1 Introduction

One might ask why a complete issue of ZDM dedicated to yet another theory in mathematics education. In particular, we ask why an issue on enactivism as a research methodology in mathematics education?

Reflecting on the papers in this issue we note that as researchers (who are teachers) we have an emotional orientation (Maturana, 1988a) to observing learners engaged in knowing actions that can be seen as mathematical; for explaining such actions; and (perhaps more relevant) for acting with pedagogical intent. Enactivism, as a theory that explains human cognition as involving recurrent sensorimotor patterns that enable action to be perceptually guided (Varela, 1999, p. 12), has captured our attention. More explicitly, Maturana and Varela (1987/1991) claim that “Knowing (mathematics, or how to teach mathematics, or how to know mathematics as discussed in this issue) is effective action, that is, operating effectively in the domain of existence of living (human) beings” (p. 29). Enlarging upon this they claim to be explaining the phenomenon of effective action of a living (here human) being in its environment by observing them as having autonomous organizations and structurally coupling with the environment and others in it. This involves other phenomena: behavioural coordination in interactions recurring between human beings and recursive behavioural coordination upon behavioural coordination. With respect to the latter (and highlighted in Reid and Mgombelo, 2015) not only is there consensual coordination of actions but languaging—the consensual coordination of consensual coordination of action and as seen in many of these research projects the consensual coordination of languaging. This picture of human knowing and the concomitant view of knowledge, not as existing “in any one place or form but as enacted in particular situations” (Varela, Thompson and Rosch, 1991/1993, p. 196) provided us a starting place that demanded that we rethink the unit of analysis in our enactivist research. What follows is the explication of 3 moves that allow us, while not commenting in detail about the papers (they speak well for themselves on their own various research work) to provide one way to start answering the question posed above: why “enactivism” in the study of mathematics education.

2 First move: the observer

Especially evident in the papers is a move to include (in a radical way) the role of the observer. A reader of Maturana’s Foreward in Autopoiesis and Cognition: the Realization of the Living (Maturana and Varela, 1980) finds this focus on the observer as Maturana’s the first concern as evidenced by his much quoted, “Everything said is said by an observer”. This rather poetic line is only the first part of his initial offering on the observer. We take the liberty of adding the whole of his first short commentary.

Anything said is said by an observer.

In his discourse the observer speaks to another observer,

who could be himself;

whatever applies to the one applies to the other as well.

The observer is a human being, that is, a living system,

and what ever applies to living systems applies also to him

(Maturana, 1980, p. 8)

This view of the observer has many features. It suggests that observation is part of the conscious flow of the researcher’s process; that is, the researcher is possibly and indeed likely to be triggered by his/her own observation to act anew in the research situation and make further observations. More generally this view of the observer suggests that the making of an observation implies a listener who can respond to, be occasioned by, extend, test, … the observation being made. In fact, in his article in the Irish Journal of Psychology, Maturana (1988a) suggests that at the heart of any useful ‘scientific’ observation is a means (he uses the term mechanism) by which the listener to the observation is enabled to understand how the observer “sees” the situation. We understand Maturana’s claim to be that the listener to the observation can then use the provided explanation to better understand the observation of the observer. Thus the observation occasions the listener to change her or his understanding of the matter at hand. And further the observation effects a recursive change in the observer; that is, the observer is subject to the import of the observation being made.

This brief discussion of a complex topic raises questions about the ‘observer’ in the papers in this issue. Who are the observers? It is certainly fair to say that the writers of the papers here are acting as observers and presenting their observations for the reader (listener). But might one not observe the occurrence of other observers in these papers? As is often the case in research about teaching, learning or the development of mathematical understanding what we observe and say about one level in the system has a fractal-like or self-similar manifestation in other levels of the system. For example, in the paper by Coles (2015) there is a ‘cascade’ of observers (Fig. 1). The researcher observes the interaction of each of the two teachers in his study. The teachers are observing the students, students who are working on the mathematical notion of conjectures and ‘proof’’ in 2 different number theoretic problem situations. Those students are making a variety of observations that Coles and (since he provides the details) we readers can take up. Because the teachers under observation are acting with pedagogical intent, Coles points out that they provide prompts for the students to publicly make observations based on their own complex actions with the problems—actions that indeed do vary among students. With respect to two matters, in the case of Teacher A, this teacher listens to student x’s observation and highlights this by asking other students to attend to this observation, but also to respond to it in their own terms. On our reading of this is that the teacher, in occasioning other students to attend to student x’s observation, has deliberately noted that x has provided a means (a particular multiplying action and more than that a general form for that action) by which a listener (now making her/his own observation(s)) can enact through her/his own knowing action. This enables the listener not only to work on particular examples but also to be occasioned to, at the very least, sense the generality of their work. This brief discussion, that could be supplemented with examples from any of the papers in this issue, shows that there are many levels of observation written into these papers and how the writer/observer plays a vital role in occasioning the reader to observe the observations and effects occurring in the researched phenomenon.

Fig. 1
figure 1

Cascade of observers

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This raises a second dimension of the radical nature of the observer. What is observed and how is it being observed? Perhaps the paper by Maheux and Proulx (2015) provides the clearest answer to this question, the two parts of which the ‘what’ and the ‘how’ are not separate but inextricably linked for the observer. Maheux and Proulx put it this way: “although the construction of meaning has long been a leading metaphor, our enactivist perspective demands greater attention to the experiential aspect (Varela et al., 1991/1993) of mathematically engaging oneself” (to which we would add to the environment and others in it). Further they add that “one of the methodological challenges of analyzing the data collected for this study through our enactivist perspective is to avoid making assumptions about what our participants (in their case student teachers ‘reading an equation’) might or might not know—as if they were holding the knowledge in some way—and rather, attend to their actual doing|mathematics” (emphasis in original). They see this compound idea in this way: “it is both doing something (some thing) recognizable as mathematics, but also producing mathematics as this thing that we are doing when we do what we do.” They call this phenomenon “the productive circularity that is at the heart of our enactivist thinking according to which there is no objective end or start, but only an observational beginning … in which “we do not look for earlier knowledge or conventions …but focus on what we see happening.” For us, this and other papers in this issue point to an enactivist research focus which is not on ‘searching for’ preconceived elements in the data relating to student mathematical knowing actions, but on seeing our observations as a much more open ‘searching’. In this way our observations are less likely to be limited by what we are looking for and open to considering all of the actions and inter-actions occurring before us. This focus has other implications for the what and the how of the observer.

In most of the papers in this issue you will notice that the researcher is using some form which Abrahamson and Trninic (2015) describe as microgenetic analysis—“an intense investigation of a relatively brief period of rapidly changing competence, with the aim of making sense of the processes underlying this change.” In most papers in the issue we see attention to a detailed focus on small pieces of data drawn from students engaging in a particular task environment. See for another example, Khan, Francis and Davis (2015) where short video clips of children building robots are used. In other words, such analysis, like that of Abrahamson and Trninic, does not typically rely on pre-post test comparisons, but on observing in detail the doing|mathematics of individuals and groups of students and teachers or students and researchers. The data for such micro-analyses comes in a variety of forms: annotated pieces of transcripts of interactions witnessed by the authors in a wide variety of classroom circumstances; detailed study of videos of mathematics in interactions; several first hand researcher/observer accounts of “doing|mathematics” based on a variety of ‘data’ gathered by researchers in classroom and other settings. One exception to this detailed analysis is found in the work Lozano (2015) who conducted case studies of groups of students learning algebra over a two-year time span. Like the other studies mentioned above even in her study the emphasis was on effective mathematical behaviour and trying to understand its aspects and provide explanations of/for such behaviour.

This last remark calls attention to the question of where the observer is—another element in our focus on the nature of ‘the observer’ in these studies. In their discussion of doing|mathematics cited in the paragraph above, Maheux and Proulx (2015) at least indirectly assert that the observer, and particularly the researcher as observer, needs to be centred in the setting where a number of students are engaged in doing|mathematics. This is, on our reading supported by Steinbring’s (2015) interesting perspective on “enactivist” research that combines various ideas from Maturana and Varela with those of Luhmann. Steinbring prompts us to notice that one cannot directly observe the nature of the mathematical concepts ‘held’ by a student. And arguing against any idea of a transmission model in learning and teaching he further suggests that such concepts observed to be transmitted from a teacher to students or from one student to another are better understood as interactive spaces that open up possibility for understanding and meaning to develop (Sect. 2). Like most of the papers in this issue, the site for observing student mathematics and mathematics teaching and learning is in actions and inter-actions that have a concrete, sense-able quality. Using Luhmann’s ideas, related to but different from Maturana’s ideas of structural coupling, the unit of analysis focuses on the inter-action between persons as a site for observing mathematical knowing where each person, say the student and the teacher may take in the contribution of the other (words, actions on objects, or actions with symbols…) and transform them for their own use (as per von Foerster, 2003a). Such continuing inter-action may allow an observer to see in their actions and the products of those actions the doing|mathematics. Maturana (2005) would put it differently suggesting that in languaging, objects arise for the observer; that is, “Objects arise in languaging, they do not exist by themselves, they do not pre-exist to their arising as manners of flowing in recursive consensual co-ordinations of doings, and have the operational concreteness of our structural operation in the realization of our living” (Inter-objectivity, para. 1).

The implication of such ideas for the observer is that in situating her/himself, the observer needs to be centred in such inter-actions and as argued by Metz and Simmt (2015) part of such centering is having an empathic orientation to the mathematics knowers with whom we are observing (in the cascading way). This creates the possibility for openness in observing through inter-relating how the knowers know mathematics and in what ways such knowing can be explained. To emphasize, such a central position for the observer comes with the realization that in enactivist research that they, the observers are in no way separate from the phenomenon under observation. This is perhaps most evident in the paper by Abrahamson and Trninic (2015) who in many ways governed the features of the Mathematics Imagery Trainer used in their research. As they point out, “both the interview protocol (with students) and the interactive affordances of the instructional materials evolved as we progressed through the pool of participants” (grade 5 and 6 students working on an ingenious task related to recognizing equivalent ratios of heights in a electronic environment allowing for, in fact demanding a kinesthetic modeling of such ratios). Clearly the whole design of the research inter-acts with the participants (at least their actions) as well as the perceptive analyses of student work and the observation of the role of the physical properties of the knowing observed. But for us this realization of the connectedness of the observer to the phenomenon under observation is seen even when the inter-action under observation is seen to be taking place between two organizations, for example a project and a school board (Preciado-Babb, Metz and Marcotte, 2015).

This brief discussion of the observer in these studies and in enactivist research more broadly points to the centrality of the concept of observer in this research. It also points to the necessity of examining objectivity. Maturana (1988a, b) proposes two versions: objectivity without parentheses (objective in the common sense of the term) and objectivity within parentheses (multiple explanations that may be contradictory but are each valid within consensual domain). If a researcher holds her research findings as revealing objectivity without parentheses she is claiming that they represent a convincing argument for the findings (e.g., knowledge of doing|mathematics) because they reveal privileged knowledge of the way mathematics becomes known by individuals, a universalist view. The reader should be convinced by the research because that is the way the phenomenon “really is.” On the other hand objectivity in parentheses is based on a multiversalist position on the nature of the knowledge produced. While like the former form of knowledge without parentheses, the requirement is that the researcher provides adequate explanations or mechanisms whereby the readers if they follow the explanations can test the researcher’s idea of the phenomenon under investigation and research claims are ‘objective’ in that sense. Objectivity in parentheses suggests that there are other valid explanations of the phenomenon at hand (doing|mathematics), ones that the researcher’s approach or point of view could not have considered. Given the general care of the research presented in the papers and the provision of explanations for mathematics knowing in each of them, yet the diversity of the forms of the phenomenon of doing|mathematics explicated, on our reading it seems that the conclusions of these papers about mathematics knowing would fit the multiveralist position.

3 Second move: bringing forth a world of significance

The second move we observe in the enactivist literature (as illustrated in this issue) is to recognize the relationship between the learner and the environment in which the learner is seen to bring forth a world. We picture this below using a form of a model (see Fig. 2) we have used in our research over the past 15 years (e.g. Simmt, 2000; Kieren and Simmt 2002, Kieren and Simmt, 2009). This version of the model reflects Varela et al.’s, (1991/1993) view in which they assert that the knowing actions of the individual in an environment are determined by her/his structure, which is plastic and hence changes through acting (knowing) in an environment. Thus, they note, that the actions of the individual are co-determined by the selected elements of the environment in which the actions take place. Varela (1999) goes further to assert that: “At the very centre of this emerging view is the conviction that the proper units of knowledge are primarily concrete, embodied, incorporated, lived; that knowledge is about situatedness; and that the uniqueness of knowledge, its historicity and context, is not a “noise” concealing an abstract configuration in its true essence. The concrete is a not a step toward something else: it is both where we are and how we get to where we will be” (p. 7).

Fig. 2
figure 2

Interaction that brings forth world of significance

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In our work and following Maturnana and Varela’s thesis in The Tree of Knowledge (1991/1993), we have come to understand mathematics knowing as the bringing forth a world of SIGNificance with others in a sphere of behavioural possibilities that an observer sees as involving mathematics (Kieren, Simmt and Mgombelo, 1997). This expression brings into focus three things: (1) know-edge/know-ing as bringing forth (happening in inter-action and not as an acquisition held by persons); (2) the world of SIGNificance highlights the fact that the use of language and signs and languaging is implicit in such a view of knowing; and (3) the persons so engaged in bringing forth a world engage in what other papers in this issue (e.g., Reid and Mgombelo, 2015; Maheux and Proulx, 2015) suggest is the praxis of living—that is knowing is doing is living is being. It is the observers, who may be the participants themselves, who see this aspect of their living actions as mathematics or mathematical in nature.

Our connection with Maturana’s (1988b) view on knowing in an environment is strong:

Our living takes place in structural coupling with the world that we bring forth, and the world that we bring forth is our doing as observers in language as we operate in structural coupling in it in the praxis of living. We cannot do anything outside our domains of structural coupling; we cannot do anything outside our domains of cognition; we cannot do anything outside our domains of languaging. This is why nothing that we do as human beings is trivial. Everything that we do becomes part of the world that we live as we bring it forth as social entities in language (p. 40).

As Maturana (2005) notes in the bringing forth with others objects (say mathematical ideas) arise. These objects exist in our languaging (consensual coordinations of consensual coordinations of actions with others). This knowing with others generates inter-objectivity.

Objects arise in languaging, they do not exist by themselves, and they do not pre-exist to their arising as manners of flowing in recursive consensual co-ordinations of doings, and have the operational concreteness of our structural operation in the realisation of our living. In this operational concreteness, the world of objects that we live is a domain that exists in our recursive co-ordinations of doings that we experience in our feelings and sensations as a domain of independent entities. Since the entities that arise in our languaging in conversations arise in recursive co-ordinations of doings, they exist in the interplay of the members of a languaging community, and I call the domains of objects and entities that arise in our coexistence in languaging, domains of inter-objectivity. The different worlds that we live as languaging beings, are different domains of inter-objectivity that as different domains of entities that are operationally part of our domain of existence as configurations of our doings, become part of the medium in which we exist in structural coupling. In this sense none of the objects or entities that arise in our domains of inter-objectivity, no matter how abstract they may seem to a naive observer, are ever trivial because they are part of the niche in which we conserve our living. That is, all that we distinguish in our operation in language, our emotions, or our feelings, that become some kind of manipulable entity with our hands, with our thinking, or with our feelings, in the flow of our recursive co-ordinations of consensual doings, operate as part of the worlds in which we realise and conserve our living (Inter-objectivity, para. 1).

It seems to us that the model above taken with the supporting sources quoted point to our observation that seeing knowing as bringing forth a world of significance (second move in enactivism in mathematics education research) is deeply connected to all of the elements of the first move and to the third move (ethics), as will be evident below. For us, the discussion shows how looking at mathematics knowing through these enactivist lenses prompts us as observer/researchers to observe and characterize the elements of the flow of interaction pertaining to many forms of mathematics knowing (the bringing forth of mathematics in an environment); to try to understand and portray how these forms arise in inter-action; how the teachers’ and the students’ observations figure in the (mathematical) world and the inter-objects being brought forth; as well as the qualities of the domain of inter-objectivity–the qualities of the mathematical inter-objects brought forth and the nature of the actions and inter-actions that allow us, but more importantly the actors under our observation, to be positively changed by engaging in the actions that brought the inter-objects forth. The various papers in this issue are replete with opportunities to engage in such observations.

To take up a single example, from Coles’s (2015) paper one sees just how he observes the flow of inter-actions that allow the nature of mathematical proof for students to be observable as an inter-object in the remarkable flows of inter-action. The teacher uses her observations to occasion students to observe the work of one student; but beyond that to express in their own (the students’) terms why the generalization in this circumstance is true; through the reader’s observation of the researcher’s observation of the teacher’s and students’ observations (one example of the “cascade” from the first move above) that allow us to see the role of and nature of the inter-object in this setting. Because, Coles offers two examples of this kind of setting with two prompts under the guidance of two teachers this allows the reader to observe different ways in which the teacher participates in the inter-action, how the prompt comes to be used and affects the inter-action both for students and the teacher and allows the reader to understand better the process in action through which inter-objects arise.

In Metz and Simmt (2015) they observe the interactions in which students experience doubt (and certainty) as they interact with a task that has them interact with their physical environment and each other to try to explain how they can know about a past event (what time did ice start melting given water collected and the rate at which is continues to melt). Exploring the possibilities for researcher teacher to be an empathic observer we see how deeply the teacher is implicated in the worlds of significance learners bring forth. Towers and Martin (2015) take an interesting approach to view the collective co-action that brings forth a world of significance that includes mathematics. Through their treatment of the data as one voice they illustrate quite nicely how the class can be observed as a unit for studying the emergence of mathematical understanding. (We might say this is a second-order system in that the larger class unit emerges from the interactions of the individual learners.)

Perhaps the most striking and unusual example of this is in the paper of Preciado-Babb, et al. (2015) in which the authors, all related to the Galileo project allow us to see how two-third order systems, the project and the school board curriculum organization, can be observed to bring forth evaluative system inter-objects that allow both organizations to agree on a portrayal of valid student knowing that allow both organizations to observe the mathematics knowing of students in the project in a manner satisfying to both “camps.”

It seems to us, that an enactivist approach to research that prompts the researcher to closely observe this flow of consensual coordination of consensual coordinations of actions at several levels allows us to “see” just what students do as/when they “know” mathematics. A particular observation we make from the papers in this issue is how as observers we an observe bringing forth a world of significance at many different layers: from individual structural coupling with a tactile artefact (Abrahamson and Trninic, 2015; Khan et al., 2015; Maheux and Proulx, 2015), to the emergence of the pre-formulated emotional knowing (mood (Maturana, 1988a)) that orients the learner in her moment-by moment actions (Metz and Simmt, 2015), to the logical chains of reasoning that the learner can observe relating one bit of mathematics to the next (Maheux and Proulx, 2015), to the social patterns of acting that enable a group to interact or the ruptures in the patterns that generate a space for new possibilities (Coles, 2015; Steinbring, 2015), to the collective second and third order learning systems that emerge when people make space for the other (Towers and Martin, 2015; Preciado-Babb et al., 2015). All of these are observed as bringing forth a world of significance. What is striking in this regard is the great variety of levels of mathematics that form the basis of the works here; the great variety of teaching materials and prompts used by the authors across these papers; the great variety of roles of researchers in these studies. In other words this research with the commonalities reflected in moves 1 and 2 above appear to allow researchers and hence readers to see mathematics knowing as actions that take on many characters—physical, computational, logical, …—at many levels.

So what does this second move, observing mathematics knowing as bringing forth a world with others that can be observed to pertain to mathematics provide the reader, the writer researcher, the teacher, the students? As expressed above it allows one to look for mathematically intelligent behaviour in a wide variety of learning settings and hence characterize mathematics knowing in terms of actions in settings—“mathematical knowledge” seen as a verb, not as an object held by the student. This view of knowing allows the observer to ask many “in what way” questions that are pertinent to classroom research.

  • In what way were the actions of the students showing competence within a particular arena of mathematics?

  • In what way do appropriate mathematical inter-objects arise in a particular knowing setting? In what ways is the material setting used in this bringing forth? In what ways does it matter?

  • In what ways does this view of knowing impact the nature of mathematics observable itself?

  • In what way can the observer make distinctions of these inter-objects especially as they are occurring in inter-action? In particular, would we say the teacher/researcher in any of these papers see these brought forth mathematics differently than the students? Could one observe variations among students interpretations of the brought forth mathematics? What impact does this have on what would be considered as the mathematics learnt?

  • In what ways is the mathematics, as observed in action and in the inter-objects—that is the mathematics brought forth—related to matter meant, say in the curriculum, the textbook, etc.?

These questions are meant not as a critique of the research presented in this issue, because most of them at least implicitly take at least one or more of these questions into account, but as aids in reading the papers and gaining new insights into what they have to offer both at the level of research methodologies and to thinking about mathematical knowing more generally. We add to these questions ones that point out what we see as a major contribution of enactivist research to this work and of this research discussed in this issue to the testing and elaborating of enactivist theory more generally. In what ways can different patterns of recurring consensual coordinations of consensual coordinations of actions arise in mathematics knowing? How can differing flow patterns of inter-action in the flow of such consensual coordinations be observed to lead to different and different kinds of inter-objects? Maturana in his quote above makes a claim that inter-objects arise in such inter-actions. These papers in a wide variety of ways show just how such arising might happen. We have already highlighted Coles’ (2015) paper with respect to observer roles; it is also a very good source for observing recurring consensual coordination of consensual coordination of actions leading to observable inter-objects. It is also a source of studying patterns in such inter-actions; roles of the teacher and the task occasioning such inter-actions; and showing contrasts between comparable mathematical knowing settings in regards to the “flow”. But such patterns are observed in many other papers; for example, the flow of such inter-actions are clearly observable in the inter-action of the 2 students building robots reported in the Khan et al., (2015) paper. The papers by Abrahamson and Trninic (2015), as well as that of Towers and Martin (2015) read with the notion of flow and Maturanaian ideas in mind might provide further insight into the concept of co-action.

4 Third move: ethics

Through consideration of our colleagues’ enactivist research described in this issue we have noticed that their (collective) attention has been turned to how “learner-and-learned, knower-and-known, self-and-other co-evolve and are co-implicated” (Davis et al., 1996, cited in Reid and Mgombelo, 2015). In this paper we reflected on the observer, what we referred to as the first move in enactivist in research. We then turned to what we distinguished as a second move in enactivist research; i.e., the move from a view of knowledge construction to one of knowing as perceptually guided action that brings forth a world of SIGNificance that includes mathematics. Taking seriously these two moves we find ourselves with a heightened awareness of ethics and the ethical implications of an enactivist frame in our work as mathematics education researchers and teachers.

Remarkably, in this enactivist themed issue of ZDM there is no explicit discussion about ethics or the ethical (for enactivist papers that do discuss ethics see Mgombelo, 2006; Kieren and Simmt, 2009). Although we do not read specific discussions about ethics in this issue, we do “observe” the ethical in many if not all of the papers. Hence, we assert that a move of our (enactivist) collective attention towards ethics and the ethical is co-emerging with our enactivist sensibility. Maturana (1988b), Varela (1999), and von Foerster (2003a) each have been explicit about the ethical implications of the explanatory path we are calling enactivism.Footnote 1

Everything that we do becomes part of the world that we live as we bring it forth as social entities in language. Human responsibility in the multiversa is total. (Maturana, 1988b, p. 40)

At the very centre of this emerging view is the conviction that the proper units of knowledge are primarily concrete, embodied, incorporated, lived; that knowledge is about situatedness; and that the uniqueness of knowledge, its historicity and context, is not a “noise” concealing an abstract configuration in its true essence. The concrete is a not a step toward something else: it is both where we are and how we get to where we will be. (Varela, 1999, p.7)

“Objectivity, … (is a derivation) of a choice between a pair of in principle undecidable questions which are, “Am I apart from the universe? Meaning whenever I look, I’m looking as if through a peephole upon an unfolding universe; or, “Am I part of the universe? Meaning whenever I act, I’m changing myself and the universe as well. (von Foerster, 2003a, p. 293)

We are not speaking here of a technical version of ethics. Ones that are institutionalized through research ethics boards and demanded by funding agencies: things such as, informed voluntary consent, using data as it is intended, and truthful reporting of data. Neither are we speaking about morality: good and evil, right and wrong. The ethics we speak of are the implications of the moment by moment and day to day interactions we have as autopeotic and structurally coupled beings to others and otherness. These are the ethics that “must reside in the action itself” (Wittgenstein quoted in von Foerster, 2003b, p. 290).

In our reading of enactivism in mathematics education (as illustrated in this issue) we identified a move to reconceptualizing and asserting the significance of “the observer” in research (and teaching). We note two implications that the enactivist “observer” implies which we have discussed earlier but we raise here again. The first is the status of claims made by the researcher, and the second is the position of the observer in the research setting.

Illustrations of such ethics-imbued situations can be seen in most of the papers: for example when Proulx and Maheaux (2015) proposed that Maturana and Varela (1987/1992) fused epistemology and ontology with their aphorism “all doing is knowing, and all knowing is doing” and when Proulx and Maheux made a distinction between observations as the utterances and discourse and mathematical events as observations of the knowers our awareness of ethics and the ethical was occasioned, as it was when Lozano (2015) claims that it is her knowing we are observing in her reported research. In both of these papers we are asked to think about doing|mathematics in terms of the transformation of the researcher. This is quite different than thinking about the mathematical knowledge a learner is expected to have when they complete a task or program. This is a shift we must take note of since the claims being made then are not ones that point back to the participants but instead to the researchers. We recognize that if the claims from the research are about the researcher’s knowing and not the participants, we cannot immediately jump to “recommendations for teaching” or “recommendations for the classroom,” a common practice at the conclusion of most dissertations. Instead we are left with further questions, what can our research say to us about teaching? How can we use our understanding to inform practice? What practice and whose practice? We suspect there are deeper implications for the role of schooling with research based on a perspective that fuses ontology and epistemology.

The research in this issue that explores the interaction patterns in the classroom or other learning environments leads us to question what we think we are observing. Reading the research by Coles, Steinbring, Towers and Martin, and Preciado-Babb et al. (all in this issue) led us to question the individual as the unit of analysis. In their work they make some convincing arguments that the knowing is observed in the interaction and the recurrent interaction patterns of the teachers and learners. A learner’s knowledge co-dependently arises (Varela et al., 1991/1993) with their world of significance and that world includes the patterns of acting and co-acting. Again, if this is the case then what research claims can be made about an individual’s knowledge or knowing since this knowing is extended into other bodies, technologies, language and so on? How might their work inform the focus on evaluating and assessing individual student performance in mathematics? Can we speak of a learner’s mathematics without speaking about the teacher’s (observer’s) mathematics?

As we discussed in our section on the second move in enactivist research, we observe that when humans bring forth worlds of significance (including mathematics) they change the possibilities for not only themselves but for others too. Reflecting on the studies in this issue we note that the various positions the observer can take in a research study have distinct implications. Observing from the back of the classroom or from behind a screen (video) is quite different than acting as a participant observer where the observer interacts with participants in the setting. In the first case there is little or no explicit interaction with the research participants (Lozano, 2015; Steinbring, 2015) in the latter there is interaction but not deliberately instructive (Precido-Babb et al., 2015). This is quite different from when the researcher is also the teacher (Metz and Simmt, 2015) and this is different when the researcher is a colleague or a superior to the people whose classrooms are being observed (Coles, 2015). And all of these are different from a clinical interview setting where participants engage in tasks specifically for the purpose of being observed as research subjects (Abrahams and Trninic, 2015; Khan et al., 2015). Each of these settings affords a particular coupling between research and participants. When Maturana and Varela (1987/1992) suggest that knowing is effective action they do so with the structurally coupled knower-environment in mind. If we want to know if a student knows about rotations and we provide a paper and pencil task, the student is afforded a much different coupling with that environment than the student who is provided a robot to build from paper directions (Khan et al., 2015). Hence there is an ethics in the act of task selection. The same argument fits for each of the other cases. In our position as observers, whether it is observer behind the screen, participant observer, teacher, colleague, supervisor, or task designer we are part of the environment of the learner (the other). When we act, our acting changes the sphere of the possible for ourselves and for the other.

Hence our research is not (cannot be) neutral on the setting, the participants, or the overall context. Observing a unity in this enactive way changes the world at many levels. Therefore the position and role of the observer in the research setting (including responsibilities, commitments, interactions and so on) leads to questions of ethics. For example, we noted multiple roles reported in the papers in this issue: the observer behind the screen who does not interact with the participants (Towers and Martin, 2015), the observer who interacts but is not pedagogically responsible for the participants (Lozano, 2015), the researcher who is a facilitator (Coles, 2015; Preciado-Babb et al., 2015), the researcher teacher (Metz and Simmt, 2015), the teacher researcher (Maheux and Proulx, 2015); and the task/prompt provider (Khan et al., 2015; Abrahams and Trninic, 2015). These varying roles call for at least two distinguishable ethical stances. In all of these studies the researchers were working in and to an extent creating and bringing task elements to the environments that they hoped would be useful in occasioning mathematics knowing actions worth observing. In these latter circumstances one might observe the role of provisional ethics (Kieren and Simmt, 2009) suggesting that the tasks in the environment are seen to be important at two levels: they are appropriate for both the research and (more importantly) are aimed at allowing students to bring forth a world of mathematical significance. In nearly all of the research the researcher(s) needed to attend to the knowing in action and this was true for all of the members in the cascade of observers discussed under move 1. In other words, all of the actors in these various situations of mathematics knowing are viewed as non-trivial machines who must take up and transform materials from the environment to be seen to be engaging in such knowing, attending to the other and to the environment in appropriate ways is key. In other work we have observed this as an attentional ethic (Kieren and Simmt, 2009). Finally, if the flow of languaging and distinctions in languaging is critical to the kinds of inter-active knowing observed in these papers, then we must admit that whenever a person acts in those situations (whether they are conscious of this or not) those very actions and the residual artefacts of them trigger change in the environment in which they take place and create new possibilities for others sharing the environment. Varela et al. 1991 point out the ethical commitment in such knowing actions that we have highlighted with the name occasion ethics (Kieren and Simmt, 2009).

Most of the papers in this issue focused quite specifically on methodological issues that co-emerge with an enactivist orientation to research, especially in terms of how they impact the work and role of the researcher (Proulx and Maheaux, 2015; Lozano, 2015; Kahn et al., 2015; Abrahamson and Trninic, 2015) and the claims that can be made from the research. Most profound for us is the stance that an understanding of knowledge as that which co-emerges in the moment-by-moment perceptually guided action of people interacting with others and enabled and constrained by the particular tasks, questions and expectations a teacher puts in place for the learners demands that as observers seeking to explain mathematics teaching and learning or curriculum we must as Proulx and Maheaux suggest—listen for the mathematical, as Khan et al. do—observe for the sensorimotor patterns that enable action, or as Coles (2015) and Steinbring (2015) suggest—look for the patterns of interaction that are immediately observable and those that co-emerge over time within the collective. Teaching as a reciprocal activity and the emergence of new systems which may not be represented previously require observers shift their gaze from the structurally coupled individual and environment to the system that may be generated from many people interacting as one, as a single learning system. Towers and Martin (2015) explore the implications of this. They provide a means for observing co-action and the development of collective understanding. Co-action might also be observed as an ethical act, individuals making space for others beside themselves (Maturana calls this love (see Tree of Knowledge, Maturana and Varela, 1987/1992)) thus creating a consensual domain of existence.

Steinbring’s (2015) attention on communication as an autopoetic system raises for us the question about ethics that emerges from the theoretical stance a researcher takes. In our view there is an imperative that emerges from his understanding that “communication is neither directly possible nor transferable from one to another person… understanding in communication presupposes a conceptual common ground and a joint praxis of actions.” If there is no direct exchange in a communicative act then there is a pedagogical imperative which involves an attentional ethical responsibility for the teacher to listen for the understanding of the learner rather than for a particular response. Steinbring raises the notion of a trivial machine: i.e., a machine for which a particular input will generate a known (hence predictable) output. When teaching is understood as transmission of knowledge and learners as receptors of that knowledge the learner is understood as a trivial machine. With the explanatory mechanisms of autopoesis, structural determinism and coupling the learner is viewed as a non-trivial machine that acts on itself to bring forth itself and its world of significance. In this regard enactivism leads us to a critique of schooling. To the extent that in the process of schooling there is a deliberate attempt to transmit knowledge from teacher to child, trivialization of the human being is promoted (von Forester, Foerster 2003b). Steinbring uses Luhmann’s work to demonstrate how communication itself is an autopoetic system hence the direct transmission of knowledge from teacher to learner is not possible. Rather there are recurrent patters of interaction that can lead to a researcher’s observation of children behaving as if they are trivial machines. On the other hand, a history of patterns and routines enable adequate action. Varela (1999) in Ethical Know How: Action, Wisdom and Cognition, builds a strong case that if cognition is understood as moment-by-moment perceptually guided action (“it is both where we are and how we get to where we will be” (p.7)) then each act has an ethical imperative. “We always operate in some kind of immediacy. It is through recurrent interactions that we have a readiness-at-hand proper to every specific lived situation…. Thus who we are at any moment cannot be divorced from what other things and other people are to us” (p. 9).

5 Conclusion

We began this paper by asking, “Why a special issue on enactivism as a research methodology in mathematics education?” That question focused our search as we read the papers in this issue. There were many theoretical and empirical concepts and questions related to mathematics knowing and research knowing challenges that we could have addressed to answer our question but we selected what we came to see as three moves in the enactivist framed mathematics education research we found in this issue. Of course we are structurally determined beings with a long history in enactivism; our response is what it is because of who we are and who we have interacted with over the years (theorists, colleagues, learners and teachers). This paper reflects the world we continue to bring forth through observing observers (who are ourselves too).

The first move (and the one that received much of the attention in the papers) was that of the observer. Enactivism proposes the observer is one who arises in the act of observing and whose knowing is explained through the mechanism she describes. The second move is an understanding that all knowing is perceptually guided action that brings forth a world of significance. The third is a consequence of the first two: All knowing has implications. Enactivism demands that we understand that: The observer is not neutral; her observations bring forth worlds of significance that intersect with the worlds of others. Hence when the observer’s world changes, the environment of the other is altered; the other has an altered environment to select from as she brings forth her world of significance.

Enactivism as a methodological frame for mathematics education research is a form of research that is occasionally and multiversally incomplete, which implies that there is necessarily always more to be said and different grounds for the saying about the phenomena under investigation in the papers in this issue and about “doing|mathematics in general.” We accept the incompleteness of this knowing but invite and make space for others to live it with us.