JOHN MASON

ENABLING TEACHERS TO BE REAL TEACHERS: NECESSARYLEVELS OF AWARENESS AND STRUCTURE OF ATTENTION

ABSTRACT. Awareness is a complex concept comprising both conscious and unconsciouspowers and sensitivities which enable people to act freshly and creatively in the moment.In the case of mathematics teachers, and teachers of those becoming mathematics teachers,it is possible to lead students mechanically through a sequence of ritualised tasks by meansof trained and habitualised reactions. But the result is that even though student attentionis indeed directed, their behaviour trained, and their awareness educated to some extent,the students have not been taught in the fullest sense of that word. I argue that to be a realteacher involves the refinement and development of a complex of awarenesses on threelevels, and that this is manifested in alterations to the structure of attention.

The problematic nature of what students are attending to when a teacher is teachingthem, led to the conjecture that each technical term in mathematics and in mathematicseducation signals a shift in the structure of attention of people using that term, and thata corresponding shift is required for students to appreciate that term. Investigations ofattention led to the development of Gattegno’s very general but rather subtle notion ofawareness into a three-layer structure which applies both to mathematics and to teaching,and so demonstrates why becoming a teacher is such a complex matter.

INTRODUCTION

The mark of expert mathematicians is that they make problem solving andproof look easy: they are articulate with technical terms, they make thechoice and use of techniques look easy, and they are aware of connectionsbetween otherwise apparently disparate topics. The mark of expert mathe-matics teachers is that they make exposition, explanation, task-design, andrelating to students look easy. They not only have mathematical expertise,but they are also aware of the nature of that expertise, and they have waysof re-entering the state of not having that expertise. The mark of expertmathematics teacher educators is that they not only have teaching expertisefounded on mathematical expertise, but they are also aware of the natureof that teaching expertise and have ways of re-entering what it is like notto have it.

In all three cases, the expert differs from the novice in the form andstructure of their attention: what they attend to, and what they have access

Journal of Mathematics Teacher Education1: 243–267, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

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to. In these terms, the role of a teacher is to create conditions in whichstudents experience a corresponding shift in the structure of their atten-tion, in which they become aware of acts and facts of which they werepreviously unaware. Thus the key notions underlying real teaching are thestructure of attention and the nature of awareness. These are the notionsbeing developed in this paper.

Shulman (1987, p. 6) identified a taxonomy of seven types of teacherknowledge:

• content knowledge,• general pedagogical knowledge,• curriculum knowledge (of materials),• pedagogical content knowledge;• knowledge of learners and their characteristics,• knowledge of educational contexts (sociology of groups, institutional

functioning, character of communities and cultures); and• knowledge of educational ends, purposes and values, and their philo-

sophical and historical grounds.

This list is daunting in the extreme, and the many inter-connectionsbetween types mean that the taxonomy is rather unstable in practice. Thetrouble with such a list is that it comes across as factual knowledge,knowing-that, and as rigid and discrete. Indeed it has been interpreted thisway by many different authors with only a few notable exceptions (forexample, Elbaz, 1983). By thinking in terms of awareness, it is possibleto see how teaching is itself a path of personal development. Individuals,supported by colleagues, have ideal conditions to work on their own aware-ness, to re-enter and re-vivify both mathematical and pedagogic shiftsof attention which they themselves have achieved, in order to provideconditions for their students to experience them too. Teaching is funda-mentally about attention, producing shifts in the locus, focus, and structureof attention, and these can be enhanced for others by working on one’s ownawareness.

Whereas Ryle (1949) distinguished between knowing-that, knowing-how, and knowing-about (having a story to account for), Skemp (1969)indicated that there is alsoknowing-to act. Someone can be a walkingstorehouse of knowledge but not have that knowledge come to mind in themidst of some event. For example, most people have had the experienceof failing to remember something when it was needed! Whitehead (1932,pp. 1–2) called such knowledgeinert and suggested that it was the coreproblem of education. But interrogation of experience suggests that thereare varying degrees of inertia at various times depending on context, howrecently the knowledge has been employed, and so on. Knowing-to is not

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just knowing-when which has a predominate sense of academic knowledge(writing essays about it) without necessarily the practical knowledge in themoment. Knowing-to is the kind of knowledge which enables people to actfreshly and creatively. Folklore often mocks at academic knowledge whichlacks a practical base (Shah, 1978), and politicians and media punditsare likewise more taken by practical knowledge than by understanding.The Discipline of Noticing (Mason & Davis, 1989a, 1989b; Davis, 1990;Mason, 1991, 1996) was developed in an attempt to be explicit about thevarious ways people have transformed knowledge about into knowing-to.

In order to teach effectively, it is necessary to appreciate the conductof the discipline, however intuitively, and also to be aware (howeversubliminally) of the process of conducting the discipline. Without thelatter awareness, you can be an excellent mathematician but you arelikely to be a frustrating and frustrated teacher, since only those who canfollow in your wake can learn effectively in your presence. Without theformer awareness, you can at best train others in the techniques you havemastered yourself, applied to the problems you yourself can do. You mayon the surface be very effective, for your pupils may also solve the routinetasks for which they are trained. They may spot and prepare for standardquestions on an examination. But training in routines is like walking onone foot: unstable hopping. Merely being educated in ideas is equallyunsatisfactory, as it is only hopping on the other foot. Real possibilitiesemerge when awareness is educated and behaviour is trained in concert.And this is done by harnessing emotions. All this is summarised in

Only awareness is educable;

Only behaviour is trainable;

Only emotion is harnessable.

I call statements like theseprotases,because aprotasisis the first state-ment of a syllogism, and with details of your experience providing thesecond term, the syllogism action produces a conclusion inside you, andthis action can transform awareness. The first assertion is from Gattegno(1987); the other two are derived from one interpretation amongst severalof an image from various Upanishads (Mason, 1994a). Teaching is thenseen as a process of directing students in the harnessing of their emotionsto provide the energy both to train their behaviour and to educate theirawareness. This paper concentrates on the educating of awareness throughwork on attention.

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ATTENTION

Most teachers have had the experience of asking students a questionand getting an apparently totally nonsensical reply. They have also hadthe experience of trying to guide a student who apparently does not seesomething obvious such as a common factor, a rearrangement, or a simpli-fication, and finding it very hard to believe that the student does not see it.These are examples of students not seeing what the teacher sees. Althoughnot seeing is an obvious and common phenomenon, it is often obscuredbehind apparently taken-as-shared discourse.

One way to expose not-seeing is to create a supportive and conjectur-ing atmosphere in which it is normal to expose confusion and uncertaintyrather than to hide it. For example, conducting public readings of complexmathematical passages can reveal, by the ways in which the studentstresses and stumbles, what the student is aware of and what not. Robotic-Instruction-Following by a teacher, in which the teacher does exactly whata student says with intentional misinterpretations, can force the studentto describe succinctly and precisely the steps in a technique. A teacher’sresponsibility can be described in these terms as attracting students tobecome aware of, to stress (and consequently ignore), the way the rela-tive expert does. This is perhaps what enculturation into a community ofmathematicians really means.

When young children are learning English as their first language, andstart to construct past tenses of irregular verbs as if they were regular, theyreach a stage where no amount of correcting seems to make any difference;then suddenly they construct almost all of them correctly. It is as if they areattending to something else, and store away the information for when theyare ready. There are similarities with phase transitions in physics: some-times putting energy in produces no rise in temperature, then suddenly thestate changes and the temperature rises. So too, sometimes expert attentiondirected to a particular aspect seems to be ignored by students, but laterthey show that they have integrated it into their functioning.

Some confusions arise from momentary metonymies in the formof idiosyncratically triggered associations which divert attention onto adifferent plane. Classic examples are listed each year after examinations,and there are many examples in the folklore: the child who refused to sitat an odd (numbered) table because she didn’t want to be odd; interpretingvolume as a knob on a radio; even solving 3� = 36 with 6 through treating3� positionally rather than as a multiplication.

Many confusions arise not from momentary slips of attention butthrough students stressing differently from experts. For example, a travelbook stated that “there could be sudden drops of temperature by as much

LEVELS OF AWARENESS AND STRUCTURE OF ATTENTION 247

as 10◦C (50 ◦F)”: Someone apparently converted 10◦C into Fahrenheit,unaware that it was a difference, not a temperature, being discussed andthat difference makes a difference. Stress placed on the “10◦C” blockedout the presence ofdifference, which was either seen but ignored asperipheral, or not seen at all.

Students often cannot see detail because they are blinded by the whole.For example, a student may not be able to see

√5−√

3√5+√

3as the ratio of the

difference to the sum of two quantities, because the square-roots, through amixture of cognitive uncertainty, affective disharmony, and enactive lack ofcompetence, block them seeing a global pattern. The presence of a compo-nent which is not confidently manipulable (the square roots) produces aholistic seeing of the entire expression as a painful or frightening entity.Instead, a student needs to be able to see like an expert, to whom both

√5

and√

3 are merely blobs standing for numbers.Similarly, novice teachers are usually much more concerned about

exposing themselves in front of a group of people they do not alreadyknow than about philosophical issues at the heart of teaching. They arecaught up in issues of control of themselves and control of others. Sensi-tive teacher-educators are aware of what they are stressing, even thoughthe best way to address it may not be explicitly or head-on, but rather bybuilding up the novices’ personal confidence. Many pre-service teachersarrive in mathematics education courses having been convinced that theyare mathematical failures, so that an important early task is to rebuildthat confidence. Again, addressing it head-on is not necessarily maximallyeffective. But certainly it is worth being aware that novice attention islikely to be dominated by the initial traumas of teaching and may notalways be focused on the issue of current concern to the educator.

Fundamental Questions

Reflections such as these have led me to a fundamental question ofteaching:

What are students attending to?

and the related question for teacher education:

How can we enable teachers to see what students are attendingto?

I cannot expect a definitive answer to either question, nor even expect tokeep the questions at the front of my attention all the time. But I find that

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they provide a touchstone, a place to return to when something unusualhappens. The more aware I can be of what students or teachers see, ofwhat they are stressing and ignoring, the more help I can be in offeringdifferent foci for their attention. Since I have found it generally moreproductive to examine my own experience before trying to get othersto tell me about theirs, contemplating the question of what students areattending to led me very quickly to a parallel question:

What am I attending to?

If I am unaware of what I am attending to when I am expounding, explain-ing, guiding, or questioning, unaware of what I am stressing and ignoringinside me in my inner awareness, then there is quite likely to be a mismatchbetween my internal and my external stressing. And if I am outwardlystressing differently to my inner stressing, or not stressing at all because itis for me so routine, students are going to find it difficult to see what I amseeing, in the way that I am seeing it.

For example, suppose I am going through an example on the board.I might be writing down ten’s complements, adding fractions, factoring aquadratic, graphing a rational polynomial, finding the equation of a straightline or other curve, finding an area, etc. I am aware that I am applying ageneral technique, and that the particular numbers involved do not mattersince they are merely illustrative. But are the students similarly aware?How are they to know whether particular numbers, or perhaps relation-ships between the numbers, are important? For example, in addressing thequestion of how to divide£500 in the ratio of 2 to 3, is it important thatthere are only two numbers? That the 2 and the 3 add to 5 and it is£500 tobe divided? How are students to know that the context is important only inindicating the type of question – for example, that a maximum or minimummust be found – or the type of answer sought – for example, that the answermust be positive, or integral, as seems to be the case for Cardano (1545) –unless it is stressed explicitly? How are they to appreciate something as anexample if they do not already know what it is exemplifying?

Suppose I am planning a sequence of tasks, and I am tempted to choosesome specific numbers, perhaps as examples for me to work through,perhaps as exercises for them. It might look like good practice to selectcarefully, and to start from the particular. But if I always do the select-ing, if I always do the specialising from the generality in my awareness,then the people I am working with will not be called upon to use theirown powers of specialising. This applies to students learning mathematicsand to teachers working on their mathematics or working on teaching.

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Over a period of time the power to specialise is likely to atrophy, orat least to be excluded from the classroom as surplus to requirementsbecause it is not called upon. This is likely to disenfranchise students andteachers, to weaken teachers’ mathematical ability to interact with studentsin meaningful ways when unfamiliar generalities arise, and to weaken theirchances of recognising the general teaching principles being exemplified.It follows that it is important for teachers to encounter unfamiliar gener-alities themselves so that they can develop their own ability to specialise,both in mathematics and in mathematics teaching, thereby enhancing theirability to know-to employ strategies with their students.

The purpose of doing specific cases is to get a sense of generality.But if students and teachers are never explicitly invited to vary, extend,re-contextualise or generalise a question or a result, to try to articulatewhat a general question of a given type would look like, they may notappreciate the generality, may even be blissfully unaware of the existenceof an encompassing generality.

The didactic contract (Brousseau, 1984) provides an explanatory theoryfor the tension in which students and teacher are inevitably caught.Students undertake set tasks on the understanding that this will somehowproduce the required learning. They are led naturally to invest a minimumof energy and attention needed to complete the outer requirements of settasks, and never contact the inner aspects (Tahta, 1980, 1981; Mason,1993a, 1994b). The teacher is then caught in the didactic tension (OpenUniversity, 1988): “The more explicitly the teacher indicates the behavioursought, the easier it is for the students to display that behaviour, withoutgenerating it through understanding” (p. 14).

Unless their powers of generalising and specialising, of stressing andignoring, of presenting and representing, of imagining and expressing, areexplicitly provoked, students may never appreciate that those powers arerelevant to mathematics. They may form the impression that mathematicsis a smorgasbord collection of unrelated terms, facts, techniques, andproblems. For example, many students seem blissfully unaware that bothfractions and decimals are names for numbers, and that you can switchfrom one representation to another. Segmentation of mathematical topics istypical of students who have not been provoked into forming connectionsfor themselves.

If it is of assistance to teachers to work on their sensitivities, to extendtheir awareness of what students might be attending to by being explicitlyaware of what they are attending to, then it behoves teacher educators towork with their teacher-students on those sensitivities. To be consistent,teacher-educators similarly need to work on their own awareness in order

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to be in a position to support others in becoming aware of their awarenessin turn.

Structure of Attention

Structure of attentionencompasses the locus, focus, and form of attentionmoment by moment. For example, you can be fully focused on a task,such as working on a mathematical problem or expounding or explainingto students, to the extent that you are oblivious to things happening aroundyou. When working on a problem you can be unaware of someone cominginto the room; in a lesson you can forget concerns about your health orabout the latest government statements about teachers. These illustrate asingle focus and a single locus.

You can also, when working on a problem, be simultaneously awarethat you have different strategies you could be using, or that you havea main goal but that you are working on a subgoal; in a lesson you canbe simultaneously aware that you are concerned about the way certainstudents are behaving, and that you need to explain something beforegetting them to do the next task. These are examples of multiple foci, andwe shift rapidly between multiple and single foci.

You can also be aware that you will shortly have to cook dinner orunload the washing machine, and that you are not getting anywhere onyour current task, yet that awareness can be sufficiently below the surfacethat you cannot actually act to do something about being stuck; in alesson you can be simultaneously aware that you need to see your head ofdepartment about something while interacting with a child. These illustratemultiple loci, because it is as if you are simultaneously or in rapid succes-sion present in different places, living in different worlds. People often findwhen preparing a workshop that they are aware of a multitude of possibil-ities and connections, yet when they get into the session, most of thoseconnections seem to evaporate. If this awareness rises sufficiently near thesurface it can be acted upon to improve preparation for sessions. This illus-trates a difference in form between on the one hand a principal focus witha diffuse sublayer of potential connections, and a single tunnel-visionedfocus. You can gaze at a scene and have thoughts drift through your head,and you can attend to specific details, as in

√5−√

3√5+√

3. This illustrates different

forms of attention, from diffuse to focused.Thus, attention can be focused or diffuse, it can be unitary or split, it

can be centered on a single domain or else either flit between or simulta-neously locate you in different worlds; it can be multiply connected withtriggers waiting to operate, or it can be tunnel-visioned (Mason & Davis,1988).

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It is very hard to define attention, precisely because, in some veryessential sense, we are our attention and we are where our attention is.Since you cannot be aware of what you are currently not aware of, thereis an all-encompassing aspect to attention. And yet there is somethingmissing in this description. It is possible to develop an inner witness whoobserves but does not comment, who extends the structure of attention.Such an observer is independent of attention, but a component of aware-ness. Schoenfeld (1985) called it theexecutive, and Mason, Burton, andStacey (1984) called it aninner monitor.

There is a mathematical analogy which helps to distinguish betweenan endless sequence of observers in which one acts, one observes oneselfacting, one observes oneself observing, one observes oneself observingoneself observing, . . . , and the presence of an inner witness. The endlesssequence of ordinals 1, 2, 3, . . . ,each just beyond the previous, corre-sponds to the sequence of observers, and the ordinal of the set of finiteordinals,ω0, corresponds to the witness.

Where does the evidence for assertions about the structure and natureof attention come from? It comes from my own research “from the inside”(Mason, 1994b), augmented by reading of philosophical, psychological,and English literature. It is made possible by the growth of an innerwitness, a part of me which can be separate from the action and which canobserve without comment. It is something like the internal mathematicalmonitor sitting metaphorically on my shoulder who suddenly wakes upand puts into my mind questions like “Why I am doing this particularcalculation?”, “Is it supposed to be this complicated?”, and “Might I havemade an error?”. Developing an error-checking monitor is an importantpart of the meta-cognition involved in becoming an expert. Your monitoris witness to your awareness, to the structure of your attention. It is notyour attention; you cannot switch focus and dwell in that perspective. It isco-present with your attention. It sleeps, it wakes, it develops over time, asa consequence of working on awareness.

It follows that if a teacher’s attention is differently structured from thatof the students, it is very likely that students’ interpretation of what theteacher is saying and doing, and what the teacher is asking them to do, willbe very different from the teacher’s. Consequently, teachers need morethan just knowledge in its traditional sense. They need an awareness ofbeing educated, that is, awareness that attention is structured, awarenessof the structure of their own awareness, and awareness of what they arestressing and ignoring while speaking to students.

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The Shifts Conjecture

Alerted by the problematicity of what students are attending to, and withwhat structure, it is then logical to ask what the difference is betweenexperts’ and novices’ attention when a technical term is used. Technicalterms emerge or are employed when people try to express succinctly,precisely, and manipulably, something that they are seeing or sensing, inother words, when they adumbrate or assert a theorem, sinceseeingis theGreek root meaning oftheorem. Thus each technical term marks a partic-ular way of seeing. Furthermore, the termshift in the structure of attentionself-referently provides a label to crystallise or pin down an as-yet-unstableawareness (Mason & Davis, 1988). It follows that a corresponding shift ofattention must be re-experienced by each generation of students trying tomake sense of that term, and this is the shifts conjecture.

For example, if you use a term such asangle, you assume that otherpeople have the same or similar meaning brought to mind to that which youare aware of when you use it. It signals a potential awareness as a turning,and an ignoring of the lengths of the arms which specify its beginning andits end. Yet students naturally begin by associatinganglewith an awarenessof a point, of arms, and of space. Most attend at first to the lengths of thearms as an indication of the size of the angle (Balacheff, 1987; Griffin& Gates, 1989). In order to appreciate the termangleand to use it withfacility, they have to undergo the shift of attention which it signals.

Some shifts are like entering a brand new world. The mathematicalnotion of a function involves a complete change of perspective. Suchcomplete changes are often marked by people who suddenly see the newidea everywhere, and want to impose this view on everyone else. “Func-tion” is a case in point! Some shifts, indeed perhaps all shifts in somesense, involve a move to multiplicity, as in seeing an arithmetical or alge-braic expression as both a calculation to be performed and as the answerto such a calculation. This double-perspective can be a real wrench forstudents desirous of unique interpretations. In the case offunction, it isvaluable sometimes not to stress the functional view, but to be able to seethings as a more general relationship, or as a different kind of connection.Wittgenstein (Kenny, 1994) said that “By being educated in a technique,we are also educated to have a way of looking at the matter which is just asfirmly rooted as that technique” (p. 239). Put another way, when you havea hammer in your hand, the world divides itself into nails and not-nails.

Each person’s concept image (Tall & Vinner, 1981), procept (Gray &Tall, 1994), reification (Sfard, 1991), and encapsulation (Piaget, 1977;Dubinsky & Levin, 1986) has both idiosyncratic aspects and more-or-less shared aspects. The idiosyncratic aspects are the personal connections

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set up through metonymic triggers below the level of consciousness dueto personal propensities and past experiences, and through metaphoricresonances induced by both frozen and active metaphors embedded inthe discourse. The more-or-less-shared aspects arise through continuedinteraction with others both in language and in actions, so that similaractions and similar utterances follow similar stimuli amongst members of acommunity of shared practice. Where these things are discussed explicitlyit is possible to have a negotiation of meaning, not in the sense of finding acompromise, but in testing out each other’s uses in different contexts andso seek agreement as to how a term is to be used.

Re-Entering a Shift of Attention

Once a shift of attention has taken place, it can be very difficult to recallwhat it was like not to see things that way, not to have that flexibility,not to be aware of connections. How frustrating it is (on both sides) whensome people are aware of multiplicity and others seek or desire simplicity:educators and politicians are perhaps doomed to be forever divided on thisdimension. It can be profoundly irritating when someone suddenly seesthings freshly, makes a shift of attention, and imposes it on everyone else.

It is particularly difficult to recall a pre-shift world when the shift takesplace below the level of consciousness or without much struggle. Thusteachers who themselves have struggled with topics are sometimes morehelpful to their students; but not always. I suggest that when their ownexperience enables them to be aware of the components and constituentsof that shift, then they can be helpful, but where their experience is domi-nated by their own struggle, and particularly by affective dimensions ofthat struggle, which can block out flexibility and connections, they are aslikely to be unhelpful as helpful to students.

Moshovits-Hadar (1988) argued that it is both possible and desirableto re-awaken the novelty, the surprise, which is present in many technicalterms. Isn’t it quite extraordinary that the angle subtended in a circle onone side of a chord is constant or that the difference of two squares alwaysfactors? If a teacher cannot re-enter a semblance of the state of fascina-tion of wonder which accompanies a discovery, then students are likely toconclude that there is no place for personal creativity in mathematics. Inthe case of factoring, one approach is to become aware of the use offactorwith its roots in the wordfactory as a label for a place providing a factor(a warehouse) with goods to distribute (literally, to factor), for it not onlyprovides links to experience which can enrich meaning, but offers entryinto the way that originators of the term might have been thinking. Thetrouble is that merely saying it to students does not recreate the insight,

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the release of energy, the shift of attention in them; at best it offers only apossibility.

Furthermore, the process of generalising, of seeing how a number ofpreviously disparate items can be subsumed under one generality, is theoccasion for a sense of pleasure, of expansion of awareness, of release ofenergy. That too can be re-experienced freshly on each encounter, as longas the expert places him- or herself in the state of being open to, rather thandismissive of, re-constructing that generality.

The shifts conjecture is something that cannot be proved universally assuch, but rather has to be verified by each individual. It provides a poten-tially useful way of thinking, indeed, it self-referently offers indicationof a shift of attention. Technical terms could become triggers to becomeaware of shifts associated with them (Mason, 1996). This involves takingeach particular term, locating what associations the term triggers, whatimagery and confusions, what particular discourses and techniques, whatproblems and what contexts (Griffin & Gates, 1989). This is the structureof a concept-image, an encapsulation, a reification. The shifts conjecture isvalid in a particular instance if such enquiry provides access to your ownawareness of what it means to understand the term.

A concept image can be verified or augmented against what variousauthors offer, including, ultimately, scholarly research into the historicalorigins of the term. The insights afforded by this contemplation providea framework for “Preparing To Teach a Topic” (Griffin & Gates, 1989;Mason 1993b).

AWARENESS

Being aware is a state in which attention is directed to whatever it is thatone is aware of. However, this idiom is used to refer both to being explicitlyaware and to being potentially aware. Thus when driving a car we areaware of what is happening around us, yet it is when something changesthat we become explicitly aware. Before that the awareness was potentialrather than actual. One of the key features of learning mathematics andof learning to teach mathematics seen as a social activity is that whensomeone else points something out, our being aware alters subtly. It israther as if there are different degrees of attention, and someone else’sremark can alter those degrees slightly, so that we become more explicitlyaware of some features, and less aware of others.

Gattegno (1970, 1987) used the termawarenessto refer to that whichenables powers that have been integrated into one’s functioning to beemployed. Once integrated into our functioning, we are usually no longer

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aware of exercising those powers, nor of the awarenesses which enablethose powers to be employed. To distinguish Gattegno’s use of aware-ness, I shall refer toawareness-in-action, following a theme introducedby Vergnaud (1981) when talking about children displaying a theorem-in-action even though they were not explicitly aware of it.

The shifts conjecture, and observations about the structure of attention,provide entry into the language of awareness as used by Gattegno both tocapture what is meant by the notion of mathematics, say, as a discipline,and to indicate what is required in order to be a teacher in the full senseof that word. In the next section I elaborate on my interpretation of hisuse of the term (try to enter the shift of attention which it signals), andsuggest that to become an expert it is necessary to develop and articulateawareness of your awarenesses-in-action; to become a teacher in the fulland most appropriate sense of that word, it is necessary to become awareof your awareness of those awarenesses-in-action.

Three Forms of Awareness

Gattegno (1987) suggested that each discipline emerges through peoplebecoming aware of the subtle and often subconscious awarenesses whichenable them to act in certain ways. Although the termawarenessis oftenused as a synonym for consciousness (Crick, 1994), Gattegno used theterm in a much more encompassing and subtle way.

I can become aware that my eyes move or, more precisely, that I can act on some of myocular muscles to move my eyes from left to right or the other way around and that thatmovement can be made very gradual and slow, and through generating such actions I canbecome aware of my will as it commands my eyeballs to move. Therefore I am not onlywith my eyes, I am with my will as well. (Gattegno, 1987, pp. 38–39)

Moving the eyeballs is an example of what I shall call anawareness-in-action, because it is an awareness, albeit below the surface, which makescertain actions possible; being aware of that movement is a further form ofawareness, and being aware of the will which moves those eyeballs is yeta further form of awareness.

Continuing the ocular context, if there is a sudden movement in periph-eral vision, our attention is attracted. There is some part of us which attendsto movement in peripheral vision of which we are unconscious. Since it isessential to our functioning, it seems sensible to include it as an awarenesswhich lies below the surface of consciousness as an awareness-in-action.

Although peripheral vision is physiological, it provides a vibrantanalogy for the structure of attention more generally. Thus although youare at one moment focused on a specific calculation, you have periph-eral attention of various kinds (patterns previously met, connections with

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other topics, resonance with previous experiences which can be activatedwith more or less ease). It is precisely these awarenesses which a teacherwants students to bring closer to the surface of consciousness, in orderto educate and develop them into more substantial powers, on the way tobeing integrated into their functioning.

The sections which follow identify and develop three forms ofawareness, namely

awareness-in-action(the powers of construal and of acting inthe material world);

awareness of awareness-in-action, orawareness-in-discipline,which enables articulation and formalisation of awarenesses-in-action, and is closely linked to one form of shift of attention;

awareness of awareness-in-discipline orawareness-in-counsel,which is the self-awareness required in order to be sensitive towhat others require in order to build their own awarenesses-in-action and -in-discipline.

Awareness-in-action.Some of the things we do, like responding to move-ment in peripheral vision, are consequences of our genetic structure.Others, like walking, talking, reading, and counting, are learned, but oncemastered, we do them with very little explicit attention. We have integratedthe functioning, subordinating it in attention so that we can attend to thepurposes and goals of activity, to the overall direction, and not to detailsof the acts themselves. Imagine two people walking: one attends to whereeach foot will go, the other strides ahead with his or her eyes on the distanthills. Both are extremes. In any particular situation attention is on overalldirection until the going becomes uncertain, at which point some attentionmay be diverted to support the details of walking.

Amount and degree of attention is highly personal and context specific.For some people, adding fractions always requires considerable attention;for others, usually very little. For some people, manipulating algebraicsymbols always requires considerable attention, while for others it usuallyrequires very little. Awarenesses-in-action are the subconscious foci of ourattention (whether directed inwardly or outwardly) which enable thoseactions to be performed. By their nature they are largely implicit. Theyonly become explicit when we become specifically aware of them due tothe context.

The term awareness-in-actiondraws metonymically on Vergnaud’snotion of theorem-in-action(Vergnaud, 1981; Binns & Mason, 1993) andencompasses a wide range of experiences. For example:

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I can count without being aware of one-to-one correspondence; I can add, subtract,multiply, and divide, without being explicitly aware of my awarenesses-in-action ofnumerals, place-value, routines, the role of order, etc., which make that arithmetic possible.I can form and detect patterns and locate formulae which generalise specific cases withoutbeing explicitly aware of my awarenesses-in-action of same and different, relatedness,induction, stressing and ignoring, that make generalisation possible.I can engage in resolving triangles and calculating trigonometric ratios without beingexplicitly aware of my awarenesses-in-action of invariance expressed by similarity, equiv-alence expressed by congruence, and trigonometric functions as forms of angle measure-ment, that lie at the heart of such activity.I can combine fractions according to rules, without being aware of how fractions relate todecimals and to integers, and how they generalise number, without being aware of the slidebetween operator and object, without relating them to a number-line. However, I can’t getfar without these underlying awarenesses-in-action becoming explicit, as is evidenced bythe ongoing and universal struggle teachers have to teach fractions to children.

Take, for example, counting as an act. It uses the counting poem “one,two three, . . . ” and callsupon the power of making distinctions so asto identify distinct entities, the power of simultaneous acts of attention(uttering, pointing, attending) which comprise the notion of one-to-onecorrespondence. But that only happens when you become aware of yourawarenesses-in-action which constitute what others call counting. Throughsubordination of the act of counting, or as Gray & Tall (1994) would haveit, through merging process and object asprocept, attention is freed to beplaced on the count, and through undertaking many counts, to numerals asthe product of a count.

The basis of learned awarenesses-in-action are powers of construalwhich everyone who walks and talks has displayed in profusion:

• the power to select, distinguish, demarcate, discern, detect difference;• the power to see (construct) similarity, commonality, genericity;• the power to see (construct) something as an example of something

else, and hence to generalise;• the power to abstract (going beyond generalisation by asserting that

certain properties shared by many examples are to be the definingproperties, leading to axioms and rules);

• the power to imagine, both the past and the future, to manipulate thoseimages, combining, ordering, classifying, and juxtaposing them;

• the power to connect, link, associate, which has linguistic, visual, andaural roots;

• the power to express some aspects of inner experience;• the power to fit in with and adopt-adapt social practices for our own

use;• the power to decide, including to decide not to decide; and so on.

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These are fundamental awarenesses-in-action, for they are the basis of allactions. They are also the basis for mathematics, and indeed other disci-plines, when they are applied in certain ways and in certain domains.It is the identification of those ways which constitutes a discipline.Awarenesses-in-action are the sensitivities to certain situations whichprovoke and enable action.

The behaviours to which awarenesses-in-action contribute can to someextent be trained without explicit reference to awareness, for it is notnecessary to be explicitly aware as long as situations remain routine anddo not require innovation, novel interpretation, or creativity. Short termsuccess can indeed be achieved in getting children through assessmenthurdles, but education is about more than training of behaviour in routineactions on symbols on paper, and teaching is about more than carryingout sequences of instruction. We want students to be able to re-construct,to modify, to adapt, and above all, to know-to act when it might beappropriate. Teaching is about inducting children into disciplined formsof thinking and perceiving, and these emerge when awarenesses-in-actionare brought into explicit awareness and formalised. Teaching people to beteachers involves inductingthem into disciplined forms of thinking andperceiving pedagogically as well as mathematically in a classroom.

Awareness-in-discipline.Gattegno (1987) suggested that a discipline ariseswhen people become aware of the awarenesses which enable them tofunction. In becoming aware, they distinguish and label both actions andawarenesses, enabling them to study the properties and uses of thoselabeled actions and awarenesses-in-action. “Sciences are born when some-one states that what occupies his mind IS, and, because of that, is part ofreality and worth being considered by others” (Gattegno, 1987, p. 21).

Mathematics arises as a discipline when we become aware ofawarenesses-in-action such as those that constitute counting, ordering,classifying, and relating, and start to formalise these in the languagesof algebra and geometry. Gattegno suggested that algebra emerges whenwe become aware of our awarenesses-in-action involved in the dynamicsof relationships, and geometry through awareness of the awarenesses-in-action involved in the dynamics of our minds (of which mental imageryis a major part). Thus when we attend to how we order, how we classify(the notion of equivalence, of same and different), how we count (throughone-to-one correspondence), and formalise these in general statementswe produce algebra, and, simultaneously, the rules of algebra. When weattend to the powers of mental imagery to conceive mentally of points,lines, curves, surfaces, and movements and inter-relations between them,

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when we formalise them into assertions of what must, might, and maynot happen (Mason, 1989), and justify these assertions, we develop thediscipline of geometry.

The term discipline is important, for it includes not only encounteringfacts and techniques, but also habits of thought, forms of fruitful ques-tions, and methods of resolution of those questions. However unpopularthe termdiscipline is in current discourse, progress in meaning-makingand creativity both require constraint and structure in order to succeed.As creative people in the arts know very well, creativity is a responseto constraint: the sonnet and the sonata form release creativity throughconstraint, in a way that free form does not. Discipline is an essential aspectof thinking, and this is why I refer to awareness of awarenesses-in-actionas awareness-in-discipline.

Gattegno’s approach also provides insight into inter-relationshipsbetween geometry and algebra: where relationships and their dynamics areinvolved, you really have algebra; where the dynamics of mental imageryis involved, you have geometry. Thus, as Tahta (1989) remarked, “Thegeometry which can be spoken is in fact algebra” (p. 20).

Students asked to “do the exercises” are likely to devote much oftheir attention to each exercise in turn. They workthrough the tasks, noton them. Often they display no recognition of commonality, of type, orof general method, this despite authors and teachers believing that it isthrough doing plenty of exercises that students practice to mastery, thatthey integrate technique into a skill to be used in other situations, thatthey shift from awarenesses-in-action to awarenesses-in-discipline. Sets ofroutine exercises actually attract attention to the doing and away from theconstruing, strengthen awarenesses-in-action to the detriment of awarenessof those awarenesses, despite 3000 years of pedagogical practice of settingstrings of exercises for students to carry out.

In order to force attention away from the doing, sets of rehearsal exer-cises have to be done against the clock, in a time which seems impossibleif full attention is given to each step. Since this sort of practice hasfallen out of favour and is inappropriate in most Western cultures, othermethods have to found. For example, tasks can be set which provokestudents into rehearsing or exercising skills, but which at the same timeattract their attention away from the skill to be automated. As Hewitt(1994) showed, such tasks are much more likely to support studentsin integrating techniques into their repertoire of skills. Primary teachershave done this for years, inventing all sorts of games whose executiondemands a variety of important skills. Hewitt developed Gattegno’s notionof integration through subordination and showed how it can become a prin-

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ciple of economy with application to many different aspects of teachingmathematics, including the design of software and of teaching gambits.

Awareness of awarenesses-in-action, when recognised, enables peopleto act upon those awarenesses, to denote and manipulate them, to separatethemselves from them. Awarenesses-in-discipline are the sensitivitieswhich enable us to be distanced from the doing sufficiently to instructothers, to give orders, literally, for doing things. The ability to becomerelatively expert in mathematics requires awareness-in-discipline. Theability to be enabled as a mathematics teacher requires even awareness ofthis awareness, so that he/she can make choices about which aspects ofhis/her awareness of the structure of his/her attention to stress explicitlywith students.

Awareness-in-counsel.Awareness-in-discipline is what constitutes thepractice of an expert. But as many people have discovered, it is only whenyou come to teach something that you really come to understand it. Thefact of an audience produces a relationship with ideas, skills, and purposes,which often transforms my appreciation of the topic. You become aware ofawarenesses-in-discipline, and this is what supports effective teaching ofthat discipline. It is not sufficient to be expert as a creative researcher in thediscipline, to be a good teacher. Although some people can learn a greatdeal from being in the presence of an expert who is not a good teacher,most people require the presence of awareness of awareness-in-disciplinefor structuring tasks and encounters from which they can learn.

Dewey (1902) distinguished between the expert immersed in advancinga subject, and the teacher who is concerned with how “the subject mattermay become a part of experience” (p. 22). He called this thepsychologisingof the subject matter: “to see it is to psychologise it” (p. 22), and thisformed the basis for Shulman’s categories listed earlier. Dewey goes on tosay

It is the failure to keep in mind the double aspect of subject-matter which causes thecurriculum and the child to be set over against each other.. . . The subject-matter, just as itis for the scientist, has no direct relationship to the child’s experience. We are practicallythreatened on all sides. Textbook and teacher vie with each other in presenting to the childthe subject matter as it stands to the specialist. (p. 22)

In responding, the specialist is inevitably sucked into thetransposi-tion didactiqueidentified by Chevellard (1985) and used by Kang andKilpatrick (1992), in which expert awareness (subject matter as it standsto the specialist) is transposed into instruction in behaviour (as construedby the student). Dewey, and every real teacher and author before and since,constantly struggles against this degeneration. Gattegno actually addressed

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the transposition. He spoke in rather different, but I believe correspondingterms (I repeat the first paragraph quoted above and add a second):

I can become aware that my eyes move or, more precisely, that I can act on some of myocular muscles to move my eyes from left to right or the other way around and that thatmovement can be made very gradual and slow, and through generating such actions I canbecome aware of my will as it commands my eyeballs to move. Therefore I am not onlywith my eyes, I am with my will as well.

Since I am aware of my will while it moves my eyes slowly from left to right, can Ileave my eyes altogether and be aware of my will as such? For that I have to acknowledgethat within my awareness my self can simultaneously find present a sensation of my eyes,of the movement, of slowness, of direction, of my will, and of its presence in the eyes –and of additional observations if they appear. Then, I can observe that I can ignore some ofthese components and only be aware of the presence of the will in the act of stressing andignoring some contents. In focusing my awareness on my will, awareness of awareness [ofawarenesses-in-action] has been achieved. (Gattegno, 1987, pp. 38–39)

I interpret this as offering a route through and beyond thetranspositiondidactique, by working to support students in educating their awareness.Awareness of awarenesses-in-discipline provides access to sensitivitieswhich enable us to be distanced from the act of directing the actionsof others, in order to provoke them into becoming aware of their ownawarenesses-in-action and awarenesses-in-discipline. I have called this,awareness-in-counsel, drawing on the sense of a counsel as a sourceof collective wisdom, separated from immediate action and transcendingspecific disciplines.

Bennett (1978, pp. 126–127) used the language of energies to describesimilar distinctions to those I am describing as awareness, but which gowell beyond what I am trying to locate and express here. He referredto automatic energy, which corresponds to awareness-in-action, beingessentially mechanical, subordinated, and unconscious. Drawing on olderpsychological traditions, Bennett stressed that much of what we thinkof as conscious is in fact more akin to sleep, being automatic, habitual,reactive, and not actually within our direct control in the moment. Whenwe awaken from this sleep, we become aware of awareness, throughdrawing uponsensitiveenergies. These energies are what make possibleawareness-in-discipline, and so response rather than reaction, choice ratherthan habit. Awareness of awareness of awareness-in-action draws uponconsciousenergy which, without the construction of a means of holdingand employing it, is transitory and fleeting.

Although knowledge of mathematics is important to a teacher, as isknowledge of the psychology of individual others and knowledge of thesociology of groups of others and of communities in which mathematicalpractices arise or are used, it is essential to develop awareness of aware-ness of awareness. It is this three-leveled awareness which distinguishes

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the teacher from the novice, from the expert practitioner of the discipline,and from the trainer. In his comprehensive review of teacher knowledge,Shulman (1986) reminds us that the original meaning of higher degrees(masters and doctorate) was recognition of the ability to teach the subject,not just prosecute it as an apprentice (as a bachelor). He quotes Aristotlein a translation by Wheelwright (1951):

We regard master-craftsmen as superior not merely because they have a grasp of theory andknow the reasons for acting as they do. Broadly speaking what distinguishes the man whoknows from the ignorant man is the ability to teach, and this is why we hold that art andnot experience has the character of genuine knowledge (episteme) – namely that artists canteach and others (i.e. those who have not acquired an art by study but have merely pickedup some skill empirically) cannot (p. 69).

Awareness of Teaching

Adults in the presence of young children often drop into a parent-childor teacherly mode of interaction such astelling in an instructing toneof voice, andasking questions to which they themselves clearly knowthe answer (Ainley, 1987). These enculturated awarenesses-in-action are,naturally, present in teachers, alongside basic powers, and in company withawarenesses at all levels concerning the discipline such as mathematics.

A teacher with active awareness-in-discipline and awareness-in-counsel in mathematics can inspire students, but they may not be ableto support colleagues in working on their teaching. Teachers function-ing on automatic pilot without awareness of the awarenesses-in-actionto do with teaching itself can at best exchange materials and good ideas(awarenesses-in-action) with colleagues. Without being aware of theirteacherly awarenesses, their gambits and principles, their ideals and prac-tices, they are in no position to be sensitive to the needs of colleagues.They can at best reproduce themselves. Short term success can be achievedin getting teachers through in-service events on reform topics like inves-tigations, co-operative learning, structured lesson plans, constructivism,planning, assessment, etc., but teaching is about more than carrying outsequences of instruction, and teaching teachers is about more than handingout tips and purveying practices.

Each school needs teachers who are more than excellent teachers, whodo more than dwell in their awarenesses-in-action of teaching. The forcesand pressures from outside the institution, together with social forces aris-ing from collectivity, and psychological forces arising from individualresponses to pressures, all act to increase habituation and automaticity,and to decrease awareness-in-discipline and awareness-in-counsel of bothsubject matter and teaching. In order to develop and sustain vibrant aware-

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ness which will support an ethos in which student and teacher awarenessescan thrive, the institution needs to maintain a community of active enquiryand personal and professional development, what Schön (1983) called acommunity of reflective practitioners.

Just as teaching a discipline such as mathematics requires awareness-in-discipline and awareness-in-counsel in the subject matter, so inducting newteachers into the profession and providing on-going support for profes-sional development requires more than just awareness-in-action regardingteaching. It depends on both awareness-in-discipline in order to generate adiscipline of mathematics education, and awareness-in-counsel in order tosupport teachers in their personal and professional development.

Awareness of teacherly awarenesses-in-action generates the disciplineof what some call pedagogy and others call didactics. When focusedon subject awarenesses, such as those associated with mathematics, adiscipline of mathematics education emerges. Gambits and principlesconcerned with evoking and provoking mathematical awarenesses areidentified and labeled so as to become objects of study, refinement andmodification and for promulgation. The form and method of promulgationcan itself be studied. For just as students do not necessarily get muchfrom simply being told, or from simply exploring, so teachers and would-be teachers do not necessarily get much from being told or exposed togambits and principles. There are both psychological and socio-culturalissues. How is it that someone comes to identify a possibility, whether agambit or a principle, a strategy or an ideal, and how might that possi-bility come to be manifested in their practice? In other words, how dothey become aware of their teacherly awarenesses-in-action in conjunctionwith their subject matter awarenesses-in-discipline, and in-counsel? Howcan an ethos be fostered and sustained in which colleagues support eachother in practitioner enquiry (e.g., Gates, 1989)? Awarenesses-in-counselwith respect to teaching are required.

Without developing awarenesses-in-counsel with respect to teaching, itis difficult to get beyond cause-and-effect conceptions of the process ofteaching and learning, because they are subject to thetransposition didac-tique. As in the case of subject matter, awareness-in-counsel with respectto teaching is the requisite separation required to respond to student-teacher-colleague needs rather than their desires, informed by the wisdomwhich comes from intentional reflection and awareness of awareness ofawareness in both subject matter and teaching. Work on them generates adiscipline, or science, of education, whose pursuit Gattegno urged on allof his audiences.

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SUMMARY

Starting from the pragmatic question of what students are attending to atany particular moment, I have been led, via the corresponding questionof what I myself am attending to, to locating shifts in the locus, focus,and structure of attention as being at the heart of educational success,and signaled by fluent and appropriate use of technical terms. I illustratedthese in terms of the arising of mathematics as a discipline, and suggestedthat there are parallel forms of awareness to do with teaching itself. Theresult is a multi-layered structure of awareness, whose complexity suggestswhy it is that politicians find teacher-education so frustrating (it requiresmore than training in procedural skills), why pre-service teachers find it somystifying (learning to teach is as much about them as it is about children),and why an integrated pre-service and in-service programme of profes-sional development is essential (for without it, all the forces push towardshabituated automaticity and absence of requisite awareness). Thus teachereducation has a three-fold ongoing task:

• working with teachers to help them educate their awareness of theirown, and hence of their children’s awareness-in-action;

• working with teachers to educate their awareness-in-discipline sothey are sensitised mathematically to work with their students in amathematically informed and appropriate fashion; and

• working with colleagues to educate their own awareness-in-counsel.

As we become more aware of awarenesses-in-action, thus develop-ing the discipline of mathematics, and more aware of the discipline ofmathematics itself, a discipline of mathematics teaching emerges. Thisis the discipline that constitutes a possible foundation for teacher educa-tion and offers substance for those who would enable teachers to becomereal teachers. Mathematical thinking lies at the heart of teacher-pupildiscourse, but can only be conducted effectively if it is informed bypersonal awareness-in-discipline and in-counsel. Awarenesses-in-action todo with the psychology of working with another person, and with thesociology of working with groups of others, interact with awarenesses-in-action and awarenesses-in-discipline of mathematics, to produce acollection of awarenesses-in-action for teaching both in schools and inteacher education programs. Awareness of these generates a discipline ofmathematics teaching, and awareness of these in turn generates the disci-pline of mathematics education, including the teaching of others to teachmathematics.

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