Educ Stud Math (2016) 91:149–163 DOI 10.1007/s10649-015-9666-3

Epistemology and networking theories Ivy Kidron 1

Published online: 2 December 2015 # Springer Science+Business Media Dordrecht 2015

Abstract A theoretical reflection on epistemology is presented. The important role of epistemological analysis in research in mathematics education is discussed. I analyze the epistemological evolution as a consequence of the changes in the mathematical culture and demonstrate how the epistemological analysis is tightly linked to the cultural dimension. Then I analyze the connection between epistemology and networking of theories. Different meanings of the word “epistemic” are observed as well as the role of epistemology in the networking of theories. Keywords Cultural dimension . Epistemological analysis . Epistemology . Networking theories . Social dimension

1 Introduction At the colloquium in honour of Artigue which was held in Paris in 2012, Luis Radford asked the following questions: 1. What can be the interest of epistemological analysis to mathematics education? 2. What is meant by epistemic in a given theory and how much does it differ from other meaning in other theories? 3. How can epistemological analysis take into account the social and cultural dimension of knowing? These challenging questions stimulate me towards a theoretical reflection on epistemology. This reflection is presented in this paper. In section 2, the interest of epistemological analysis to mathematics education (Question 1) is analyzed. Section 3 deals with the need for different theories. In section 4, the coordinating of the different theories is discussed. Section 5 is the core of the paper. It offers a

* Ivy Kidron [email protected] 1

Department of Applied Mathematics, Jerusalem College of Technology, Havaad Haleumi Str. 21, POB 16031, Jerusalem 91160, Israel

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reflection on epistemology and networking theories. This section is an effort to answer first Question 2 and then Question 3. The concluding remarks in the last section refer to the question of how the networking of theories offers a new viewpoint on epistemology.

2 The interest of epistemological analysis to mathematics education 2.1 The role of epistemology The important role of epistemology is discussed in details in Artigue (1990, 1995). Artigue offers a detailed analysis of the role played by epistemological analysis in didactic theory and practice. She first points out that epistemology permits us to look at the mathematics education world from the outside. The “prise de distance” or “vigilance épistémologique” are the expressions used by Artigue to mark a distancing attitude which helps the mathematics educator to differentiate between the mathematical knowledge and the taught knowledge, and to understand the historical way the mathematical concepts were developed. Artigue also points out the importance of understanding the historical way in which meta-mathematics notions were developed, for example, the notion of rigor. The epistemological analysis teaches us that the problems of rigor, of the foundations of mathematics were not the first step in the historical development of mathematical concepts. For example, in analysis, the definitions of key concepts such as derivative as a limit, appeared a long time after these concepts were used and developed. The epistemological analysis permits us to therefore understand how the mathematical concepts were formed. For Artigue, the teaching of mathematics aims not only in the transmission of mathematics knowledge; it also aims towards a mathematical culture. The epistemological analysis has a central role in formulating important questions which permit the mathematics educator to decide which elements of the mathematical culture will be reproduced in the teaching of mathematics. Another important aspect of the role of epistemology is discussed in Artigue (1990): the analysis of epistemological obstacles. The debate concerning the notion of epistemological obstacles, as it appears in the last decades, is described in Schneider (2013). Schneider also refers to the questions that arise in Artigue’s (1990) analysis in characterizing the concept of epistemological obstacle: Can we talk about epistemological obstacles when there is no identification of errors but simply of difficulties? .. Should we look for their unavoidable character in the students’ learning process?..Can we talk, in certain cases, about a reinforcement of epistemological obstacles due to didactical obstacles? Schneider (2011), commenting on the entire paper by Artigue (1990), points out that the conclusions of the paper are still valid. Furthermore, Schneider adds that the important relation between epistemology and didactics should be reinforced today. Asking what could be the interest of epistemological analysis in different traditions of research on the teaching and learning of mathematics, Radford (2015) answers that the reasons already given by Artigue in the early 1990s seem to remain valid. He adds, “These reasons can undoubtedly be refined. This refinement could be done through a reconceptualization of knowledge itself, reconceptualization that might

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consider the political, economical and educational elements that.. come to give their strength and shape to knowledge in general and to academic knowledge in particular”. Ernest (1999, p. 80) points out the important role of epistemology: “it can be said that an epistemological perspective on mathematics and school mathematical knowledge foregrounds assessment and the warranting of knowledge”.

2.2 Epistemological considerations in the early research studies: the case of calculus I refer to the case of calculus because it is one of my main domains of interest, but one can read this section thinking about other domains. In Kidron (2014) I described research on teaching and learning calculus. Beginning with the early research studies in learning calculus, I mentioned the cognitive difficulties that accompany the learning of central notions in calculus. These difficulties are well described in the literature (see for example Sierpinska, 1985; Tall, 1992). These early researches were essentially cognitive and were accompanied by epistemological considerations. We find in these early studies an analysis of the dynamic interaction between formal and intuitive representations. We also read that some of the cognitive difficulties that accompany the understanding of mathematical concepts such as the concept of limit might be a consequence of the learners’ intuitive thinking (see for example, Fischbein, Tirosh, & Hess, 1979; Tall & Vinner, 1981). I have been involved for many years in research concerning students’ understanding of such mathematical concepts as real numbers and limits. Like other mathematical education researchers, I tried again and again different directions to help students overcome the conceptual difficulties that accompany the understanding of the limit concept. In my early research, I used a theoretical perspective that highlights the cognitive dimension and concerns about students’ cognition led me towards the different paths of my research. Later, I felt a need for other theoretical frames. This need is demonstrated in Kidron (2008). This need is not a personal need in my own research; I will refer to this need in a more general way in the next section.

3 The need for different theories The notion of theory is, in its own, a crucial question. Frank Lester (2005) wrote that too often researchers ignore, misunderstand, or misuse theory in their work and that it is time for a serious examination of the role that theory should play in the formulation of problems, in the design and methods employed and in the interpretation of findings in education research. The importance of the role of theory is well demonstrated in the French school. For example, dealing with the importance of epistemological considerations, Artigue (1990) points out that the work of the mathematics educator is not limited to integrate the questions of epistemological nature in his activity. It also consists of building theoretical frames which will permit the work on these questions. Indeed, the recourse to epistemology characterizes the main theoretical frameworks of the French school of didactique des mathématiques. In the following, we analyze the need for theories that highlight different approaches such as the institutional, social and cultural approaches.

3.1 Theories that highlight the institutional approach The question of how we, as a research community, decide if a research project in mathematics education is successful is of importance. Let us consider, for example,

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the large number of research projects which aim to use technology towards constructing mathematical knowledge. Even if some of these projects seem very successful, success should not be only local. This is well expressed in the following sentence by Artigue: “success required a radical change in the institutional status given to the graphical semiotic register…a cultural and systemic change was needed that certainly would need time, and much more than what our isolated piece of research could allow” (Artigue, 2010, p. 465). Artigue makes it clear that the institutional approach should be taken into account towards a successful technological integration at large scale level. I would like to add that technological integration is only one among other examples which illustrate the role of the institutional dimension. The crucial role of the institutional approach is explicit in the Anthropological Theory of the Didactic (Bosch & Gascón, 2014).

3.2 Theories that highlight the social and cultural approaches Socio-cultural approaches view mathematical objects not as absolute objects, but as entities which arise from the practices of given institutions. These practices are described in terms of tasks in which the mathematical object is embedded, in terms of techniques used to solve these tasks and in terms of discourse which both explains and justifies the techniques. The nature of mathematical objects is a domain which illustrates the importance of the social and cultural approaches. I would like, for example, to consider the addition of these two approaches, social and cultural, in connection to the previous analysis in the early research studies with the epistemological and cognitive lenses which permitted to notice the difficulties that accompany the learning of calculus. Some of the efforts which were done these last decades to overcome these difficulties include the use of technology and the use of history of mathematics as a kind of epistemological laboratory. In both cases, in the use of technology and in the use of history, the cultural dimension has an important role. As pointed by Radford (2008b), it is “not enough to declare our aim to put technological advance at the service of the improvement of mathematics education”. Radford adds (p. 440): as the anthropologist Clifford Geertz suggested, the human brain is thoroughly dependent upon cultural resources for its very operation; and those resources are, consequently, not adjuncts to, but constituents of, mental activity.. Culture as well as its artifacts are said to play a cognitive role. Radford (2008b, p. 451) wrote that instead of conceiving of artifacts as mere aids to thinking “I conceive of artifacts as co-extensive of thinking: we act and think with and through artifacts.”. Radford (1997, p. 27) also points out that our social and cultural conceptions have their influence on the way we use history of mathematics: “what we are “seeing” is necessarily mediated by our own modern social and cultural conceptions of the present and the past”. Radford (1997) describes how the notion of “epistemological obstacles” were revisited by Sierpinska (1989) and notes the following sentence by Artigue (1995, p.16): “epistemological obstacles identified in history are only candidates for obstacles in the present day learning processes”. In section 5, trying to answer Radford’s third question, I reflect on this sentence nearly 20 years after those “present day learning processes” mentioned by Artigue.

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4 Coordinating different theories Each theory has its own focus and characteristics. Moreover, different theories may agree on the importance of a specific approach such as social interactions but, at the same time, the way the different theories consider social interactions may be very different. Researchers in mathematics education devote efforts to understand how theories can be connected successfully while respecting their underlying conceptual and methodological assumptions, a process called ‘networking theories’. Comparing different theories, not only were connections and contrasts identified between the frameworks but also additional insights, which each of these frameworks can provide to each of the others. The need for different lenses, which was mentioned in the previous section, justifies the interest in the effort of networking theories. Another substantial reason is to explore ways of handling the diversity of theories, exploring the insights offered by each theory to the others, and at the same time exploring the limits of such an effort (Artigue & Mariotti, 2014; Kidron, Lenfant, Artigue, Bikner-Ahsbahs, & Dreyfus, 2008; Lagrange & Kynigos, 2014; Radford, 2008a). For about 15 years, the diversity of theories has been intensively discussed in the mathematics education research community (Sriraman & English, 2010). My own journey on networking theories started at the CERME 4 working group on Theories (Artigue, Bartolini-Bussi, Dreyfus, Gray, & Prediger, 2006), and is still ongoing (Kidron, BiknerAhsbahs, Monaghan, Radford, & Sensevy, 2011; Kidron, Bosch, Monaghan, & Radford, 2013). The aim of this paper is a theoretical reflection on the role of epistemology. I will use my experience on networking theories in my effort to answer Radford’s second and third questions.

5 Epistemology and networking theories We first refer to Radford’s questions 2 and 3 and then analyze the connection between epistemology and networking theories.

5.1 What is meant by epistemic in a given theory and how much does it differ from other meaning in other theories? I will refer to this question in two parts. The first part relates to the differences of the meaning of the word epistemic in different theories. In the second part, epistemology is used as a system of reference for the networking of theories.

5.1.1 Different meanings of the word “epistemic” In Schwarz, Dreyfus and Hershkowitz’s book Transformation of Knowledge through Classroom Interaction (Schwartz et al., 2009), Kidron and Monaghan wrote a commentary on those chapters concerning the construction of knowledge as seen by different theoretical frameworks. All the chapters were concerned with knowledge in different educational settings. The authors realized that knowledge construction in educational settings “requires an embedded, though usually implicit, epistemology: a teacher literally cannot teach without an epistemology. It is, then, not surprising that epistemic issues are central to all four chapters, but the epistemic issues they address differ” (Kidron & Monaghan, 2009, p. 84). Indeed, one of these chapters on construction of knowledge referred to epistemological discontinuities and tool use, another chapter used an epistemological categorization in the dissection of knowledge into

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‘taught knowledge’ and ‘knowledge to be taught’. In a third chapter the embedded epistemology was that of Davydov’s ascent to the concrete, and epistemological considerations were confined to the formation of mathematical abstractions, not learning in general. Analyzing the similarities and differences between the chapters, Kidron and Monaghan realized that they addressed similar issues in different ways. With regard to epistemic issues, the authors realized that in the different chapters there are connections between learners’ past knowledge and the framing of future knowledge construction, but these connections were addressed in different ways. Another research study (Kidron et al., 2008) illustrates an example of networking between three theoretical frames. In this study, the authors focus on how each of the different theoretical frames takes into account social interactions in learning processes. In order to better understand the different meanings of the word “epistemic” in the different theories, I will shortly introduce the three theories. The first theory belongs to the French school. It is the Theory of Didactical Situations (Artigue, Haspekian, & Corblin-Lenfant, 2014) named TDS. TDS began to develop in the 1960s in France, initiated by Guy Brousseau (1997). Artigue et al. (2014) focus on three characteristics that create the specific lens through which TDS considers the teaching and learning of mathematics: the systemic nature of teaching and learning; the epistemology of mathematical knowledge; and the vision of learning as a combination of adaptation and acculturation. The theory is structured around the notions of a-didactical and didactical situations, and includes concepts relevant for teaching and learning in mathematics classrooms. The social dimension also has an important role in TDS. In essence, the central object of the theory, the situation, incorporates the idea of social interaction. The second theory which is considered in Kidron et al. (2008) is the theory of Abstraction in Context (Hershkowitz, Schwarz, & Dreyfus, 2001; Schwarz et al., 2009) named AiC. This framework has been developed over the past 15 years with the purpose of providing a theoretical and methodological approach for researching, at the micro-level, learning processes in which learners construct deep structural mathematical knowledge. AiC posits three phases: the need for a new construct, the emergence of the new construct, and its consolidation. The second phase constitutes the core of the process of abstraction, namely the emergence of a new construct. The associated process of knowledge construction is expressed by means of three observable and identifiable epistemic actions, Recognizing, Building-with, and Constructing. Constructing refers to the first use of a new knowledge element and is largely based on vertical reorganizing of existing knowledge constructs in order to create a new one. Recognizing takes place when the learner recognizes that a specific knowledge construct is relevant to the problem she or he is dealing with. Building-with is an action comprising the combination of recognized knowledge elements, in order to achieve a localized goal, such as the actualization of a strategy, or a justification, or the solution of a problem. By means of these epistemic actions, AiC proposes tools that allow the researcher to infer learners’ thought processes. The third theory which is considered in Kidron et al. (2008) is the Theory of Interest-Dense Situations (Bikner-Ahsbahs & Halverscheid, 2014) named IDS. IDS provides a frame for how interest dense situations and their epistemic and interest supporting character are shaped through social interactions in mathematics classes. Analysis from the epistemic point of view shows that the epistemic processes in interest-dense situations are built by three different epistemic actions: collectively gathering and connecting mathematical meanings, and structure seeing. A group of students gather mathematical meanings if the students in the group gather single units of a mathematical content such as examples, counter examples, ideas, formulas, … They collectively connect mathematical meanings if they, as a group, put pieces of knowledge

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together to make sense of connections. Structure seeing means perceiving a pattern or a rule, which is tested, proved, verified, validated or confirmed. Kidron et al. (2008) realized that social interactions constitute the epistemic process in IDS, which is not the case in TDS and AiC. Moreover, the authors mentioned that TDS offers another perspective than the one offered by IDS to reflect on social contracts, on the dynamics of the epistemic process, and on the building of situations reflecting in-depth the mathematical epistemology of a given domain. The three theories use different categories in relation to the analysis of the development of mathematical knowledge. AiC approaches abstraction through three types of epistemic actions: recognizing, building-with, constructing; TDS distinguishes between three functionalities of mathematical knowledge: for acting, for communicating, for proving, which serve to organize the development of students’ conceptualizations through appropriate situations. Moreover, although epistemic actions are used by both AiC and IDS frameworks, not only are they different actions but they are viewed in different ways. The AiC approach puts the cognitive aspects in the center, considering social aspects as important but secondary, whereas the IDS approach considers the social aspects to be of primary importance in constructing knowledge. Therefore, there are differences between the natures of the sets of epistemic actions, which turned out to provide complementary insights as described in Kidron et al. (2008).

5.1.2 Epistemology as a system of reference for networking theories The role of epistemology in the networking of theories is an explicit focus in the paper by Ruiz-Munzón, Bosch and Gascón (2013). It is addressed through the necessity for research to elaborate its own particular “vision” of the mathematical contents involved in research and to use this “vision” (conception or model) as a reference point from which to observe teaching and learning practices. This idea of a “reference epistemological model” (REM) is used in Ruiz-Munzón et al. for networking Chevallard’s Anthropological Theory of the Didactic (ATD) and Radford’s Theory of Knowledge Objectification (TKO). Radford and Chevallard offer two different research programs that deal with school algebra as a research domain. Ruiz-Munzón et al. consider Radford’s approach as a representative of research dealing with ‘algebraic thinking’ while Chevallard’s work corresponds to investigations carried out within ATD that consider the “process of algebraization of mathematical activities”. The two approaches asked different questions. Using TKO, the questions are summarized around algebraic thinking, for example, “What are the relationships (filiations and ruptures) between numerical or arithmetical and algebraic forms of thinking?”; “What is the evolution of the different components of algebraic thinking in young students”? Using ATD, the questions are, for example, “What conditions are required for elementary algebra to normally exist as a modelling tool in an educational institution (for instance at lower secondary school), so that the school mathematical organisations can be progressively algrebraized”? The paper by Ruiz-Munzón et al. is an invitation for a dialogue between these two approaches: When comparing both approaches, the first obvious observation is that they question and problematize different aspects of the ‘didactic reality’ they wish to study.. our postulate is that the differences in how problems are formulated are deeply dependent

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on the way of interpreting and describing algebra in each framework, that is, the reference epistemological model of school algebra used…the distance between the two ways of approaching the problem of school algebra can be explained by the differences between the epistemological models assumed (Ruiz-Munzón et al., 2013, p. 2871). The differences in the reference epistemological models are used towards the networking of the two approaches.

5.2 How can epistemological analysis take into account the social and cultural dimension of knowing? Trying to answer this question required several phases of reflection. In order to describe the evolution of my thinking, I first analyze the increasing influence of socio-cultural approaches towards learning processes. Then, being aware that the social and cultural dimension of knowing might be viewed differently in different theories, I try to connect between the “cultural dimension of knowing” and “networking theories”.

5.2.1 Increasing influence of socio-cultural approaches towards learning processes Socio-Cultural studies and reconceptualization of theoretical constructs There is an increasing influence taken by socio-cultural and anthropological approaches towards learning processes in different domains of mathematics. As an example, in Calculus, even the construct ‘concept image and concept definition’ which was born in an era where the theories of learning were all cognitive and epistemological was revisited (Bingolbali & Monaghan, 2008) and used in interpreting data in a socio-cultural study. Bingolbali and Monaghan’s research study investigated students’ conceptual development of the derivative. This was done with particular reference to rate of change and tangent aspects. The participants in Bingolbali and Monaghan’s research study were influenced in their concept images of derivative by differences in the teaching practices of mathematics and engineering departments. For mathematics students, the focus was on tangent aspects of derivative whereas for engineering students, the focus was on rate of change aspects. This study is an example how a theoretical construct which appeared in the early research studies with the epistemological and cognitive dimensions was reconceptualized in order to “re-embed it in the remarkably richer contemporary theoretical landscapes of the field” (Nardi, 2006, p. 188). Socio-Cultural approaches and the historical development of the mathematical concepts Epistemological analysis permits us to understand how mathematical concepts are formed. But as Radford (1997, p. 27) states”even the most titanic effort of putting away all our modern knowledge in order to see the historical event in its purity will not succeed: we are damned to bring our modern socio-cultural conceptions of the past with us”. Discussing the effects of culture and society on the way in which knowledge is produced leads us to a revised “epistemological obstacles” perspective in which these obstacles are viewed as “cultural obstacles”. Radford (1997, p. 32) notes that “historico-epistemological analyses may provide us with interesting information about the development of mathematical knowledge within a culture and across different cultures”.

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The influence of the social and cultural approaches on the epistemological analysis has increased over the years. We pointed to the refinement of the “epistemological obstacles” and to Artigue’s remark (1995, p. 16)”epistemological obstacles identified in history are only candidates for obstacles in the present day learning processes”. Today, Artigue (February 2015, private communication) would prefer the terms “cultural dimension of the epistemological obstacles”.

Changes in the mathematical culture and epistemological evolution of mathematical domains towards new approaches Let us reflect on Artigue’s example describing her mathematical research in differential equations and the way she notes the epistemological inadequacy of teaching in this area, for students in their first two years at university (Artigue, 1995, p. 16). By means of epistemological analysis, Artigue described how historically the differential equations field had developed in three settings: the algebraic, the numerical and the geometric settings. For many years, teaching was focused on the first setting due to epistemological and cognitive constraints. Reflecting on these constraints was a starting point towards building new teaching strategies which better respect the current fields’ epistemology. By means of the epistemological analysis, one can see the epistemological evolution of the field towards new approaches (in the case of differential equations, towards geometrical and numerical approaches). The important point that we learn from this example, is that this epistemological evolution is a consequence of the changes in the mathematical culture. In this example we note how the epistemological analysis highlights the crucial role of the cultural dimension of knowledge. The technological aspect of students’ learning being taken as a culture Changes in the cultural context influenced my own research on the way students conceptualize central notions that relate to the continuous such as the limit notion in the definition of the derivative. I noted the influence of modern results in research Mathematics on Mathematics education. In Kidron (2008), I illustrate how deep and modern results in research mathematics (for example, the fact that small variation may indeed result in large change) has entered the realm of mathematics education in a significant way. We also note how computer packages have changed Mathematics over the past decades (Borwein, Borwein, & Straub, 2012). In the following, I demonstrate how these changes in the cultural context influenced my own research in mathematics education and how the technological aspect of students’ learning was being taken as a culture. In my previous research, I was aware of the cognitive difficulties relating to the understanding of the definition of the derivative as the “limit of the quotient Δy/Δx as Δx approaches 0”. Students viewed the limit concept as a potential infinite process and this was a possible source of difficulties. Moreover, previous researches (Tall, 1992) expressed students’ belief that any property common to all terms of a sequence also holds of the limit. I therefore realized that this natural way in which the limit concept is viewed might be an obstacle to the conceptual understanding of the limit notion in the definition of the derivative function f ′(x) as lim Δy=Δ x. In particular, the derivative might be Δx→0

viewed as a potentially infinite process of Δy=Δ x approaching f ′(x) for decreasing Δx. As a result of the belief that any property common to all terms of a sequence also holds of the limit, the limit might be viewed as an element of the potentially infinite process. In other words, limΔx→0Δy/Δx might be conceived as Δy/Δx for a small Δx. I therefore looked for a counterexample that demonstrates that one cannot replace the limit” lim Δy=Δ x” by Δy/Δx for Δx very small. Δx→0

Finding such a counterexample was crucial to my research focus.

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Such a counterexample demonstrates that the passage to the limit leads to a new entity and that therefore omitting the limit will change significantly the nature of the concept. It demonstrates that the limit could not be viewed as an element of the potentially infinite process (Kidron, 2008, p. 202). In Kidron (2008) I explain that such counterexample exists in the field of dynamical systems. In particular, I demonstrated how the counterexample is given by the logistic equation. A dynamical system is any process that evolves in time. In a dynamical process that changes with time, time is a continuous variable. The mathematical model is a differential equation dy/dx=y’=f(x,y) and we encounter again the derivative y0 ¼ lim

Δx→0

Δy

=Δ x. Using a

numerical method to solve the differential equation, there is a discretization of the variable “time”. My aim was that the students will realize that in some differential equations the passage to a discrete time model might totally change the nature of the solution. I also aimed to help students realize that gradual causes do not necessarily have gradual effects, and that a difference of 0.001 in Δx might produce a significant effect. In the counterexample (the logistic equation), the analytical solution obtained by means of continuous calculus is totally different from the numerical solution obtained by means of discrete numerical methods. The essential point is that using the analytical solution, the students use the concept of the derivative as a limit lim Δy=Δ x but, using the discrete approximation by means of the Δx→0

numerical method, the students omit the limit and use Δy/Δx for small Δx. I examined students’ reactions when they realize that the approximate solution to the logistic equation by means of discrete numerical methods is totally different from the analytical solution. My aim was that the students will reach the conclusion that passing to limits—or passing to discrete approximations—may change the nature of a problem significantly (Peitgen, Jurgens, & Saupe, 1992, pp. 295–301). Reflecting on my previous work in relation to Radford’s question, I became conscious that the changes in the cultural context influenced my own research in mathematics education by means of changes in the didactical designs. The new cultural means permit a new epistemic status of the artifacts (Kidron, 2015). The artifacts used in my research permitted the learning experience. They were mediators of activity. In the following, I reflect on the exact meaning of the mediating nature of the artifacts in the learning experience described in Kidron (2008, 2015). The numerical, graphical and symbolical capabilities of the software were used to apply both the continuous analytical approach and the discrete numerical approach to the same problem. In other previous research I used the same software and its capabilities to help students visualize and understand some central notions in analysis. In this previous use of the software, it acts as an aid for the students, for example, to visualize formal definitions. In the learning experience described in Kidron (2008) the situation is different. Reflecting on this difference helped me realize the epistemic status of the artifact. Without advanced software, it would have been very difficult and probably impossible to perform the research study. The software allowed performing a very large number of iterations before looking at the results graphically and numerically. It permitted, for example, the possibility to iterate 1000 steps without writing or plotting any points, and then to continue on the resulting initial point for another 30 steps and to analyze qualitatively the long term behavior of the solution. This capability is impressive, but this is not the crucial point

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in my present analysis of the epistemic status of the artifact. The essential role of the artifact which I emphasize is its deep cognitive role while learners interacted with the artifact changing very gently the value of a parameter and realizing in a concrete way that some of their beliefs (such as “gradual changes in a cause necessarily produce gradual changes in its effect”) are false. An essential point is that the mental images created by means of interaction with the artifact influence the students’ thinking even when the computer was turned off. For example, the understanding that small changes in a cause can produce large changes in its effect was accomplished when students remembered a previous experience with the artifact in which a change of 10−6 in the initial condition of the differential equation might cause a large change in the analytical solution. In summary, in the research study by Kidron (2008, 2015) the role of the artifact corresponds to the description by Radford (2012, p. 285): In this view, artifacts are not only facilitators of knowledge acquisition. They become part of the way in which we come to think and know… artifacts are considered cultural devices that modify our cognitive functioning…As artifacts change, so do our modes of knowing.

5.2.2 Social cultural approaches and the networking of theories The examples in the previous subsection demonstrate how the epistemological analysis is tightly linked to the cultural dimension of knowledge. Even so, answering Radford’s third question required an additional phase of reflection especially when we realize that the nature of mathematical knowledge is objective yet, at the same time, embedded in different ways in different cultures. We may note how each culture has its own view of how mathematical knowledge is constructed. Since there are different cultures, sometimes different historical periods of the same society, there are different regimes of knowledge. Therefore, the cultural perspective is itself a view of what knowledge is and how it is acquired. My experience in networking theories taught me how different theories that deal with construction of mathematical knowledge have different views about what constitutes this mathematical knowledge. To the extent that a theory determines what a mathematical object is and how it can be known, it is a bit like a culture. There are different cultures; therefore, there is a need for networking the different cultural perspectives. Both the “social cultural approaches” and the “networking of theories” approach imply a new view of epistemology where different cultures with their different views of what is a mathematical object live together.

5.3 The connection between networking theories and epistemology In order to further understand the connection between networking theories and epistemology, I present two examples of my networking experience. The first example, presented in Kidron et al. (2014), illustrates how focusing on epistemological considerations allows a beginning of dialogue which permits the networking of theories. The example relates to the networking of the three theories: TDS, ATD and AiC with their different foci. AiC focuses on the learner and his cognitive development. TDS and ATD focus on didactical systems. The three theoretical approaches are sensitive to issues of context but, due to these differences in focus, context is not

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theorized and treated in the same way. Kidron, Artigue, Bosch, Dreyfus, and Haspekian (2014) explain the meanings of the terms context (for AiC), milieu (for TDS) and media-milieus dialectic (for ATD), each of them being a cornerstone for the theory while all of them try to theorize specific contextual elements. As a first step, the networking efforts start by analyzing the episodes in line with the way each theory views the role of context: for AiC, the focus is on the influence of the context on the learners’ process of constructing knowledge; for ATD the focus is on the media-milieus dialectic; while TDS researchers are interested in the potential and limitation of the milieu. The complexity being addressed by the notion of context is well known. A first problem is that what is considered as a part of the context in one theory is not necessarily considered as that in another theory. For example, in AiC the focus of analysis is on the learner, therefore all other factors such as the task, the computer, the teacher, and the learning goals are considered as contextual factors. As a consequence, the notion of context for AiC might include notions which are not necessarily considered as part of the milieu for TDS or ATD. The authors expected some complexity in the effort of creating a dialogue between the three theories in relation to constructs such as context, milieu, and media-milieus dialectic. The three theories share the aim to understand the epistemological nature of the same learning episode but in each of the three theories different questions were asked. Questions for analyses in AiC stressed the epistemic process itself, whereas researchers in TDS and ATD asked how this process is made possible. Nevertheless, these questions indicated that the researchers were able to build on the other analyses in a complementary way. The dialogue between the different approaches was possible because a point of contact was found. A new term was introduced: the common epistemological sensitivity of AiC, TDS, and ATD, which can be noticed in the a priori analyses provided by each frame. This initial proximity was essential for the dialogue to start. The authors observed how the dialogue between the three theories appears as a progressive enlargement of the focus and how it becomes productive, showing the complementarity of the approaches and the reciprocal enrichment, without losing what is specific to each one. The three concepts, context, milieu and media- milieus dialectic were accessed by different data or different foci on data in a complementary way sharing epistemological sensitivity, which facilitated establishing connections and reflecting on them. This common epistemological sensitivity of AiC, TDS, and ATD, was noticed in the a priori analyses provided by each frame. These analyses are the starting point of the dialogue between the approaches and it is not by chance that the common epistemological sensitivity was noticed in the a priori analyses: the reason is that the a priori analyses take into account the mathematical epistemology of the given domain. Focusing on epistemological considerations permits the beginning of dialogue. The second example is a consequence of the networking experience between AiC and TDS researchers which resulted in AiC researchers’ decision to implement the idea of a priori analysis in an explicit way. The networking experience is described in Kidron et al. (2008) in which three theories were involved: TDS, AiC and IDS. In a previous section, I already described this networking experience with its focus on how each of these frameworks is taking into account social interactions in learning processes. In the following, we focus on a specific kind of insights: the epistemological concerns which were highlighted as a consequence of the networking between TDS and AiC. We first characterize the epistemological dimension in the two theories:

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TDS provides a frame for developing and investigating didactical situations in mathematics from an epistemological and systemic perspective. TDS combines epistemological, cognitive, and didactical perspectives. TDS focuses on the epistemological potential of didactical situations. AiC focuses on the students’ reasoning; mathematical meaning resides in the verticality of the knowledge constructing process and the added depth of the resulting constructs. An epistemological stance is underlying this idea of vertical reorganization but AiC analysis is essentially cognitive. Focusing on epistemological concerns, we characterize the insights offered by TDS to AiC as described by Kidron et al. (2008, p. 254): According to Hershkowitz et al. (2001), the genesis of an abstraction originates in the need for a new structure. In order to initiate an abstraction, it is thus necessary (though not sufficient) to cause students’ need for a new structure. We may attain this aim by building situations that reflect in depth the mathematical epistemology of the given domain. This kind of epistemological concern is very strong in the TDS, and the notion of fundamental situation has been introduced for taking it in charge at the theoretical level. It could be helpful for AiC. This was an invitation for AiC researchers to build an a priori analysis that reflects in depth the mathematical epistemology of the given domain. Strong epistemological concerns in TDS were integrated in AiC in a way that reinforces the underlying assumptions of AiC.

6 Concluding remarks Reflecting on the connection between epistemology and networking theories, we noticed how epistemological considerations enrich the networking and vice versa, how by means of networking strong epistemological concerns in one theory might be integrated in another theory in a way that reinforces the underlying assumptions of this other theory. Different theories agree on the importance of the socio-cultural dimension but this dimension is viewed differently in the different frames. Two views are reported in Radford (2015): in the first view, “in the epistemological analysis, the center of interest revolves around the content itself. Social and cultural dimensions are not excluded, but they are not really organically considered in the analysis”. In the second view, “we cannot conduct an epistemological analysis without attempting to show how knowledge is tied to culture and without showing the conditions of possibility of knowledge in historical-cultural layers that make this knowledge possible”. The theoretical reflection on epistemology described in this paper supports this second view. I analyzed examples which demonstrate the epistemological evolution as a consequence of the changes in the mathematical culture and how the epistemological analysis is tightly linked to the cultural dimension. I also analyzed the connection between social cultural approaches and networking. These analyses and the connection between epistemology and networking theories offer a new view on the importance of the question of epistemology. The importance of the question of epistemology is especially relevant nowadays in which different theories exist in mathematics education and, in addition, we have become aware that the nature of mathematical knowledge is embedded in different ways in different cultures and periods of history.

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