education (p. 12).
In the first TME conference, theory was an important topic, especially the distinction between socalled borrowed and homegrown theories. Borrowed theories are taken from outside mathematics education whereas homegrown theories are those developed within mathematics education. With respect to this distinction, Steiner’s complementary view made him point to the danger of onesidedness. In his view, borrowed theories are not just transferred and used but rather adapted to the needs of mathematics education and its specific contexts. Homegrown theories, however, are able to address domainspecific needs but are subjected to the difficulty of establishing suitable research methodologies on their own authority. The interdisciplinary nature of mathematics education requires regulation among the different perspectives but also regulation of the balance between homegrown and borrowed theories (Steiner 1985; Steiner et al. 1984).
So, what is Steiner’s specific contribution to the discussion of theories and theory development? Like other colleagues, such as Bigalke, he has pointed to the role of theories as being in the core of mathematics education as a scientific discipline, and he proposed the notion of theory developed by Sneed and Stegmüller (cf. Jahnke 1978; pp. 70–90; see also Bigalke in this article) as being suitable for such an applied science. Steiner proposed complementarity to be a guiding principle for the scientific field and required investigating what complementarity means in each case of the field’s topics. In this respect, the dialectic between borrowed theories and homegrown theories is an integral part of the field that allows the discipline to develop from its core and to be challenged from its periphery. In addition, Steiner emphasized that mathematics education as a system (see Steiner 1987b) should reflect on its own epistemological basis, its own theory concepts and theory development, the relation between theory and practice, and the interrelation among all its perspectives. He has added that the specific view of mathematics education always incorporates some epistemological model of how mathematics and teaching and learning of mathematics are understood, and that this is especially relevant for theories in mathematics education.
7.4 PostTME Period
 (1)
Methodology: methodological and thus theoretical aspects in interpretative research (Beck and Jungwirth 1999), interviews in empirical research (Beck and Maier 1993), multimethods (Wellenreuther 1997); explaining in research (Maier 1998), methodological considerations on large scale assessments such as e.g. Third International Mathematics and Science Study (TIMSS) (Knoche and Lind 2000);
 (2)
Methods in empirical research: e.g., two special issues of ZDM in 2003 edited by Kaiser presented a number of methodical frameworks;
 (3)
Issues on metaresearch about what mathematics education is, can, and should include: considerations on paradigms and the notion of theory in interpretative research (Maier and Beck 2001), comparison research (Kaiser 2000; Maier and Steinbring 1998; Brandt and Krummheuer 2000; Jungwirth 1994), and mathematics education as design science (Wittmann 1995) and as a text science (Beck and Maier 1994).

Kirsch (1977/2000) and Becker (1978) are some of the rather sparse examples of metatheoretical reflection on Stoffdidaktik (“subject matter analysis”) from proponents of this traditional strand of mathematics education research in Germanspeaking countries. Both the notion of “concentration on the mathematical heart of the matter” (Kirsch 1977, 2000) and the sense and purpose of working out mathematically elaborated background theories for school mathematics (Becker 1978) have been questioned from a systems theory perspective in Steinbring (1998) and Steinbring (2011).

In their discussion of the use of interviews in interpretative research, Beck and Maier (1993) also presented an account of ‘understanding’ in mathematics classrooms (as process and product) developed according to the interpretative paradigm. Weigand (1995) contrasts this view with more traditional, normative accounts of ‘understanding’ developed just within the aforementioned framework of Stoffdidaktik. Weigand raises the question whether interpretative notions of ‘understanding’, originally developed in social science and cultural contexts, can in principle meet the particularities of mathematical thinking and learning, and stresses the complementarity of interpretative and Stoffdidaktikapproaches.

Knoche and Lind (2000) introduced models of item response theory which were used within the TIMSStudy (Trends in International Mathematics and Science Study) and subsequently were and are used in the Programme for International Student Assessment (PISA) to a broader audience of mathematics education researchers in German speaking countries. Since then, these models have become more widely adopted, and their benefits for assessing and analyzing students’ mathematical competence have been discussed, e.g. in Knoche et al. (2002), Büchter and Pallack (2012) and Leuders (2014). On the other hand, the appropriateness of these models for conceptualizing mathematical learning and the theoretical assumptions related to mathematical learning and student performance underlying these models have been challenged fundamentally, e.g. in Meyerhöfer (2004), Bender (2005), Vohns (2012) and Wuttke (2014)—some of the articles leading to rebuttals and rejoinders.
To reiterate, these are just some cursory examples of theoretical discussions across different strands of mathematics education research, and the reader may again be referred to the other articles in this volume for a more complete and balanced view on theoretical issues that have arisen and been discussed within and between the respective strands.
In order to provide a deeper insight into theory strands of Germanspeaking countries, we summarize two examples presented during the ICME13. Both theoretical approaches are then reconsidered and linked in an analysis of an empirical example, as it is usually done in the Networking of Theory strands to show how different theories may be used to better grasp the complexity of teaching and learning mathematics. Referring back to Steiner and his TME program, we will use the insight gained from this exercise to describe how mathematics education as a scientific discipline could reflect on its own epistemological basis, and do metaresearch as Steiner proposed to clarify the specificities and roles of its theories and their relations to practice.
7.5 Two Theories, Their Origins and Their Purposes
In the survey on theory strands in Germanspeaking countries (BiknerAhsbahs and Vohns 2016), two theories are described in detail and used for the analysis of an empirical example. The first theory, presented by Bruder and Schmitt (2016), is that of Learning Activity, originally developed by Joachim Lompscher in the German Democratic Republic (GDR). The second approach, presented in the same survey by Dörfler (2016), is an example of theorizing mathematics as a semiotic way of doing mathematics by referring to the concept of diagram introduced by Peirce and relating this to the idea of semiotic games by Wittgenstein (1999). For the purpose of this article, we will give a brief overview of both approaches.
7.5.1 Learning Activity
Bruder and Schmitt (2016) discuss the theory of Learning Activity developed by Lompscher within the theory culture of activity theory introduced by soviet psychologists, e.g. Vygotski, Leont’jev and Luria (Lompscher 2006). This theory culture takes activities as meaningful, purposeful, culturally and historically coined components of an activity system. Driven by a general motive, an activity brings itself about collectively by actions which are goal oriented and linked to the individuals’ psychological development. These actions are influenced by the social and cultural environment in which they are conducted. They are mediated by practical or mental tools available in the cultural environment and directed towards goals; they consist of operations determined by the specific situated conditions (Giest and Lompscher 2006, p. 39), and are often conducted unconsciously (Hasan and Kazlauskas 2014, p. 10). The relation between subject and object is at the core of any activity. This relation, together with actions, goals and available means, structure the activity (Giest and Lompscher 2006, pp. 37–41). Through activities, the subject actively acquires cultural knowledge and knowing, and in the same process this cultural knowledge and knowing is transformed by the individual. Thus, internalisation and externalisation are mutual processes of transformation (Lompscher 1985a, p. 25). Examples of activities are playing activity, learning activity, and working activity (Giest and Lompscher 2006, p. 55).
Lompscher has applied this theoretical view on teaching and learning in school (see Bruder and Schmitt 2016; Lompscher 1985a, b, 1989a, pp. 23–32; Giest and Lompscher 2006, pp. 67–106). Through a learning activity, a student acquires societal knowledge and cognitive competencies by interacting with other individuals and the environmental conditions. Lompscher (1989a) emphasizes that knowledge and competencies are related to “segments of societal experience of the world” (p. 29, translated). The general motive of a learning activity is selfdevelopment according to the specific cultural requirements (Giest and Lompscher 2006, p. 83), an aspect that distinguishes learning activity from other activities (p. 93). The teacher is crucial for constituting a learning activity: he/she arranges the learning conditions as tasks and provides the means to solve them. The learning activity on a topic is achieved by learning actions. These are arranged in steps, building a pathway for a learning trajectory to shape suitable learning conditions, providing resources for a sequence of learning actions which are supposed to lead to the desired learning goal. Subtasks are to be arranged in a way that the learner can adopt these tasks and their subgoals as his/her own. As Lompscher puts it: The outcomes of the individual learning is only achieved by “the intensity and quality of the learner’s own activity on and with the learning object, the adequately using resp. shaping or transforming of the learning conditions, the employment of available learning means resp. changes according to adequate aims and conditions” (Lompscher 1989a, p. 32, translated, emphasis in the original).
According to Bruder and Schmitt (2016), Giest and Lompscher (2006) distinguish three parts of a learning action: the orientation, the performance and the control part (p. 197), and three types of orientations a student may be able to conduct (Giest and Lompscher 2006, p. 192; see also Bruder and Schmitt 2016, pp. 16–18): trial orientation (driven by some kind of trial and error), pattern orientation (a sensitivity to patterns can be followed in a focused area), and field orientation (knowledge can be acquired and transferred in a complete knowledge field). A general motive for a learning activity is the development of field orientation, but this is not so easy to achieve. Bruder and Schmitt (2016, p. 16) refer to Davydov’s (1990) idea to start within an initial abstract feature as a means for orientating, exploring and enriching the abstract with the concrete. Ascending from the abstract to the concrete is regarded as a strong approach to reach field orientation as early as possible (see Lompscher 2006, 131–205, 1989b).
Lompscher’s research group has undertaken empirical studies in close connection with the teaching and learning practice in several school domains (Giest and Lompscher 2006). Mathematics was just one of them. The theory of learning activity has been intensively applied, adapted and further developed in research and development for teaching and learning mathematics in various directions (see Bruder and Schmitt 2016): for example, specifying elementary mental operations by Bruder and Brückner (1989), developing a comprehensive model for competence development for modelling, problem solving and argumentation (Bruder et al. 2003), investigating mathematical problem solving (Collet and Bruder 2008; Bruder and Collet 2011), developing learning tasks (Bruder 2010), and difficulties in representing functions (Nitsch 2015), to name just four.
7.5.2 A Semiotic View on Mathematics: Sign Use and Semiotic Game
The second example, presented by Dörfler (2016), is a specific semiotic view referring to Charles Sanders Peirce and Ludwig Wittgenstein. In the 1990s, Michael Otte introduced Peirce’s semiotics as an important view on mathematics to the German community of mathematics educators (see for example Fischer 2005, p. 375; Dörfler 2016, p. 23; Otte 1997). In the subsequent years, Peirce’s theory of semiotics has also been taken up by several researchers for different purposes, for example to develop a semiotic theory on learning (Hoffmann 2001), to illustrate its epistemological nature (Hoffmann 2005), to include the view on diagrams in the mathematics classroom (Dörfler 2006), for analysing chatcommunication (Schreiber 2006) or investigating the epistemic role of gestures (Krause 2016).
Dörfler’s theoretical view is rooted in a dynamic understanding of mathematics itself (Dörfler 2004, 2006, 2008, 2013a, 2016). Similar to Hoffmann (2005), Dörfler takes the concept of diagram introduced by Peirce as a starting point and describes doing mathematics as diagrammatic reasoning. However, the specificity in Dörfler’s elaboration is abstaining from the view on mathematical activity as a mental activity building abstract objects in the individual learner.
A sign, or representamen, is something which stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea, … (CP 2.228, emphasis in the original)
However, an interpretant does not necessarily need to be produced by a human being, it can also be produced in the physical world (Nöth 2000, p. 227). But in any case, the interpretant is the part of the sign that points to meaning. Peirce distinguishes between three kinds of signs in relation to the object: a sign can be an icon, an index, or a symbol. An icon, such as a photo of a person, is a sign that resembles the object: the material person. An index is a sign that refers to another sign because of its direct connection to it, like smoke refers to fire. A symbol is a conventionalized sign or a habitualized sign like the equivalent sign. It links the sign to the object by some kind of regularity or law (Nöth 2000, p. 66).
All necessary reasoning without exception is diagrammatic. That is, we construct an icon of our hypothetical state of things and proceed to observe it. This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. (CP 5.162)
Diagrammatic reasoning has been worked out more clearly by Bakker and Hoffmann (2005) for mathematics education. As indicated in the quote from Peirce (CP 5.162), they distinguish three steps of diagrammatic reasoning (pp. 340–341): (1) constructing a diagram to represent relations (diagrammatization); (2) experimenting with diagrams based on rules of the specific sign system, rules that tell us what can and what cannot be done with the diagram; and (3) observing the results of the experimentation and reflecting on them (cf. Hoffmann 2005, p. 129). The latter may lead to the discovery of patterns of relations, “which we may or may not be able to formulate with precision, and proceed to inquire whether it is true or not” (CP 5.162).
Dörfler (2016, p. 23) precisely describes how his theoretical view on working with mathematical diagrams represents doing mathematics. He argues that language is a sign system that just mediates between individuals and diagrams. In his view, diagrams are “extralinguistic signs” (Dörfler 2006, p. 27) with a spatial structure representing relations and providing rules for inventing, exploring and transforming them. As these rules are taken to be without contradictions, mathematical inferences appear consistent and strict. Mathematical meanings are at stake in these transformations as transforming rules. These rules can be exposed linguistically, but their meanings are more directly expressed in the relations of the diagrammatic inscriptions. However, individuals can build a relationship with these diagrams, while exploring, perceiving or talking about them. According to Dörfler, diagrammatic reasoning expresses the nature of doing mathematics, and it is highly creative. Dörfler rejects the existence of mathematical objects as abstract mental objects. Instead, mathematical objects, in his view, manifest in the relations of the diagrams and the rules of their transformations. Thus, “Diagrammatic reasoning is a rulebased but inventive and constructive manipulation of diagrams for investigating their properties and relationships” (Dörfler 2016, p. 26). Hence, it is at the core of the dynamic semiotic view on mathematics, for example when equations are produced they can be transformed into other equations by transformation rules and allow features to be observed and rules to be identified in the diagrams.
Thus for Peirce, to learn mathematics would be to acquire expertise in diagrammatic reasoning, and for Wittgenstein, it would be to participate in the many various sign games and their techniques. In both cases, which are closely related, it is of great importance to stick meticulously to establish rules… mathematics is thereby fundamentally shown to be a deeply social and socially shared cultural activity and product: sign activity can be executed with others and shown to others in public form. This is very different from imagining mathematics as a kind of abstract and mental activity. (Dörfler 2016, p. 30)
7.6 Reconsidering the TME Program by Networking the Two Theoretical Views
In line with the TMEprogram, we will now present a piece of metaresearch to clarify the nature of the two theories above and their relation to inform practice and to raise the awareness of the epistemology on hand. “(…) Steiner (1985) has emphasized the need for the field to become aware of its own processes of development of theories and models and investigate its means, representations, and instruments” (BiknerAhsbahs and Vohns 2016, p. 9). This kind of awareness can be achieved by metaresearch: that is, research on the research. To do so, we will use the Networking of Theories approach developed since 2006 (see BiknerAhsbahs et al. 2014, 2017; Dreyfus 2009). The Networking of Theories approach also emphasizes metaresearch. However, it does not explicitly want to advance the field, although this may happen in small steps during the practical process in research. Its main aim is to show a way to solve complex problems for which more than one theory is needed, and reflect on the very process. In order to include metaresearch as an additional practice into research, research practices have to be broadened to address also the theories themselves, their methodologies, and the research practices as research objects. The purpose for this kind of metaresearch may vary, for example it may be important to obtain methodological or theoretical clarity in a multitheoretical approach (Kidron et al. 2014; BiknerAhsbahs and Kidron 2015), to solve an apparently contradicting problem (Sabena et al. 2014), to clarify the nature of research results or the specificity of the particular epistemology in the study. A nice example for exploring the complementary relation of individual and social processes in an inquirybased classroom has been presented by Tabach et al. (2017).
In the following section, we will explain what we mean by networking theories. To undertake a networking case, we will present a small piece of data: a students’ group solution for a mathematical problem. This data set will be analysed from Lompscher’s perspective of Learning Activity and from Dörfler’s perspective of semiotic game and diagrammatic reasoning according to the common question of how the process of problem solving yields the result. By comparing and contrasting the two theoretical views rooted in the analyses, and the analyses presented, we want to contribute to the TME program and show the added value of the networking of theories for obtaining an indepth understanding of the two theories as well as the manner in which they inform teaching and learning practice.
7.6.1 Networking Theories Approach
Each theory provides particular knowledge to the field, paying attention to some aspects while leaving other aspects aside. Therefore, the main assumption in the Networking of Theories approach is to respect the diversity of the theories in the field as richness (BiknerAhsbahs 2009). Neither unifying theories nor ignoring other theories should be part of this practice. The Networking of Theories, say for example the two approaches above, is more a dialogue between theory cultures in multitheoretical research. This ‘dialogue’ (Kidron and Monaghan 2012) can be approached by the four pairs of networking strategies (Prediger et al. 2008) positioned in between the two poles of the landscape in Fig. 7.2 and ordered according to their degree of integration. Networking of theories begins with understanding the other theory and making one’s own theory understandable. What does this mean? For example, it means that assumptions which often are implicit should be explicated, or that historical roots as well as paradigmatic empirical cases can offer access to clarify the essential concepts of the theory (cf. BiknerAhsbahs and Prediger 2014). However, sometimes there are limits. If concepts emerge within an educational culture, it may be difficult or even impossible to explain them to another culture (BiknerAhsbahs et al. 2017, p. 2689). By comparing and contrasting theories, their similarities, commonalities and differences can be identified, hence, contribute to deepening the understanding of both theories. The intermediate step to integration is combining and coordinating the theories. This step is not always possible; for instance, when theories cannot be combined in a compatible way because this would lead to contradictory results in research. But for instance if the theories address complementary views on the teaching and learning processes (Tabach et al. 2017), the process of improving mutual understanding may progress. The final step is the strategy pair of local integration and synthesizing. Local integration sometimes can be achieved when boundary concepts (cf. Akkerman and Bakker 2011), which can be understood from both theories, are identified (Sabena et al. 2014), or when theoretical concepts of two theories can be integrated into a new theoretical framework (Shinno 2017). As Shinno has shown, the step of integration may have losses and gains: the concepts in the integrated framework may change their notion, but open up new directions of research.
This landscape of networking strategies will now be used to network the two theories, Learning Activity and doing mathematics as a semiotic game of diagrammatic reasoning.
7.6.2 Try to Find a Fraction Representing \( \sqrt 2 \)
7.6.3 The Semiotic Game Analysis
We first theorize the solution process in Fig. 7.3 by applying Dörfler’s elaboration on doing mathematics as a semiotic game of diagrammatic reasoning and learning mathematics as gaining expertise therein. To do so, we have to analyse the diagrams as they are transformed step by step, and identify the rules represented explicitly or implicitly in the transformations and relations expressed in the diagrams.
The students begin solving the task with the statement that \( \sqrt 2 \) has to be bigger than 1 but it is not clear where this comes from. They start with the fraction \( \frac{5}{4} \) being bigger than one (line 1), as a kind of tentative true rule that ‘this is taken as being equal to \( \sqrt 2 \)’.
In step 1 the tentative equation is transposed by conventionalized transformation rules of equations. The equation obtained is 32 = 25, which is wrong. The inequality is recognized by the students; but their inference ‘the fraction is too big’ is also wrong (line 2), since the original fraction is smaller than \( \sqrt 2 \). The implicit rule ‘taken as equal’ was too vague. This kind of reasoning ‘building a tentative equation for \( \sqrt 2 \), changing it to remove the square root and interpreting the result’ is repeated in the following steps but with creative changes in constructing arithmetic equations as diagrams.
In step 2, both the nominator and the denominator are changed at the same time by increasing both by 1. Since the nominator is bigger than the denominator, the new fraction has become smaller, but we cannot assume that the students know this. The new inference from the resulting equation 50 = 36, that the fraction is “too small”, is now correct (line 3).
In step 3, the diagram is worked out according to the same rules as before, but this time only the denominator is reduced by 1. If we take this as an interpretant of the previous inference, then the underlying rule is to make the next fraction slightly bigger. From the resulting equation 32 = 36, the students infer now that it is “still” too small (line 4), but this is wrong. The new fraction has indeed become bigger than \( \sqrt 2 \).
Step 4 reacts to the previous false inference, since the fraction is now made even bigger by increasing only the nominator by 1. The result 32 = 49 “still does not fit” (line 5) and it is even “worse than before” (line 6). “Worse” seems to indicate that the difference between 32 and 49 is bigger than the one between 32 and 36 taken from step 3. That the fraction now is bigger than \( \sqrt 2 \) does not appear as an interpretant.
Meanwhile a number of rules have emerged: increasing the nominator of the fraction by 1 and reducing the denominator of the fraction by 1 make the fraction bigger, reducing the nominator by 1 and increasing the denominator by 1 make the fraction smaller. The tentative rule ‘take the two numbers as being equal’ is a pragmatic rule which can be falsified by the inequality of the result. However, the inferences about the kind of inequality are inconsistent. From step 3 onwards, the rule for changing the fractions seems to be ‘change either the nominator or the denominator by 1 according to the previous result’.
In step 5, we would expect that either the nominator is reduced by 1 or the denominator is increased by 1. Reducing the nominator would reveal the previous fraction, hence, this transformation does not make sense. In fact, the students increase the denominator by 1. Since the manipulation of the equation now leads to the two numbers 50 and 49 close to each other the students’ result is \( \sqrt 2 \approx \frac{7}{5} \). The approximatelyequal sign and doubling the underlining indicate that an approximate result is accepted.
Through diagrammatic reasoning, two kinds of rules are put into effect: (1) if the equation is true, then the manipulation of it will lead to an equation which is also true. Otherwise the result will indicate how to approach the next iteration. (2) Finding an iteration of fractions to box \( \sqrt 2 \) is a quasisystematic way to determine a fraction close to \( \sqrt 2 \).
The students’ interpretations are expressed in linguistic terms, taken as inferences or interpretants, which show that they sometimes interact with the diagrams in an ambiguous way (line 2 to line 4). The visible transformations, the rules used and produced, are not precisely expressed. Conventionalized rules for transforming equations are used as routine actions not addressed in the students’ comments. Only the results are interpreted, but partly ambiguously. It turns out that the mistakes in steps 3 and 4 are not relevant because the underlying rule to change either the nominator or the denominator in an opposite direction revealed a result where the mistakes did not harm the process. The final strategy of approximating \( \sqrt 2 \) by boxing it through an iteration of fractions emerged as a heuristic rule that resonates well with the students’ overall strategy “to make [the] fraction fit” (line 9).
7.6.4 The Learning Activity Analysis
Let us now add the analysis from the perspective of the theory of Learning Activity. In contrast to Dörfler’s semiotic approach, this theory addresses the complete course of learning, from the teacher’s planning to the goals, whether they are achieved and what comes next. This planning already starts with the question of which culturalhistorical knowledge should be learned, whether this knowledge is already accessible, and how the goal should be approached. Specifically the history of teaching and learning in the class has to be considered in the preparation of this course. The teacher in our example has initially constructed \( \sqrt 2 \) on the number line. His next goal is the proof of the irrationality of \( \sqrt 2 \) as a prerequisite for achieving his final goal: the introduction of real numbers. In this teaching course, the task above is a subtask with the subgoal yielding the insight that a fraction which exactly represents \( \sqrt 2 \) cannot possibly exist.
Lompscher has emphasized the mutual dependency of the leaning actions, the learning goals, and the learning objects in the learning conditions that together provide an arrangement in which the students may constitute their own learning activity. Not surprisingly, this open task has produced three more types of solutions in the class. One student pair used their calculator to find an approximate fraction. A second pair tried to find a finite decimal fraction to represent \( \sqrt 2 \) but failed, and therefore showed by the last digits that this does not work: they got stuck. A third pair used the factorizing of prime numbers in a fraction to find a representation for: they also failed, but tried to find a reason why. Given this situation, only the solution above would prepare the teacher’s intended proof, although at this stage the prooflesson could be prepared in a way that also builds on the students’ diverse solutions.
In contrast to Dörfler’s view, the interplay of the subject and the object is at the core of the learning activity leading to the individual student’s personal development. Therefore, we have to ask, what kind of knowledge and competencies have the students previously built, and prospectively are to build in the future. In the solution presented in Fig. 7.3, two elementary acquisition actions (stressed by Lompscher) are shown, identifying and realizing: The students identify a fraction close to \( \sqrt 2 \), they realize transpositions of equations and build an iteration of fractions for approximating \( \sqrt 2 \). They use heuristic strategies and transforming equations as heuristic means, and thus realize an argumentation similar to that of a proof of contradiction. In their task solution, the two students show trial orientation at the beginning including errors. But through heuristic strategies (equations as heuristic means and systematically changing the starting conditions of the next step) they quickly begin to systematically build an approximation boxing \( \sqrt 2 \) into subsequent fractions, probably not yet conducted quite consciously. However, the way they transform the diagrams systematically depicts their ability of pattern orientation in the way boxing is realized, based on the interpretation of previous inequality. Field orientation does not seem to be touched yet because the theme of irrational numbers has just started to be in the scope of learning.
Can we finally confirm that the students have built their own learning activity through changing the conditions and resources given? We cannot exactly answer this question, but we may find indicators for this outcome. The students used heuristic strategies that are not required, such as equations as heuristic means, and asifactions as a heuristic strategy to reveal necessary conditions. They systematically scrutinized the manipulation of equations and checked the results to continue with a slight change of the conditions in the next step. Through heuristics, they constructed conditions which enabled them to proceed in the solving of the problem. In fact they show quite proficient problem solving actions leading to a result that could raise the question as to whether it would be possible to represent \( \sqrt 2 \) by a fraction, and whether or not a final solution could be reached algorithmically. All these aspects indicate that the students really have established their own learning activity yielding their solution of the task. However, they might not be aware yet that representing \( \sqrt 2 \) by a fraction is impossible. The theory of Learning Activity would now focus on the teacher’s actions of how to systematize all the students’ results and provide further tasks and resources to prepare the intended proof.
7.6.5 Undertaking a Networking Analysis
Networking both theories based on the empirical analyses confirms the results already achieved by the analyses done with another empirical case presented in the ICME13 survey (BiknerAhsbahs 2016).
The semiotic approach elaborated by Dörfler starts from a specific homegrown account of the dynamics of doing mathematics and presents an approach which describes this doing, however, by adapting the work of two philosophers. The theory of Learning Activity is a more comprehensive theory elaborated for many subjects, borrowed and applied to mathematics to develop students’ competencies in doing mathematics. It explicitly includes learning goals to be achieved. In terms of the semiotic game view, the students and their mental activities are not at the core of the analysis. The process of diagrammatic reasoning and the transformation rules expressed in the diagrams are addressed rather independently of the students’ individual way of interpreting the situation. The inferences can be taken as a mathematical part of the diagrams, thus, of diagrammatic reasoning. We may even state that the relationships shown in the diagrams, in which the next step can be regarded as an interpretant to the previous one, advance the transformation process and constitute the rules. In contrast, the Learning Activity analysis focuses more on the learners, the culturalhistorical conditions and the context in the course of teaching and learning in which the students may be able to develop themselves by creating an own learning activity.
Whereas diagrammatic reasoning and the rules obtained belong to the kernel of the theory’s identity of Dörfler’s semiotic approach, the individual students and their abilities belong to the theory’s periphery. In Lompscher’s theoretical view, this is the other way round: the students’ development is at the core of the theory of Learning Activity, whereas the diagrams are resources belonging to the conditions of this development. This has considerable consequences for research: the research question posed must be interpreted differently by the two approaches, and the methodological and conceptual tools used to gain scientific knowledge in research also differ. However, the two approaches could be used in a complementary way.
This complementarity (see Steiner 1985, 1987a) can be described with the metaphor of “zoomingout and zoomingin” (Prediger et al. 2010, p. 1533, referring to Jungwirth) when looking at the grain sizes of relevant processes. This is possible because both approaches share a certain sensitivity towards acting or doing. Coming from the teacher’s long term planning, we would zoom in on Dörfler’s view to observe and analyze the diagrammatic reasoning on a micro level, in order to reconstruct the rules shown in the semiosis. The students’ interpretation may indicate aspects of their development when the Learning Activity theory is considered. We then would have to zoom out again in order to take the whole course of teaching and learning as a complementary view into account. If Lompscher’s view is considered, we would ask what was learned before the task is posed, what kinds of resources are available, and which resources have to be made available for the students to reach the subgoals, which conditions are to be met, and how they can be changed to accomplish the overall goal. Most importantly, the aim would be to construct the course of subgoals and subactions in a way that constitutes a suitable learning activity for reaching the learning goal and revealing field orientation.
7.7 Conclusions
What can be learnt from this networking case for advancing the field by metaresearch in the sense of the TME program?
The debate surrounding the developmental stage of mathematics education as a scientific field in the 1980s already showed contrasting views. While the analysis of Burscheid based on the model of Kuhn and Masterman indicates that a monotheoretical view was desirable for advancing the field, Bigalke and Steiner emphasized the multitheoretical or even the interdisciplinary character of mathematics education as a scientific field with a specific focus on complementarity, in which the practice of teaching and learning of mathematics plays a significant role. If research is used to inform the practice of teaching and learning or to address diverse cultures, multitheoretical views may be much more useful to grasp the complex nature of the settings in the field. Such an approach could help to gain complementary knowledge to inform practice regarded from different angles, as Steiner has pointed out. In this sense, the networking of theories is a kind of metaresearch and a challenging way of research practice, when added to normal research. Its purpose is to contribute to the improvement of solving problems in the field of mathematics education. For that, it is necessary to advance theoretical and methodological clarity on the one hand, and the communication among the theory cultures and among theory and practice on the other. The previously presented networking example shows that the metaphor of ‘zooming in and zooming out’ may guide research with complementary theoretical approaches of different grain sizes heuristically.
In the TME program, Steiner has elaborated a more general topdown view for advancing the field but it does not show how this program can be implemented; that is, how metaresearch can be conducted in a way to advance the field. The TME program could rather serve as an orientation scheme, whereas the Networking of Theories regarded as an additional research practice provides examples of concrete metaresearch showing how it improves solving problems in the field and why this kind of metaresearch is useful. The Networking of Theories approach has been predominantly developed by several European researchers (see BiknerAhsbahs and Prediger 2014), but there are forerunners in the theory tradition of Germanspeaking countries, for example interesting cases of the networking of theories were presented by Bauersfeld (1992a, b) and Maier and Steinbring (1998). Advancing the field as a scientific domain as Steiner has attempted may be the byproduct of such deep and careful casebased metaresearch.
The Networking of Theories strand has started to provide concrete examples for such a research practice, pointing to its benefit and being at the same time sensitive about the difficulties and limits a multitheoretical approach may bring with it (see for example BiknerAhsbahs et al. 2017).
Footnotes
 1.
This chapter presents the ICME13 Topical Survey ‘Theories in and of Mathematics Education’ (BiknerAhsbahs et al. 2016) in a shorter, partly reworked version: Sects. 7.1 and 7.2 are slightly revised versions (see ibid. pp. 1–9), Sect. 7.3 has been reworked and expanded (see ibid. pp. 10–11). Sections 7.4 and 7.5 present a summary and intensified rework of BiknerAhsbahs et al. (2016, pp. 13–42).
 2.
 3.
Fischer also feared that once mathematics education would develop towards a unifying paradigm, the field of mathematics education were more concerned with its own problems like physics and, finally, would develop separating its issues from societal concerns.
 4.
We will not further elaborate on the theory concept by Stegmüller and Sneed as we wish to focus on the debate conducted at that time.
 5.This program was later reformulated by Steiner (1987a, p. 46; emphasis in the original):

Identification and elaboration of basic problems in the orientation, foundation, methodology, and organization of mathematics education as a discipline.

The development of a comprehensive approach to mathematics education in its totality when viewed as an interactive system comprising research, development, and practice.

Selfreferent research and metaresearch related to mathematics education that provides information about the state of the art—the situation, problems, and needs of the discipline while respecting national and regional differences.

References
 Akkerman, S., & Bakker, A. (2011). Boundary crossing and boundary objects. Review of Educational Research June 2011, 81(2), 132–169.Google Scholar
 Bakker, A., & Hoffmann, M. (2005). Diagrammatic reasoning as the basis for developing concepts: A semiotic analysis of students’ learning about statistical distributions. Educational Studies in Mathematics, 60, 333–358. https://doi.org/10.1007/s1064900555368.Google Scholar
 Bauersfeld, H. (1992a). Activity theory and radical constructivism—What do they have in common and how do they differ? Cybernetics and Human Knowing, 1(2/3), 15–25.Google Scholar
 Bauersfeld, H. (1992b). Integrating theories for mathematics education. For the Learning of Mathematics, 12(2), 19–28.Google Scholar
 Bauersfeld, H., Otte, M., & Steiner, H.G. (Eds.). (1984). Schriftenreihe des IDM: 30/1984. Zum 10jährigen Bestehen des IDM. Bielefeld: Universität Bielefeld.Google Scholar
 Beck, Ch., & Jungwirth, H. (1999). Deutungshypothesen in der interpretativen Forschung. Journal für MathematikDidaktik, 20(4), 231–259.Google Scholar
 Beck, Ch., & Maier, H. (1993). Das Interview in der mathematikdidaktischen Forschung. Journal für MathematikDidaktik, 14(2), 147–180.Google Scholar
 Beck, Ch., & Maier, H. (1994). Mathematikdidaktik als Textwissenschaft. Zum Status von Texten als Grundlage empirischer mathematikdidaktischer Forschung. Journal für MathematikDidaktik, 15(1/2), 35–78.Google Scholar
 Becker, G. (1978). Über Hintergrundtheorien geometrischer Schulkurse. Mathematica Didactica, 1(1), 13–20.Google Scholar
 Bender, P. (2005). PISA, Kompetenzstufen und MathematikDidaktik. Journal für MathematikDidaktik, 26(3), 274–281.Google Scholar
 Bigalke, H.G. (1974). Sinn und Bedeutung der Mathematikdidaktik. ZDM, 6(3), 109–115.Google Scholar
 Bigalke, H.G. (1984). Thesen zur Theoriendiskussion in der Mathematikdidaktik. Journal für MathematikDidaktik, 5(3), 133–165.Google Scholar
 BiknerAhsbahs, A. (2009). Networking of theories—Why and how? Special plenary lecture. In V. DurandGuerrier, S. SouryLavergne, & S. Lecluse (Eds.), Proceedings of CERME 6, Lyon, France. http://www.inrp.fr/publications/editionelectronique/cerme6/plenary01bikner.pdf. Accessed: July 23, 2010.
 BiknerAhsbahs, A. (2016). Networking of theories in the tradition of TME. In A. BiknerAhsbahs, A. Vohns, R. Bruder, O. Schmitt, & W. Dörfler (Eds.), Theories in and of mathematics education. ICME13 Topical Surveys (pp. 33–42). Switzerland: SpringerOpen.Google Scholar
 BiknerAhsbahs, A., & Kidron, I. (2015). A crossmethodology for the networking of theories: The general epistemic need (GEN) as a new concept at the boundary of two theories. In A. BiknerAhsbahs, Ch. Knipping, & N. Presmeg (Eds.), Approaches to qualitative methods in mathematics education—Examples of methodology and methods (pp. 233–250). New York: Springer.Google Scholar
 BiknerAhsbahs, A., & Prediger, S. (2010). Networking of theories—An approach for exploiting the diversity of theoretical approaches; with a preface by T. Dreyfus and a commentary by F. Arzarello. In B. Sriraman, & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (Vol. 1, pp. 479–512). New York: Springer.Google Scholar
 BiknerAhsbahs, A., Prediger, S., & The Networking Theories Group (Eds.). (2014). Networking of theories as a research practice in mathematics education. New York: Springer.Google Scholar
 BiknerAhsbahs, A., & Vohns, A. (2016). Theories in mathematics education as a scientific discipline. In A. BiknerAhsbahs, A. Vohns, R. Bruder, O. Schmitt, & W. Dörfler (Eds.), Theories in and of mathematics education. ICME13 Topical Surveys (pp. 3–11). Switzerland: SpringerOpen.Google Scholar
 BiknerAhsbahs, A., Bakker, A., Haspekian, M., & Maracci, M. (2017). Introduction to the thematic working group 17 on theoretical perspectives in mathematics education research. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1–5, 2017) (pp. 2683–2690). Dublin, Ireland: DCU Institute of Education and ERME Dublin (Ireland). http://www.mathematik.unidortmund.de/~prediger/ERME/CERME10_Proceedings_2017.pdf. Accessed: May 2, 2018.
 BiknerAhsbahs, A., Vohns, A., Bruder, R., Schmitt, O., & Dörfler, W. (Eds.). (2016). Theories in and of mathematics education. ICME13 Topical Surveys (pp. 3–11). Switzerland: SpringerOpen.Google Scholar
 Brandt, B., & Krummheuer, G. (2000). Das Prinzip der Komparation im Rahmen der Interpretativen Unterrichtsforschung in der Mathematikdidaktik. Journal für MathematikDidaktik, 21(23/4), 193–226.Google Scholar
 Bruder, R. (2010). Lernaufgaben im Mathematikunterricht. In H. Kiper, W. Meints, S. Peters, S. Schlump, & S. Schmit (Eds.), Lernaufgaben und Lernmaterialien im kompetenzorientierten Unterricht (pp. 114–124). Stuttgart: W. Kohlhammer Verlag.Google Scholar
 Bruder, R., & Brückner, A. (1989). Zur Beschreibung von Schülertätigkeiten im Mathematikunterricht  ein allgemeiner Ansatz. Pädagogische Forschung, 30(6), 72–82.Google Scholar
 Bruder, R., & Collet, C. (2011). Problemlösen lernen im Mathematikunterricht. Berlin: Cornelsen Scriptor.Google Scholar
 Bruder, R., & Schmitt, O. (2016). Joachim Lompscher and his activity theory approach focusing on the concept of learning activity and how it influences contemporary research in Germany. In A. BiknerAhsbahs, A. Vohns, R. Bruder, O. Schmitt, & W. Dörfler (Eds.), Theories in and of mathematics education. ICME13 Topical Surveys (pp. 13–20). Switzerland: SpringerOpen.Google Scholar
 Bruder, R., Krüger, U.H., & Bergmann, L. (2003). LEMAMOP  ein Kompetenzentwicklungsmodell für Argumentieren, Modellieren und Problemlösen wird umgesetzt. https://eldorado.tudortmund.de/bitstream/2003/33436/1/BzMU144ESBruder191.pdf. Accessed: April 28, 2017.
 Büchter, A., & Pallack, A. (2012). Methodische Überlegungen und empirische Analysen zur impliziten Standardsetzung durch zentrale Prüfungen. Journal für MathematikDidaktik, 33(1), 59–85.Google Scholar
 Burscheid, H. J. (1983). Formen der wissenschaftlichen Organisation in der Mathematikdidaktik. Journal für MathematikDidaktik, 3, 219–240.Google Scholar
 Collet, C., & Bruder, R. (2008). Longtermstudy of an intervention in the learning of problemsolving in connection with selfregulation. In O. Figueras, J L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PMENA XXX (Vol. 2, pp. 353–360). Morelia: CinvestavUMSNH.Google Scholar
 Davydov, V. V. (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Soviet studies in mathematics education (Vol. 2). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
 Dörfler, W. (2004). Diagrams as means and objects of mathematical reasoning. In H.G. Weigand (Ed.), Developments in mathematics education in GermanSpeaking countries. Selected Papers from the Annual Conference on Didactics of Mathematics 2001 (pp. 39–49). Hildesheim: Verlag Franzbecker.Google Scholar
 Dörfler, W. (2006). Diagramme und Mathematikunterricht. Journal für MathematikDidaktik, 27(3/4), 200–219.Google Scholar
 Dörfler, W. (2008). Mathematical reasoning: Mental activity or practice with diagrams. In M. Niss (Ed.), ICME 10 Proceedings, Regular Lectures, CDRom. Roskilde: IMFUFA, Roskilde University.Google Scholar
 Dörfler, W. (2013a). Bedeutung und das Operieren mit Zeichen. In M. Meyer, E. MüllerHill, & I. Witzke (Eds.), Wissenschaftlichkeit und Theorieentwicklung in der Mathematikdidaktik (pp. 165–182). Hildesheim: Franzbecker.Google Scholar
 Dörfler, W. (2013b). Impressionen aus (fast) vier Jahrzehnten Mathematikdidaktik. Mitteilungen der Gesellschaft für Didaktik der Mathematik, 95, 8–14.Google Scholar
 Dörfler, W. (2016). Signs and their use: Peirce and Wittgenstein. In A. BiknerAhsbahs, A. Vohns, R. Bruder, O. Schmitt, & W. Dörfler (Eds.), Theories in and of mathematics education. ICME13 Topical Survey (pp. 21–31). Switzerland: SpringerOpen.Google Scholar
 Dreyfus, T. (2009). Ways of working with different theoretical approaches in mathematics education research: An introduction. In V. DurandGuerrier, S. SouryLavergne, & S. Lecluse (Eds.), Proceedings of CERME 6, Lyon, France. http://www.inrp.fr/publications/editionelectronique/cerme6/plenary01bikner.pdf. Accessed: March 18, 2016.
 Fischer, R. (1983). Wie groß ist die Gefahr, daß die Mathematikdidaktik bald so ist wie die Physik?—Bemerkungen zu einem Aufsatz von Hans Joachim Burscheid. Journal für MathematikDidaktik, 3, 241–253.Google Scholar
 Fischer, R. (2005). An interview with Michael Otte. In M. Hoffmann, J. Lenhart, & F. Seeger (Eds.), Activity and sign. Grounding mathematics education (pp. 361–378). New York: Springer Science + Business Media.Google Scholar
 Freudenthal, H. (1974). Sinn und Bedeutung der Didaktik der Mathematik. ZDM Mathematics Education, 6(3), 122–124.Google Scholar
 Giest, H., & Lompscher, J. (2006). Lerntätigkeit – Lernen aus kulturhistorischer Perspektive. Ein Beitrag zur Entwicklung einer neuen Lernkultur im Unterricht. In H. Giest & G. Rückriem (Eds.), International Culturalhistorical Human Sciences (ICHS), Band 15. Berlin: Lehmanns Media.Google Scholar
 Griesel, H. (1974). Überlegungen zur Didaktik der Mathematik als Wissenschaft. ZDM Mathematics Education, 6(3), 115–119.Google Scholar
 Griesel, H. (2001). Scientific orientation of mathematical instruction—History and chance of a guiding principle in East and West Germany. In H.G. Weigand (Ed.), Developments in mathematics education in Germany. Selected papers from the Annual Conference on Didactics of Mathematics Leipzig, 1997 (pp. 75–83). Hildesheim: Franzbecker.Google Scholar
 Hasan, H. & Kazlauskas, A. (2014). Activity theory: Who is doing what, why and how. In H. Hasan (Ed.), Being practical with theory: A window into business research (pp. 9–14). Wollongong, Australia: THEORI. http://eurekaconnection.files.wordpress.com/2014/02/p0914activitytheorytheoriebook2014.pdf. Accessed: May 5, 2017.
 Hoffmann, M. H.G. (2001). Skizze einer semiotischen Theorie des Lernens. Journal für MathematikDidaktik, 22(3/4), 231–251.Google Scholar
 Hoffmann, M. H.G. (2005). Erkenntnisentwicklung. Ein semiotischerpragmatischer Ansatz. Frankfurt a. M.: Klostermann.Google Scholar
 Jahnke, H. N. (1978). Zum Verhältnis von Wissensentwicklung und Begründung in der Mathematik  Beweisen als didaktisches Problem. Bielefeld: Universität Bielefeld.Google Scholar
 Jungwirth, H. (1994). Die Forschung zu Frauen und Mathematik: Versuch einer Paradigmenklärung. Journal für MathematikDidaktik, 15(3/4), 253–276.Google Scholar
 Kaiser, G. (2000). Internationale Vergleichsuntersuchungen – eine Auseinandersetzung mit ihren Möglichkeiten und Grenzen. Journal für MathematikDidaktik, 21(3/4), 171–192.Google Scholar
 Kaiser. (Ed.) (2003). Qualitative empirical methods in mathematics education—Discussions and reflections. Zentralblatt für Didaktik der Mathematik, 35(5 and 6).Google Scholar
 Kidron, I., & Monaghan, J. (2012). Complexity of dialogue between theories: Difficulties and benefits. In Preproceedings of the 12th International Congress on Mathematical Education. Paper presented in the Topic Study Group 37. (pp. 7078–7084). COEX, Seoul (Korea): ICME.Google Scholar
 Kidron, I., Artigue, M., Bosch, M., Dreyfus, T, & Haspekian, M. (2014). Context, milieu and mediamilieus dialectic: A case study on networking of AiC, TDS, and ATD. In A. BiknerAhsbahs, S. Prediger (Eds.) & The Networking Theories Group, Networking of theories as a research practice in mathematics education (pp. 153–177). New York: Springer.Google Scholar
 Kirsch, A. (2000). Aspects of simplification in mathematics teaching. In I. Westbury, S. Hopmann, & K. Riquarts (Eds.), Teaching as a reflective practice—The German didactic tradition (pp. 267–284). Mahwah: Lawrence Erlbaum Associates. (Reprinted from: Proceedings of the third international congress on mathematical education (pp. 98–120), by H. Athen, & H. Kunle (Eds.), 1977, Karlsruhe: Zentralblatt für Didaktik der Mathematik.Google Scholar
 KMK – Kultusministerkonferenz der Länder. (1964/71). Abkommen zwischen den Ländern der Bundesrepublik zur Vereinheitlichung auf dem Gebiete des Schulwesens. Beschluss der KMK vom 28.10.1964 in der Fassung vom 14.10.1971. http://www.kmk.org/fileadmin/Dateien/veroeffentlichungen_beschluesse/1964/1964_10_28Hamburger_Abkommen.pdf. Accessed: 10 March 2016.
 Knoche, N., & Lind, D. (2000). Eine Analyse der Aussagen und Interpretationen von TIMSS unter Betonung methodologischer Aspekte. Journal für MathematikDidaktik, 21(1), 3–27.Google Scholar
 Knoche, N., Lind, D., Blum, W., CohorsFresenborg, E., Flade, L., Löding, W., et al. (2002). (Deutsche PISAExpertengruppe Mathematik, PISA2000) Die PISA2000Studie, einige Ergebnisse und Analysen. Journal für MathematikDidaktik, 23(3), 159–202.Google Scholar
 Krause, C. M. (2016). The mathematics in our hands: How gestures contribute to constructing mathematical knowledge. Wiesbaden, Germany: Springer Spektrum.Google Scholar
 Kuhn, T. S. (1970). The structure of scientific revolutions (2nd ed.). Chicago: University of Chicago Press.Google Scholar
 Leuders, T. (2014). Modellierungen mathematischer Kompetenzen – Kriterien für eine Validitätsprüfung aus fachdidaktischer Sicht. Journal für MathematikDidaktik, 35(1), 7–48.Google Scholar
 Lompscher, J. (1985a). Die Lerntätigkeit als dominierende Tätigkeit des jüngeren Schulkindes. In J. Lompscher, L. Irrlitz, W. Jantos, E. Köster, H. Kühn, G. Matthes, & G. Witzlack (Eds.), Persönlichkeitsentwicklung in der Lerntätigkeit (pp. 23–52). Ein Lehrbuch für die pädagogische Psychologie an Instituten für Lehrerbildung. Berlin: Volk und Wissen.Google Scholar
 Lompscher, J. (1985b). Die Ausbildung von Lernhandlungen. In J. Lompscher, L. Irrlitz, W. Jantos, E. Köster, H. Kühn, G. Matthes, & G. Witzlack (Eds.), Persönlichkeitsentwicklung in der Lerntätigkeit (pp. 53–78). Ein Lehrbuch für die pädagogische Psychologie an Instituten für Lehrerbildung. Berlin: Volk und Wissen.Google Scholar
 Lompscher, J. (1989a). Aktuelle Probleme der pädagogischpsychologischen Analyse der Lerntätigkeit. In Psychologische Analysen der Lerntätigkeit. Beiträge zur Psychologie. In J. Lompscher, G. Hinz, W. Jantos, B. Jülisch, L. Komarowa, I.P. Scheibe, & C. Wagner (Eds.), Psychologische Analyse der Lerntätigkeit. Beiträge zur Psychologie (pp. 21–50). Berlin: Volk und Wissen, Volkseigener Verlag.Google Scholar
 Lompscher, J. (1989b). Lehrstrategie des Aufsteigens vom Abstrakten zum Konkreten. In J. Lompscher, G. Hinz, W. Jantos, B. Jülisch, L. Komarowa, I.P. Scheibe, & C. Wagner (Eds.), Psychologische Analyse der Lerntätigkeit. Beiträge zur Psychologie (pp. 51–90). Berlin: Volk und Wissen, Volkseigener Verlag.Google Scholar
 Lompscher, J. (2006). Tätigkeit. Lerntätigkeit. Lehrstrategie. Theorie der Lerntätigkeit und ihre empirische Erforschung. In H. Giest & G. Rückriem (Eds.), International CulturalHistorical Human Sciences (ICHS), Band 19. Berlin: Lehmanns Media.Google Scholar
 Maier, H. (1998). „Erklären“: Ziel mathematikdidaktischer Forschung? Journal für MathematikDidaktik, 18(2/3), 239–241.Google Scholar
 Maier, H., & Beck, Ch. (2001). Zur Theoriebildung in der Interpretativen mathematikdidaktischen Forschung. Journal für MathematikDidaktik, 22(1), 29–50.Google Scholar
 Maier, H., & Steinbring, H. (1998). Begriffsbildung im alltäglichen Mathematikunterricht – Darstellung und Vergleich zweier Theorieansätze zur Analyse von Verstehensprozessen. Journal für MathematikDidaktik, 19(4), 292–330.Google Scholar
 Masterman, M. (1970). The nature of a paradigm. In I.A. Lakatos & A. Musgrave (Eds.), Criticism and the growth of knowledge. Proceedings of the International Colloquium in the Philosophy of Science, London, 1965 (pp. 59–90). London: Cambridge U.P.Google Scholar
 Masterman, M. (1974). Die Natur des Paradigmas. In I.A. Lakatos & A. Musgrave (Eds.), Kritik und Erkenntnisfortschritt (pp. 59–88). Braunschweig: Vieweg & Sohn.Google Scholar
 Meyerhöfer, W. (2004). Zum Kompetenzstufenmodell von PISA. Journal für MathematikDidaktik, 25(3/4), 294–305.Google Scholar
 Müller, G. N., & Wittmann, E. (1984). Der Mathematikunterricht in der Primarstufe: Ziele. Inhalte. Prinzipien. Beispiele (3rd ed.). Wiesbaden: Vieweg + Teubner.Google Scholar
 Nitsch, R. (2015). Diagnose von Lernschwierigkeiten im Bereich funktionaler Zusammenhänge. Wiesbaden: Springer Spektrum.Google Scholar
 Nöth, W. (2000). Handbuch der Semiotik. Stuttgart und Weimar: Metzler.Google Scholar
 Otte, M. (1974). Didaktik der Mathematik als Wissenschaft. Zentralblatt für Didaktik der Mathematik (ZDM), 6(3), 125–128.Google Scholar
 Otte, M. (1997). Mathematik und Verallgemeinerung – Peirce’ semiotischpragmatische Sicht. Philosophia naturalis, 34(2), 175–222.Google Scholar
 Peirce, C. S. (1931–1958). Collected papers (Vol. I–VIII). Cambridge: Harvard University Press.Google Scholar
 Prediger, S., BiknerAhsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches—First steps towards a conceptual framework. ZDM—The International Journal on Mathematics Education, 40(2), 165–178.Google Scholar
 Prediger, S., Bosch, M., Kidron, I., Monaghan, J., & Sensevy, G. (2010) Different theoretical perspectives and approaches in mathematics education research—Strategies and difficulties when connecting theories. In V. DurandGuerrier, S. SouryLavergne, & F. Arzarello Lecluse (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 1529–1544). Lyon: Institut national de recherche pédagogique.Google Scholar
 Sabena, C., Arzarello, A., BiknerAhsbahs, A., & Schäfer, I. (2014). The epistemological gap—A case study on networking of APC and IDS. In A. BiknerAhsbahs, S. Prediger (Eds.) & The Networking Theories Group, Networking of theories as a research practice in mathematics education (pp. 165–183). New York: Springer.Google Scholar
 Schreiber, C. (2006). Die Peirce’sche Zeichentriade zur Analyse mathematischer ChatKommunikation. Journal für MathematikDidaktik, 27(3/4), 240–264.Google Scholar
 Shinno, Y. (2017). Metatheoretical aspects of the two case studies of networking theoretical perspectives: Focusing on the treatments of theoretical terms in different networking strategies. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1–5, 2017) (pp. 2670–2571). Dublin, Ireland: DCU Institute of Education and ERME Dublin (Ireland). http://www.mathematik.unidortmund.de/~prediger/ERME/CERME10_Proceedings_2017.pdf. Accessed: May 2, 2018.
 Steinbring, H. (1998). Mathematikdidaktik: Die Erforschung theoretischen Wissens in sozialen Kontexten des Lernens und Lehrens. Zentralblatt für Didaktik der Mathematik (ZDM), 30(5), 161–167.Google Scholar
 Steinbring, H. (2011). Changed views on mathematical knowledge in the course of didactical theory development: Independent corpus of scientific knowledge or result of social constructions? In T. Rowland & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp. 43–64). New York: Springer.Google Scholar
 Steiner, H.G. (1983). Zur Diskussion um den Wissenschaftscharakter der Mathematikdidaktik. Journal für MathematikDidaktik, 3, 245–251.Google Scholar
 Steiner, H.G. (1984). Topic areas: Theory of mathematics education. In M. Carss (Ed.), Proceedings of the Fifth International Congress on Mathematics Education (pp. 293–298). Boston, Basel, Stuttgart: Birkhäuser.Google Scholar
 Steiner, H.G. (1985). Theory of mathematics education (TME): An introduction. For the Learning of Mathematics, 5(2), 11–17.Google Scholar
 Steiner, H.G. (1986). Topic areas: Theory of mathematics education (TME). In M. Carss (Ed.), Proceedings of the Fifth International Congress on Mathematical Education (pp. 293–299). Boston, Basel, Stuttgart: Birkhäuser.Google Scholar
 Steiner, H.G. (1987a). A systems approach to mathematics education. Journal for Research in Mathematics Education, 18(1), 46–52.Google Scholar
 Steiner, H.G. (1987b). Philosophical and epistemological aspects of mathematics and their intersection with theory and practice in mathematics education. For the Learning of Mathematics, 7(1), 7–13.Google Scholar
 Steiner, H.G. (1987c). Implication for scholarship of a theory of mathematics education. Zentralblatt der Didaktik der Mathematik, Informationen, 8(4), 162–167.Google Scholar
 Steiner, H. G., Balacheff, N., Mason, J., Steinbring, H., Steffe, L. P., Cooney, T. J., & Christiansen, B. (1984). Theory of mathematics education (TME). ICME 5—Topic Area and Miniconference. Occasional Paper 54, Arbeiten aus dem Institut für Didaktik der Mathematik der Universität Bielefeld. Bielefeld: IDM. http://www.unibielefeld.de/idm/serv/dokubib/occ54.pdf. Accessed: April 8, 2016.
 Tabach, M., Rasmussen, C., Dreyfus, T., & Hershkowitz, R. (2017). Abstraction in context and documenting collective activity. In T. Dooley & G. Gueudet (Eds.), Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1–5, 2017) (pp 2692–2699). Dublin, Ireland: DCU Institute of Education and ERME Dublin (Ireland): http://www.mathematik.unidortmund.de/~prediger/ERME/CERME10_Proceedings_2017.pdf. Accessed: May 2, 2018.
 Toepell, M. (2004). Zur Gründung und Entwicklung der Gesellschaft für Didaktik der Mathematik (GDM). Mitteilungen der Gesellschaft für Didaktik der Mathematik, 30(78), 147–152.Google Scholar
 Vohns, A. (2012). Zur Rekonstruierbarkeit impliziter Standardsetzungen zentraler Prüfungen mit Hilfe des RaschModells. Journal für MathematikDidaktik, 33(2), 339–349.Google Scholar
 Walsch, W. (2003). Methodik des Mathematikunterrichts als Lehr und Wissenschaftsdisziplin. Zentralblatt für Didaktik der Mathematik (ZDM), 35(4), 153–156.Google Scholar
 Weigand, H.G. (1995). Interpretatives oder normatives Paradigma? — Anmerkungen zum Artikel von Chr. Beck u. H. Maier: Das Interview in der mathematikdidaktischen Forschung, JMD, 14 (1993), H. 2, S. 147–179. Journal für MathematikDidaktik, 16(1), 145–148.Google Scholar
 Wellenreuther, M. (1997). Hypothesenbildung, Theorieentwicklung und Erkenntnisfortschritt in der Mathematikdidaktik: Ein Plädoyer für Methodenvielfalt. Journal für MathematikDidaktik, 18(2/3), 186–216.Google Scholar
 Wittgenstein, L. (1999). Bemerkungen über die Grundlagen der Mathematik. Werkausgabe Vol. 6. Frankfurt: Suhrkamp.Google Scholar
 Wittmann, E. Chr. (1974). Didaktik der Mathematik als Ingenieurwissenschaft. Zentralblatt für Didaktik der Mathematik (ZDM), 6(3), 119–121.Google Scholar
 Wittmann, E. C. (1995). Mathematics education as a ‘design science’. Educational Studies in Mathematics, 29(4), 355–374.Google Scholar
 Wuttke, J. (2014). RaschModell, suffiziente Statistik, Transformationsgruppen und Methodenkritik: Anmerkungen zu Büchter & Pallack (2012/13) und Vohns (2012). Journal für MathematikDidaktik, 35(2), 283–293.Google Scholar
List of References for Further Reading
 Batanero, M. C., Godino, J. D., Steiner, H. G., & Wenzelburger, E. (1992). An international TME survey: Preparation of researchers in mathematics education. Occasional Paper 135, Arbeiten aus dem Institut für Didaktik der Mathematik der Universität Bielefeld. Bielefeld: IDM.Google Scholar
 Giest, H., & Lompscher, J. (2003). Formation of learning activity and theoretical thinking in science teaching. In A. Kozulin, B. Gindis, V. S. Ageyev, & S. M. Miller (Eds.), Vygotsky’s educational theory in cultural context (pp. 267–288). Cambridge: Cambridge University Press.Google Scholar
 Lompscher, J. (1999a). Activity formation as an alternative strategy of instruction. In Y. Engeström, R. Miettinen, & R.L. Punamäki (Eds.), Perspectives on activity theory (pp. 264–281). Cambridge: Cambridge University Press.Google Scholar
 Lompscher, J. (1999b). Learning activity and its formation: Ascending from the abstract to the concrete. In M. Hedegaard & J. Lompscher (Eds.), Learning activity theory (pp. 139–166). Aarhus: Aarhus University Press.Google Scholar
 Lompscher, J. (2002). The category of activity—A principal constituent of culturalhistorical psychology. In D. Robbins & A. Stetsenko (Eds.), Vygotsky’s psychology: Voices from the past and present (pp. 79–99). New York: Nova Science Press.Google Scholar
 Mühlhölzer, F. (2010). Braucht die Mathematik eine Grundlegung? Ein Kommentar des Teil III von Wittgensteins Bemerkungen über die Grundlagen der Mathematik. Frankfurt: Vittorio Klostermann.Google Scholar
 Steiner, H.G., & Vermandel, A. (Eds.). (1988). Foundations and methodology of the discipline mathematics education. Didactics of mathematics. Proceedings of the second TMEconference. Antwerp: University of Antwerp.Google Scholar
 Vermandel, A. (1988). Theory of mathematics education. Proceedings of the third international conference. Antwerp: University of Antwerp.Google Scholar
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