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“Well, it’s no use your talking about waking him,” said Tweedledum, “when you’re only one of the things in his dream.”

— Lewis Carroll, Through the Looking Glass, and What Alice Found There.

In this final chapter I attempt to describe Enactivism, and to relate it to my research.  I begin by describing some of the key ideas of Enactivism and giving examples from my studies of these ideas in action.  I then develop a theory of the development of deductive reasoning, based on Enactivist principles.  This is followed by a description of an Enactivist methodology for research in education, which reflects the current state of development of the underlying methodology of my studies.  Finally I make a few comments on the relationship between my research and this dissertation and on the teaching of proving in schools.

1. Enactivism

“Enactivism” is used by Bateson, Maturana, Rosch, Thomson, and Varela to label their theories.  The “Experientialism” of Lakoff and Johnson is closely related to Enactivism, and I will not distinguish them here.  Enactivism is a theory of mind, but, as Bateson (1987) notes, from an Enactivist perspective “epistemology and theories of mind and theories of evolution are very close to being the same thing” (p. 38) so discussions of Enactivism range through the traditional disciplines of philosophy, psychology, and biology.  Elements of the psychology of Piaget and Vygotsky are compatible with Enactivism, and I will draw on their writings occasionally, especially in considering Enactivism in relation to learning.  The philosophical basis of Enactivism can be found, with some effort, in the writings of Wittgenstein on the philosophy of psychology, and I will make connection with his work wherever possible.

A good starting point to understanding Enactivism is the problem of the relationship between an entity and its surroundings.  The first part of this problem is specifying what it is that makes us see the entity as separate from its surroundings.  The term organization is used to describe those features of an entity which allow an observer to distinguish it from everything else.  Note that this implies that an entity’s organization varies from observer to observer.

The participants in the studies, the pens they wrote with, the tables they write on, all have particular organizations that make them people, or pens, or tables.  These entities have fairly stable organizations, but other entities do not.  During the problem solving sessions many groups worked together as a group.  The groups themselves can be distinguished as entities, which had a certain organization in the problem solving situations.  In most cases however, those entities no longer exist.  The organization that defined them no longer relates the people involved.  Stacey and Kerry were an interesting case, partly because the organization that defined them as a pair existed before they became participants in my studies.  This implies a

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 different organization for the Stacey-Kerry pair than for the others, one that continued to exist. 

Some entities have an organization that is complex.  Complexity is a term borrowed from complexity theorists (e.g., Kauffman, 1993, see also the previous chapter).  A system is complex if “a great many independent agents are interacting with each other in a great many ways” (Waldrop, 1992, p. 11).  Complex systems create themselves, in the sense that they come into being and remain in existence through their own internal interactions. 

Systems that continually create themselves are referred to in Enactivism as autopoetic.  The components of autopoetic systems “must be dynamically related in a network of ongoing interactions” (Maturana & Varela, 1992, pp. 43-44).  That is, the components interact in ways which are continually changing, but which at the same time allow for the continuation of interactions so that the system continues to exist.  In addition, the interactions of the components of an autopoetic system are responsible for the production of the components themselves.  In summary, an autopoetic system is an emergent phenomenon arising from the interaction of components which, by way of these interactions, give rise to new interactions and new components, while preserving the system’s autopoetic character.

Human beings, and living beings in general, are autopoetic.  I change continuously, but at the same time all these changes permit me to continue existing as me.  In the past three years many of the changes in me have involved the evolution of my ideas on proof, proving, and thinking.  These changes have been a part of my continuing existence as a Ph.D. candidate.  The interactions which make me a Ph.D. candidate are now changing as well as the complex system that is me prepares to orient itself to a different environment in a way that will permit my continued existence.

Adapting involves changes to a system’s structure.  It is important to distinguish between the structure of a system and its organization.  A system’s organization includes the invariant features without which it would cease to be what it is.  An autopoetic system must maintain its organization.  The structure of a system includes all its features at a given moment.  Interactions with its environment and within the system itself result in a continuous modification of a system’s structure.

The problem is how to handle the problem of structural change and to show how an organism, which exists in a medium and which operates adequately to its need, can undergo a continuous structural change such that it goes on acting adequately in its medium, even though the medium is changing.  Many names could be given to this; it could be called learning.  (Maturana, 1987, pp. 74-75)

In the problem sessions the participants’ structures were changing continuously as their understandings of the situations changed and as they reasoned in different ways.  At the same time they remained themselves.  When Eleanor switched from reasoning inductively with Ben and Wayne to proving with Rachel she did not become a new person.  Nor was there any danger in confusing who was Eleanor and who was Rachel when they were thinking in the same way.  The way they were thinking was a part of their structure.  That they could think is a part of their organization.  I have been careful not to say: “Rachel is an deductive thinker,” which implies thinking deductively is a part of her organization.  If this 

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were so then she would cease to be the entity she is whenever she thought in another way.  Thinking deductively was a part of her structure for much of the time she was investigating the Arithmagon situation, but that is as much about the situation and the social context as it is about her.

Living systems achieve autopoesis by acting in some way to adjust to local conditions.  It is this acting that indicates cognition, so in Enactivism, cognition is a feature of all living systems.  This idea is encapsulated in the phrase “Knowing is being is doing.”  The word “enactivism” is derived from this idea of knowing in action. The way a living system comes to know about the medium it is in is through interaction with that medium.  This implies that the system’s knowledge of its world depends not only on the medium, but also on the actions the system is capable of.

To take an example from the Arithmagon situation, some of the participants who knew how to solve systems of equations acted in the Arithmagon situation by solving it using such a system.  In doing so they learned that the Arithmagon is, in a sense, about a system of equations.  That is, for those participants who solved the puzzle using a system of equations, the Arithmagon is a system of equations.  For the participants who could not act in that way, either because they did not know how, or for some other reason, the Arithmagon is not about a system of equations.  What they could know about the medium they were in, the Arithmagon situation, depended on what they could do.

An autopoetic system is “an active self-updating collection of structures capable of informing (or shaping) its surrounding medium into a world through a history of structural coupling with it” (Varela, 1987, p. 52).  As noted above, a system only knows about those aspects of its medium with which it can interact in some way.  This means that in being, doing, and knowing, a system defines the world in which it lives. 

As I noted above, for those participants who solved the Arithmagon using a system of equations, the Arithmagon situation included systems of equations.  For others it included other features.  When Eleanor showed her solution method to the rest of us, the Arithmagon included a “middle number” that was simultaneously the sum of the corners, half the sum of the sides, and the sum of a corner and the side opposite.  For the participants in other sessions, the Arithmagon did not have a “middle number.”  For us it did.  Similarly other groups were in an Arithmagon situation which included only the sum of the side and the corner opposite or which included the relationship between the sums of the sides and corners.  For Wayne, the Arithmagon situation included properties of triangles, and he explored the Arithmagon using those properties. 

The activity of coming to know, of learning, is a modification of structure.  At the same time it is the system’s structure that limits what actions it can take in the environment, and therefore what it can come to know.  This limitation of a system’s possible actions is called structure determinism.  What a system does in response to a trigger from its medium is determined entirely by its structure. 

When I asked Laura if she was sure her formula worked for all Arithmagons, I asked because I wanted to know if she was sure.  Her reply “Oh, you want me to prove it” indicates that her structure was such that the trigger provided by my question did not produce the effect I expected.  My question was just a trigger.  Laura’s response was a product of her structure.  If she has spent 

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some period of time in a medium where the question “Are you sure?” is used as a trigger for proving in a teacher-game, for example in a mathematics class where proving is thought to be the only way to be sure, then Laura would have had to modify her structure to remain viable in that medium.  This modified structure then determined her response to my question.  Of course the action of producing her “proof” changed her structure, and the realization she related later, that she could prove her formula in a better way which explained it, changed her structure again, and so I could not predict how she would respond in a similar situation now, just as I could not predict how she would respond when I asked the question in the first place.  Because she has a complex structure her actions may be determined from moment to moment, but they are never predictable.

If I have a living system ... then this living system is in a medium with which it interacts.  Its dynamics of state result in interactions with the medium, and the dynamics of state within the medium result in interactions with the living system. What happens in interaction?  Since this is a structure determined system ... the medium triggers a change of state in the system, and the system triggers a change of state in the medium.  What change of state?  One of those which is permitted by the structure of the system.  (Maturana, 1978, p. 75)

In this passage Maturana introduces a central idea of Enactivism: co-emergence.  The interaction between a system and a medium (which may include other autopoetic systems) is the mechanism by which both the system and the medium change.  As long as a system and a medium continue to be able to interact they are said to be structurally coupled and they co-emerge.  It should be emphasized that co-emergence does not imply that the system and the medium are becoming more fully adapted to each other.  All that is certain is that their structures allow them to interact.  It is possible that a history of structural coupling may lead to a situation in which the system and the medium are no longer able to interact.  In this case they cease to be structurally coupled.  This may be because the system migrates to another medium or because the interaction between the medium and the system disrupts the organization of one or the other, and it dies. 

In any of the sessions the participants and the observers co-emerged with each other and the situation.  Tom’s presence changed Rachel’s actions, and hence her being, her structure.  At the same time her actions changed his structure, as he learned about mathematical understanding, among other things, in the Arithmagon situation.  What Eleanor knew at the end of the session developed through her interactions with everyone else, and what the rest of us knew at the end of the session developed in part through her interactions with us.  As what we know is embodied in our structures, our structures co-emerged throughout the time we were interacting; that is to say, throughout the time we were structurally coupled.

In describing the relationship between an entity and its environment, the mistake is sometimes made of seeing the environment as prescribing the structure of the entity.  For example, in the popular understanding of Darwin’s theory of evolution animals are seen as having certain features because their environment requires that feature.  So polar bears are white, unlike most other bears, because they live in snowy surroundings.  The enactivist view of evolution is one of natural drift, based on an animal’s environment proscribing certain features.  This proscription is simply another way of looking at the breakdown of the structural coupling between the animal and its environment.  If the animal’s structure does not 

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allow for interaction with its environment, then it dies.  In effect it is not allowed to have that structure.  This is not the same as the environment requiring that it have a certain structure, and in fact many different structures are possible within the constraints imposed by the need to remain structurally coupled.  The full range of possible structures defines a sphere of behavioral possibilities within which animals can act.

The problem situations the participants investigated defined a sphere of behavioral possibilities for them.  The Arithmagon situation can be investigated in many ways, but not, as far as I know, by singing arias.  If the participants had chosen to sing arias during a problem situation, then I would judge that they were no longer in the situation.  Their behavior would break the structural coupling between them and the situation.  When Kerry began to investigate the negative Fibonacci sequence, Tom intervened and said the sequence did not work like that.  The Fibonacci situation, which for Kerry included Tom, was proscribing the investigation Kerry had attempted to initiate. 

2. Enactivism and reasoning

Enactivism is, among other things, a theory of learning, and the following is my attempt to use the ideas of Enactivism to elaborate a theory of the development of deductive reasoning in children growing up in a Rationalist society.  This is a theory in progress, as all good theories should be, but this one is so early in its progress that I expect its structure to undergo some serious modifications in the future.  I include it here both as an application of Enactivism, and as an indication of the theoretical perspective which co-emerged with the methodology and results of my research.  It is also a hint of future research I plan, involving studies of younger students reasoning deductively, with the aim of tracing the development of deductive reasoning more closely.

In developing a theory of the development of deductive reasoning I have drawn on the ideas of Piaget, Vygotsky, Wittgenstein, and the Enactivists.  Piaget’s ideas form the basis of constructivism, a theory of learning that informs my work and the work of many others in educational research.  The central idea of constructivism is that the individual constructs the world in which he or she lives. 

...when language and thought begin, [the child] is for all practical purposes but one element or entity among others in a universe that he has gradually constructed himself and which hereafter he will experience as external to himself.  (Piaget, 1967, p. 9)

For Piaget the world is constructed by the individual, and the influence that the world has on the individual’s development is not a major focus.

For Vygotsky, the external world, especially the social world, plays a central role in the development of the individual.  This is especially evident in the relationship between thought and language. 

Thought development is determined by language, i.e., by the linguistic tools of thought and by the sociocultural experience of the child.  (Vygotsky, 1986, p. 94)

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Vygotsky holds that language development and thought development occur separately in very young children.  At a certain point however, these two paths of development meet, and “thought becomes verbal and speech rational” (1986, p. 83); that is, thought and language develop together from that point on.  The development of language cannot depend solely on the individual, as language takes place in a social context.   The linking of thought and language means that thought is similarly constrained by social context.

Wittgenstein also considered the relationship between thought and language, and contributed the important idea of a “language game”.  A language game is the context, the “form of life” in which words are spoken, and actions are made. 

...we make a radical break with the idea that language always functions in one way, always serves the same purpose: to convey thoughts—which may be about houses, pains, good or evil, or anything else you please.  (Wittgenstein, 1958, §304)

The way we use language in a particular context determines the meaning of the words we use in that context.  Meaning cannot be established, once and for all, as dictionaries attempt.  The best we can hope for is to be aware of language games, and perhaps make use of the “family resemblances” between language games to know what meanings are in play. 

Varela and Maturana introduced the Enactive approach to cognition to escape from the “chicken and egg” situation of trying to decide whether the individual constructs a world or if the world constructs the individual.  In a sense, they agree with everyone.  The material world and the social world do affect the cognitive structures of the individual, as they must if the individual is going to survive embedded in a material and social context.  At the same time, the cognitive structures of the individual guide the individual’s actions and interactions in and with the material and social worlds.  “World and perceiver, specify each other” (Varela, Thompson, and Rosch, 1991, p. 172).  We as beings co-emerge with the worlds we inhabit.

The mechanism by which this co-emergence takes place has two parts:

... the structured nature of bodily and social experience and ... our innate capacity to imaginatively project from certain well-structured aspects of bodily and interactional experience to abstract conceptual structures.  (Lakoff, 1988, quoted in Varela, Thompson, and Rosch, 1991, p. 178)

Lakoff and Johnson give the name “experiential gestalt” to our perception of structures in our material and social worlds.   They call our ability to “imaginatively project” from our experiential gestalts to abstract concepts “metaphoric projection.”  I believe these two mechanisms can be used to provide a theoretical path from human experience, at its most basic level, to the production of formal proofs.

The perception that lies at the base of deductive reasoning is coincidence.  When two unusual events happen at roughly the same time, or in the same place, we perceive them as linked.  For example, Freudenthal (1973) tells a story of walking with a child past a railway crossing.  The previous day, when they had passed the crossing, there had been a friendly dog there.  On this day they came to 

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the crossing, and the child asked “But where is the dog?”  The child had perceived a link in the coincidence of the dog and the crossing.

The capability to notice and remember coincidences is vital for perception.  Without this capability, babies, for example, could not learn that the particular sensory stimulations associated with faces often coincide with those associated with voices, a necessary step to the perception of voices emanating from faces. 

The studies of the behaviorist psychologists seem to have established that mammals and birds are capable of noticing and remembering coincidences.  It is the coincidence of two stimuli that allows the transference of a response from one to the other.  Note that noticing and remembering coincidences is an act of perception, not one of conscious, thoughtful, action.  Given the empirical evidence, it seems reasonable to accept that observing coincidences is a biological feature of human beings. 

One coincidence that babies might observe and remember is that shaking a rattle, and the sound of a rattle coincide.   In fact, babies not only observe and remember such coincidences, they practice them (Lakoff & Johnson, 1980, p. 70).  These coincidences are important for babies, and so they shake their rattles, not continuously, but in short bursts, pausing to delight in another occurrence of the coincidence. 

Bruner notes research showing that babies have a sense of causation (1986, p. 17).  Given that such a sense exists, how might it have come to exist?   Lakoff and Johnson (1980) consider causation to be an experiential gestalt.  Just as coincidence involves observing and remembering when events occur, an experiential gestalt involves observing and remembering how events occur.  The event of shaking a rattle shares an experiential gestalt with many other coincidences in babies’ lives.  Pulling blankets, dropping things, throwing things, all share features with shaking a rattle.

Lakoff & Johnson (1980) give a list of the features of the prototypical experiential gestalt for causation:

The agent has as a goal some change of state in the patient.

The change of state is physical.

The agent has a “plan” for carrying out this goal.

The plan requires the agent’s use of a motor program.

The agent is in control of that motor program.

The agent is primarily responsible for carrying out the plan.

The agent is the energy source (i.e., the agent is directing his energies toward the patient), and the patient is the energy goal (i.e., the change in the patient is due to an external source of energy).

The agent successfully carries out the plan.

The change in the patient is perceptible.

The agent monitors the change in the patient through sensory perception.

There is a single specific agent and a single specific patient. (pp. 70-71)

In the case of the rattle the agent is the baby; the patient is the rattle; the energy transfer is the shaking; and the goal is the sound.

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While the experiential gestalt of causation has been presented here in a propositional form, babies, of course, do not think of causation in this way.  I would characterize babies’ pre-verbal thinking as sensing, and speak of a sense of causation.  A parallel might be made between this sense of causation and the sense of direction we derive from hearing sounds:

I may be able to tell the direction from which a sound comes only because it affects one ear more than the other, but I don’t feel this in my ears; yet is has its effect: I know the direction from which the sound comes;  (Wittgenstein, 1958, IIviii, p. 185)

Similarly, when we know that an event caused some other event, it may be because we recognize that the condition for the experiential gestalt of causation are present, but we do not know it in that way.  We just know it, as a sense.  In the case of the pre-verbal baby this sense can exist, even though the concept of causation cannot.

The fact of becoming conscious of a category will alter its actual nature....  When the child “is cause,” or acts as though he knew one thing was the cause of another, this, even though he has not consciously realized causality, is an early type of causality, and, if one wishes, the functional equivalent of causality.  (Piaget, 1959, pp. 229-230)

When a baby begins to learn language, the sense of causation the baby has developed from coincidences is changed by the way language talks about causation.  “The rattle made noise because I shook it” seems to be nothing more than an expression of the sense the baby already possessed, but the very act of expressing that sense changes it.  As Vygotsky (1986, p. 219) puts it: “It does not merely find expression in speech; it finds its reality and form.” 

Talking about causation is an act with causation itself as its object.  The very act of talking about causation makes causation an object in our world.  “As language arises, objects also arise as linguistic distinctions of linguistic distinctions that obscure the actions they coordinate” (Maturana & Varela, 1992, p. 210).  In talking about causation we are constrained by our language.  A child’s idea of causation, once articulated, becomes subject to the rules of already existing language-games.  These language-games are the context in which verbal thought develops.  It is important to consider, as well, that this development has effects on both the individual’s thinking, and the language-game. Verbal thought and language-games coemerge.  So, even as we become able to think about causation by becoming able to talk about causation, what “causation” could mean to us is changed by our new ability to think about it.

What is the “new form” of causation?  We can now say “We will eat now because it is six o’clock.”  Such a sentence casts time as an agent, and ourselves as patients, changing who we are.  Such a sentence is an example of the metaphoric projection of causation.  The casting of an experiential gestalt into verbal form permits such projections of meaning to new domains.  This process both changes the concept we extend (causation) and the domains into which we extend it (time, and ourselves).  Such metaphoric projections can be made by individuals, but more often they are suggested to us by others.  The language used around us leads us to make certain metaphoric projections and not others.

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As we learn language we also learn to use language to refer to abstract entities like time.  These abstract entities, and the ways in which we can talk about them, are constrained by the language games in which we find ourselves embedded.  When the language game involves making links between abstract entities, analogous to causal links between material entities, we have the occasion to make inferences.  Inference is the metaphoric projection of causation to abstract entities and energies.

To emphasize the importance of language games in the development of the ability to make inferences, consider Belenky et al.’s (1986) Silent Knowers.  They experience situations of extreme social instability.  In such situations language games do not include inferences, and so making inferences is both useless and inconceivable.  It is only in retrospect that a silent way of knowing can be described, if such a description assumes a way of knowing that includes sufficient stability of abstract entities to allow inferences to be made.

The ability to make inferences probably precedes the concept of inference, just as the ability to sense causation precedes the concept of causation.  Unlike causation, however, the concept need not come after the sense.  Just as the sense of inference develops from the sense of causation, the concept of inference can develop from the concept of causation by metaphoric projection. 

The experiential gestalt of causation deals for the most part with physical agents, physical energy transfer, and physical patients.  In the example of “We will eat now because it is six o’clock” we encountered an abstract agent, time.   Such a metaphoric projection is comprehensible to us because of our ability to perceive coincidences, in this case between aspects of abstract entities and physical entities.  We cannot perceive coincidence between just any aspects; however, the aspects available to our perception are those that are reified by language.  By such a metaphoric projection the concept of inference can develop.  In the case of inference, the sense need not precede the concept, nor must the concept precede the sense.  It seems likely that when language is full of abstract entities linked by inference there would also be occasions to refer to these inferences, and in such a context the sense and concept of inference could coemerge.

The concept of inference is the basis for deduction.  Deduction involves a perception of inferences as meaningful.

The school child passes from unformulated to verbalized introspection, he perceives his own psychic processes as meaningful.  (Vygotsky, 1986, p. 170)

Being able to make inferences, being able to refer to inferences, and the use of inferences in paradigmatic language games (Bruner, 1986), gives the inferences themselves meanings, beyond those of the concepts involved.  Part of the meaning of an inference is the idea of logical necessity.  An inference with such a meaning is a deduction.  When these meaningful inferences or deductions occur in sequences they constitute what I call proving. 

An inference need not be a deduction.  In the case of narrative language games inferences are related to intention, rather than logical necessity (Bruner, 1986).  Because of this distinction, inferences about the actions of human beings and other intentional beings refer to choices made by them, rather than actions forced on them by material or logical constraints.

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Deduction permits the conscious choice to deduce, with particular ends in mind.  Deductive “chains” can be purposefully constructed.  The concept of a formal proof can be developed by metaphoric projection of deduction and chains, into an abstract entity which can be both analyzed and self-directed.

Learning to direct one’s own mental processes with the aid of words or signs is an integral part of the process of concept formation.    (Vygotsky, 1986, p. 108)

Analysis of reality with the help of concepts precedes analysis of the concepts themselves.  (Vygotsky, 1986, p. 141)

A chain consists of links, each of which is a separate entity.  The links are not materially connected to each other.  They could continue to exist outside of the context of the chain.  Their structure, however, is such that they are constrained by their neighbors.  The combined effect of these constraints can be seen by an observer to constitute a single object, a chain, where many objects exist.  Similarly the deductions in a proof can stand alone, but their structure allows them to be joined together into what seems to an observer to be a single object, a proof.

Because proofs are metaphoric projections of chains, they can be analyzed as if they were chains, one link at a time.  The strength of a proof, like the strength of a chain, is that of its weakest link.  A missing link in a proof or a chain makes it completely functionless as a way of connecting its two ends.  The making of a proof, like the making of a chain, is a self-directed activity.  At each link there is a decision made as to what link to attach next, how it should be attached, and whether there might be some shorter chains (lemmas) lying about that might be incorporated into the proof chain under construction.

Because proofs are self-directed and analyzable, they must be generated by a formulated act of proving.  Formulated proving is a sub-category of the proving which I described above as sequences of deductions.  In formulated proving the next link is chosen.  In unformulated proving the next link is whatever is at hand.  This means that an observer could judge the strength of a proof produced by unformulated proving (by analyzing it, proving in a formulated way with the proof as a guide), the person who proved could not. 

3. Enactivism and research

Enactivism is not only a logical extension of Rationalism made aware of its weaknesses, nor only a theory of learning with which to interpret the proving of students, nor only the basis for theories such as the one I just outlined.  It is also, and must also be, if it is accepted as a theory of learning, the basis of a methodology for research.  Research is learning, and educational research which employs one theory of learning to interpret a student’s actions, and a different theory to motivate a researcher’s actions, undermines its own basis.  In this section I describe my methodology, which arose out of this research and the research and teaching I have done in the past, and which co-emerged with the research methodologies of my colleagues who form a loose Enactivist research group.  Kieren, Gordon-Calvert, Reid & Simmt (1995), or Gordon-Calvert, Kieren, Reid & Simmt (1995) are examples of research done by the group as an emergent entity of structurally coupled researchers.

Basic principles

The overall methodology, which connects the various methods and analytic procedures outlined below, I call ‘bricological.’  Bricological research, in short, combines the flexibility and creativity of bricolage, with an underlying logic of inquiry.  Given the critique of Rationalism in Chapter VI, I hope you will not be surprised that some Rationalist assumptions are missing from this methodology.  The idea of objective truth, and the application of deductive reasoning to nondeterministic complex systems, like people and societies, are the two most important omissions. 

For me, the key point of Enactivism is the co-emergence of individuals with their environments.  The distinction between individual and environment must be blurred, as each is an active entity whose actions occasion modifications of the other’s structure.  A related idea is that of proscriptive constraints.  Any individual acts within a sphere of possibilities, which proscribe some actions, but dictate no action in particular.  This play in the interaction between individual and environment makes the usual assignations of cause and effect impossible.  What the individual does is ‘caused’ both by its own dynamic structure and by the constraints of the environment.  At the same time there are ‘effects’ on both the individual and the environment as their structures are changed by the (inter)action. 

Bricolage, as it is used in conceptualizing bricological research, favors the production of complex structures, theories, models, etc. because there is no need to reject possibilities that are ‘too expensive’, or ‘too long’.  It can be contrasted with a technological attitude that favors production of lots of results through straightforward, ‘clean’ techniques.  Complex theories are appropriate because the topics of my research are complex systems in and of themselves.  It is important to note that just as complex systems are self-organizing, so are complex theories.  They organize themselves in a medium that is defined by my thinking, and the thinking that takes place in the groups with whom I do research.  They also adapt as part of the process of reporting research, as writing involves a structural coupling with an imaginary reader, whose thinking joins into the theories’ medium. 

The logic of the bricological methodology comes from the questions chosen for research, and the theories and models with which the research begins.  These questions, models, and theories reflect expectations of what might be seen.  In the adaptation of Enactivism to research, these expectations correspond to the plastic structure that determines the actions of an individual in a context.  Just as an individual’s structure changes in changing the context, so our expectations change even as we observe, interview, and analyze according to our expectations. 

My favorite metaphor for bricological research is the medieval method of cathedral building.  Unlike modern office towers, the design of a cathedral was not the work of an individual whose plans determined the actions of a crew of workers.  In the construction of a cathedral every worker had a general idea of the final appearance of the building, but no single individual knew exactly what it would look like.  In the time scale of cathedral building, the master builder who sketched out the initial design might well be dead by the time the cathedral was completed, and changes in finance, style, technology, and workers might have resulted in considerable alterations.  Each worker’s contributions were to a small portion of the building and were governed by the possibilities created by the actions of previous workers.  The combined efforts of the workers constituted a bricolage of what was possible with the materials. skills, and prior work present.  At the same time their 

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work was drawn together by the idea, the logic, of the cathedral as a final form, dedicated to the glory of God. 

Some specifics

Bricological research is research in the Rationalist tradition, though much of what was wrong with Rationalism is missing, and much of what is left is changed.  All the same, I can write of my “data” and the “analysis” of it.  

The data generated in my research include field notes, video tapes, audio tapes, participants’ writings, transcripts, notes based on viewing video tapes, mathematical activity traces (MATs, see Appendix C for examples) which summarize the actions in a video taped session, research reports, conference presentations, and notes from discussions with other researchers.  These artifacts can be lumped together, as they are here, as ‘data’, but at the same time all of them record acts of interpretation.  In a sense it can be said that there is no data, only interpretations and interpretations of interpretations.  That said, I will refer to any artifact of the research process as ‘data’. 

The analysis of the data is tied up with the idea of multiple interpretations.  This means several things.  It means that the same event was interpreted in several kinds of data.  It means that the data was interpreted by several researchers.  It means that the data was interpreted many times by one researcher.  It means that the problem prompts were interpreted by a number of participants.  It means that the participants were interpreted by several problem prompts.  A rough chronology of this process follows.

The first stage of analysis in all the studies was the recording of field notes and video tapes (or in the case of the classroom observations, audio tapes) and the collection of participants’ writings.  The field notes record the initial impressions of the researcher as to what was important in the session.  The video tape records what was visible and audible from a particular point of view.  The participants’ writings record what they felt it was necessary to record during the course of their investigation of the situations. 

The second stage of analysis was the viewing of video tapes and participants’ writings by the researchers, either alone or in groups.  The notes produced through this process reflect again what was important to the researcher at the time, and the significant points introduced by other researchers.  This stage was repeated a number of times by various researchers, according to the perceived significance of the data produced.

The third stage of analysis included the production of transcripts and mathematical activity traces from the video tapes and notes taken during viewing.  Not every word spoken, nor every action taken by the participants was transcribed or entered into a MAT.  The selection of significant episodes was yet another interpretive action coming out of the viewing of the tapes.

The fourth stage of analysis was the preparation and presentation of the research.  The organization and expression of data in publications, research reports, and presentations, occasioned the modification of theories and the reevaluation of data.  The comments of respondents at presentations marked their involvement as co-participants in the research whose contributions form a part of the interpretation and data of the research.

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As is discussed in the methodology section above, the analysis of data is seen here not as the re-presentation of objective facts, but as a process of co-evolution of theory and data.  This point of view comes out of a recognition that such a process is implied in all research, and a belief that adopting a methodology that makes use of this inevitability is the best way to accommodate it.  I take seriously the claims of philosophers of science (e.g., Kuhn, 1970, and Chalmers, 1982) that no observation is free of a bias introduced by the theoretical position of the observer.  At the same time the necessity of theory to account for data results in a dialog between theory and data, with each one affecting the other.  A methodology that attempts to make use of this interaction transforms the analysis of data into a continual process of change and encourages this process as the mechanism of theory improvement.

4. A few parting words on Enactivism, proving, and teaching

I hope this chapter points out the Enactivist notion of “coemergence” in the relation between my empirical and theoretical work on reasoning, and the underlying methodology of my work.  In researching reasoning I was reasoning about research, and what I found out about reasoning and what I reasoned about what I found interacted throughout my work.  The circularity in my writing, which I noted in my introduction, is not a simple ploy to get you to read more or differently, but instead an accurate reflection of the relationship between methodology, theory, and data in my work.  Perhaps research can be done in the traditional models of picking a methodology, gathering data, and developing a theory (although Enactivism suggests that this is not so); in my research the methodology, data and theory emerged together in the interactions which define my research.

Those who have thought about the place of proving in mathematics education, and the place of mathematics in education, have typically arrived at one of two contradictory positions.  Some argue that mathematics is the best context for the teaching of rational thought.  Others have pointed to the damage done by scientific and mathematical thinking, and have wondered whether we might not all be better off not knowing how to think rationally.  I believe that both positions ignore the feature which makes rational thought so far unique.  Of the various modes of thinking we have, rational thought is the only one which has demonstrated, within its own criteria, that it has limits.  Its strength is in identifying its weakness. 

Rationalism has been a horrible choice for the status it has been given as sole mode of correct thinking.  But forgetting how to think rationally is not the answer even if it were possible.  Rationalism was embraced with enthusiasm because of its power to make predictions about the natural world.  That power is worth something in itself, and even if we never helped our children to think rationally, some would discover how to and lead humanity down the same path again.  Instead we must try to make sure that we teach deductive reasoning well enough that its limits are understood.

In the past we have been unsuccessful in teaching deductive reasoning in mathematics classes.  At the root of this failure are two misconceptions about proof and proving.  When students ask why they must prove, the common answer is that proving verifies statements.  This answer neglects both the importance of other factors in convincing us of truth, and the importance of other uses of proving.  We 

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also expect proofs to be individual works, expressed in formal language, but this expectation ignores the vital role that social interaction plays in supporting proving and its formulation.  If we can teach in a way that acknowledges that importance of explaining and exploring as motivations to prove, and that creates social contexts that allow the development of a culture of proving, then we may find that our students prove and understand proving well enough to understand that other ways of thinking are sometimes better.

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David A Reid
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