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Susan Pirie and Lyndon MartinUniversity ofBritish olumbia

The Role ollecting in the rowth Mathematical Understanding

Folding back is one of the key components of the Pirie-Kieren Dynamical Theoryfor the Growth of Mathematical Understanding. his paper looks at one aspect offolding back, that of collecting. Collecting occurs when students know what isneeded to solve a problem, and yet their understanding s not sufficient for theautomatic recall of useable knowledge. They need to recollect some inner layerunderstanding and consolidate it through use at an outer layer in the l ight of their

. now more sophisticated understanding of the concept in question. The collectingphenomenon is described and distinguished through exemplars of classroomdiscourse, and implications for teachers and learners are discussed.

i Vol. 2 No.2 127-146athematics ducation Research Journal

n recent years, there has been much interest in exploring the nature ofmathematical understanding. For examples, see the work of Bergeron andHerscovics 1989), Byers and Herscovics 1977), Cobb Yackel, and Wood 1992),Gray and Tall 1994), Hiebert and Carpenter 1992), Sfard 1991), Sierpinska 1990,1994). Skemp 1976), Tall 1978), and Walkerdine 1988). A full review of theseviews of mathematical understanding is presented in Martin 1999). This paper isconcerned exclusively with a theory advanced over the past ten years or so bySusan Pirie and TomKieren.

The Pirie-Kieren TheoryThe Pirie-Kieren Dynamical Theory for the Growth of Mathematical

Understanding differs from other views of mathematical understanding in that itcharacterises growth as a whole, dynamic, levelled but non-linear, transcendentlyrecursive process Pirie Kieren, 1991a, p. 1). This theory is compatible with theconstructivist view outlined by Von Glasersfeld 1987), according to whichindividuals must reflect on and reorganise their own personal constructs in orderto build up new conceptual structures. However, the Pirie-Kieren theory viewsunderstanding as something different from an internalised and mental process inwhich a static notion is acquired and then applied. Instead, understanding ischaracterised as occurring in action and not as a product resulting from suchactions. In particular, mathematical interactions with others and the environment)co determine that is, fully determine and are determined by the mathematicalunderstanding actions of the individual participants.

The notion of recursion embedded in the definition is fundamental to the PirieKieren view of the growth of mathematical understanding. This term is used tosuggest that understanding can be observed as complex yet levelled and that eachlevel is in some way defined in terms of itself self-referenced, self-similar), yeteach level is not the same as the previous level level-stepping) Pirie Kieren,1989, p. 8). In developing this idea of mathematical understanding as a recursive


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process, Pirie and Kieren were influenced by the work of Maturana and Tomm(1986); Tomm (personal communication, 1989); and Maturana and Varela (1987),who see knowing [as] exhibited by effective actions as these are determined by anobserver and the human knower of mathematics as self-referencing and selfmaintaining in a particular niche ofbehavioural possibilities (p. 8).The Pirie-Kieren theory provides a way of considering understanding whichrecognises and emphasises the interdependence of all the participants in anenvironment. It shares the enactivist view of learning and understanding as aninteractive process. The location of understanding in the realm of interactionrather than subjective interpretation and a recognition that understandings areenacted in our moment-to-moment, setting-to-setting movement (Davis, 1996, p.200) allows and requires the discussion of understanding not as a state to beachieved but as a dynamic and continuously unfolding phenomenon. Hence, tbecomes appropriate not to talk about unders tanding as such, but about theprocess of coming to understand and about the ways that mathematicalunderstanding shifts, develops, and grows as a learner moves within the world.Enactivism recognises the Piagetian or radical constructivist view that what alearner learns is determined by his or her individual structure, and acknowledgesthe work of Von Glasersfeld (1987) in moving away from a definition ofunderstanding as an acquisition to that of a continuing process of organising andre-organising. However, enactivism departs from constructivism in thatunderstanding is seen not only as subjective and individually unique but also assomething that can be shared through interaction. Developing the enactivist notionof cognition as an adequate functioning in an ever-changing, interactive world,understandings are seen to be not merely dynamic but also relationally,contextually, and temporally specific and thus, as one moves away from aparticular situation, one s understandings, as revealed in one s actions, may changedramatically. And so, while understandings might be shared during moments ofinteractive unity, they inevitably diverge as the part icipants come back to theirselves (Davis, 1996, p. 200).Enactivism's other major departure from constructivism is a move toacknowledge the actions of the learner and to see understanding in terms ofeffective actions. This notion of effective actions also allows for both formulatedand unformulated understandings. A learner who cannot state or verbalise theirunderstanding may still exhibit understanding through their actions. Davis (1996)talks of these as a part of our acting in the world-an acting that understands thedifference between a single or a pair of raised fingers before it can count, an actingthat understands that a sequence of two perpendicular cuts produces four piecesbefore it realizes the process is multiplicative. These are understandings that areactions of the body s doing (p. 201). Within the Pirie-Kieren theory, cognition andunderstanding are more than merely a process of reflective abstraction on mentalobjectifications of experiences. Instead, experiences and actions such as Davisdescribes themselves form part of the understanding and are enfolded andenclosed within the more formal. In the Pirie-Kieren theory, growth inunderstanding is seen as a dynamic and active process involving the building ofand acting in a mathematical world.

t is important to briefly consider the nature , purpose, and use of the Pirie-

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igure Diagrammatic representation of some features of the Pirie-Kieren theory.The first level of understanding to be built on this foundation is that which is

termed Image Making. This is the level at which learners work at tasks, mental orphysical, that are intended to foster some initial or extended conceptions for thetopic to be explored. In the case of fractions, Image Making activities wouldperhaps lead to the learner saying, Ah, fractions are what you get when you cutthings up . At this point , the theory would claim that the learner has an image (isacting within the Image Having level) for fractions-although one would hope thatthis would later become refined to the image that fractions involve thecutting ofitems in equal pieces .The above illustration of a possible path of understanding from PrimitiveKnowing through Image Making to Image Having is not meant to indicate that thegrowth of understanding moves smootWy outwards through the layers. Wecontend that growth in understanding takes place through a continual movementback and forth through the layers of knowing, as individuals reflect on andreconstruct their current knowledge. The metaphor of recursion higWights the factthat the dynamical understanding notions of a person involve states which differ incharacter but are self-similar (Kieren Pirie, 1991). A person s currentunderstanding action in some way acts to elaborate previous states and integratesthem in the sense that they are called into current knowing actions.For a more complete description of the model, see Pirie and Kieren (1994).

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olding ackA key feature of the Pirie-Kieren theory is the idea that a person functioning at

an outer level of understanding will invocatively return to an inner level. The wordinvocative (Kieren Pirie, 1992) is used to describe a cognitive shift to an inner level


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ollecting is the purpose of this paper to distinguish what we see to be a particularly

important form of folding backwhich we call collecting The process of folding backto collect entails ret rieving previous knowledge for a specific purpose and reviewing or reading it anew in light of the needs of current mathematical actions.Thus collecting is not simply an act of recall; it has the thickening effect of foldingback. In what follows we give examples of collecting, distinguish it from otherforms of understanding actions, and discuss how teachers might act to occasion

of understanding and an invocative intervention is one which promotes such ashift. An invocative shift is termed folding back when the person makes use ofcurrent outer layer knowing to inform inner understanding acts, which in turnenable further outer layer understanding Pirie Kieren, 1991b).

When faced with a problem1 that is not immediately solvable at any level, anindividual needs to return to an inner layer of understanding. The result of thisfolding back is that the individual is able to extend their current inadequate andincomplete understanding by reflecting on and then reorganising their earlierconstructs for the concept-or even to generate and create new images shouldtheir exist ing constructs be insufficient to work on the problem. However theperson now possesses a degree of self-awareness about his or her understandinginformed by the operations at the higher level. Thus, the inner layer activity cannotbe identical to that originally performed and the person is effectively building athicker understanding at the inner layer to support and extend their understandingat the outer layer that they subsequently return to. is the fact that the outer layerunderstandings are available to support and inform the inner layer actions whichgives rise to the metaphor of folding and thickening.

Although a learner may well fold back and act in a less formal, more specificway the inner layer actions are not identical to those performed previously.Folding back can be visualised as the folding of a sheet of paper in which a thickerpiece is created through the action of folding one part of the sheet onto the other.The learner has a different set of structures, a changed and changingunderstanding of the concept, and this extended understanding acts to informsubsequent inner layer actions. Folding back, then, is a metaphor for one of theprocesses of actions through which understanding is observed to grow andthrough which the learner builds and acts in an ever-changing mathematicalworld. Folding back accounts for and legit imates a return to local ised andunformulated actions and understandings in response to and as a cause of thischanging world. The Pirie-Kieren theory suggests that folding back is an intrinsicand necessary part of the process by which understanding grows and develops.

The RQle ollecting in the rowth ofMathematical nderstanding

1 We use the words problem and solve frequently in this article, but at no point are werefering to the limited, specific activity that has come to be called problem solving in themathematics education literature. To enable people to solve problems throughout theirlives, not just those contrived problems set in mathematics lessons, is the reason that allchildren are taught mathematics. For us, problem solving is simply working mathematicallywhen the route to the solution is not direct and immediately clear.


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Jasmin has no difficulty in solving th e question at all. She has th e necessaryunderstanding instantly accessible and the process she uses is essentiallyautomatic. There is no necessity for her to fold back.

John cannot deal with this probl em a t all. It is not clear whether he has met aquestion like this before but cannot now solve it, or whether subtraction questionsof this type ar e new to him. What is clear, however, is that either he doe s n ot h av eth e necessary understanding or that hi s understanding is not well enough

such folding b ack t o collect.Of particular significance in th e data relating to folding back is the occurrence

of a number of cases where, following a shift by th e learner to an inner layer ofunderstanding, there h as b ee n neither an y observable learning activity i n the senseof any visible reorganisation or reconstruction of existing constructs nor anygeneration o f who ll y new understandings. Instead of working on existing ideas,t he i nn er l ay er activity h as b ee n m or e a process of finding and collecting an earlierconstruct or understanding and then consciously using or re-reading as useful ina new situation.Before we tum to actual classroom dialogue, we as k you to consider th e threeexamples of students tackling t he q uest io n 93 - 47 ? shown in Figure 2. Thevignettes ar e based on classroom events and have been deliberately constructed toclearly illustrate and deliberately differentiate various ways of thinking about th esame problem.

Pirie Martin

So, three take seven, can t do pause nine becomes eight, thirteentake seven i s si x pause an d eight tak e fou r is four, gives forty six.

igur 2 Three classroom vignettes.

Three take seven, can t do pause no, yo u ca n do something to th enine an d t he t hree and borrow or tens it or something, lemme look.He opens his workbook and flicks through it. Yeah, that s it, make th enine an eight pause b orr ow te n so we ge t thirteen take seven is six.Now t he o th er b it is eight take four is four, forty six.

Teacher:John:Teacher:

Hmm, th re e tak e se ve n pause Hang on, seven is bigger thanthree, I ca n t do it, if was seven tak e three it would be OK. Heputs his hand up and the teacher comes over. I can t do this cos s evenis bigger than t hr ee s o y ou c an t take it away.Could you do something to t he nine an d th e three?Hmm, no , I dunno, I can t do it.OK then, I ll get the rods and blocks and w e l l m ak e ninety threeand forty seven.

They then work with the Cuisenaire rods and use these solve the problem.

Vignette 1Jasmin:

Vignette 2John:

Vignette 3Paulo:

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Collecting in the ClassroomThe rest of this paper is c once rned with illustrating the phenomenon ofcollecting as it happens in the classroom. Our i nt en ti on is to show that even brieffragments of dialogue ar e sufficient to alert us to the shifts in thinking that takeplace. The examples ar e ch osen to demon st rat e some of the key features ofcollecting and to indicate the var ie d ways in which students carry out the process

and the various teacher actions which can facilitate it.In the first of the following tw o case studies students are seen successfullycollecting inner layer understanding and using this to continue working. In thesecond case the two students are initially less successfuL Their interaction providesa valu ab le insight into their ways of thinking as they struggle to find and collectwhat they know they need.

developed to allow h im to use it. Instead prompted by the teacher he f olds bac k toperform more image making either to build a new image or to enhance an existingone perhaps by working on his image for subtractions where the unit subtrahendis lar ge r than the unit minuend. John nee ds to do more mathematical work at aninner layer before he will be able to build for himself an algorithm to answer th equestion an algorithm that he can use with understanding.We see something very different in Paulo s thinking about this problem. He toocannot immediately solve the question-he does no t have th e understanding to us ean automated process in the way that Jasmin does. Neither however does he foldback in the same wa y as John to construct or modify an image. Paulo ha s an imageinvolving the reconceptualising of th e numbers that he believes will allow im tosolve the question ..but he nee ds t o fold back to th e level of Image Having in orderto retrieve this image to re-view its properties in terms of the specific task at handand then to use it. There is a sense of him having and being a wa re t ha t he has thenecessary understandings but that they are just no t immediately accessible. Thushe needs to fold back to his mo re basic understanding and in some way recollectthat is to say re-collect-it for us e in his current thinking.

tis important to not e t ha t the process of collecting is a m enta l one. Althoughhere it is accompanied by Paulo searching his workbook this is no t essential to theidea: tcan equally be per form ed sim pl y t hr ough the conscious searching of one sthoughts. The workbook here is an aide memoir it is not in itself hisunderstandings. Although initially it may appear that he ha s a lack ofunderstanding of subtraction this is not actually the case; he was not looking for ane w idea to help him. After successfully re-collecting the image he needs he is ableto correctly complete the question using his existing understanding of the concept.His language allows us to assume t ha t h e is no t blindly applying by rote copyinga given algorithm. He has recollected the understanding process which legitimateshi s subsequent algorithmic action of subtraction. The major difference betweenthis and the folding back of John is that the inner level activity of Paulo does notinvolve a modification of his e ar lier under standings. H is w or ki ng involves himinstead in finding and recalling what he knows h e n ee ds to solve the problem. Heis consciously aware that this knowledge exists. He collects his innerunderstanding and consolidates it through intentional use.

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ase 1: Rosemary and erryThe first extract is taken from a lesson with Year 9 students (about 14 yearsold). The students, Rosemary R and Kerry K are of average abil ity and have

been set the task of finding out the area of icing on a slice of a circular birthdaycake. The teacher has introduced the task by simply drawing a circle on the board,marking a sector, and asking the students to find the area of it. This transcript isfrom when the students beginworking.

R: There must be something on t in here. (Pause as she flicks through hertextbook.) I dunno (in doubtful tone), I m looking for the area section.

K (laughs) Area is page a hundred and thirteen. (She turns to this pageheaded Area ofa Triangle in her book.)

R: Got it.K There, it s half the base t imes the height.R: No, (pause) we need .... (pause) It s pi r squared isn t it and hrnm (Pauseas she looks through book again.) Here we are, look here we are, radius and

diameter so it s .. .it s [page] a hundred and twenty one. Circumferenceequals two pi r squared. No, no, no, that s wrong, two pi r. Then areaequals pi r squared.

K No, but we don t want ....R: So, which is three hundred. (She is working with the numbers given in the

book s example. She then returns to the teacher s diagram which has no givendimensions.) No, that s wrong.

K Let s cut a quarter and make easy ....R: Just to make it easy.Here the teacher has created a situation where the students are able to begin

working in whatever way and at whatever level is appropriate for them. BeforeRosemary begins to work at making an image for the sector of a circle, she foldsback to her primitive knowing, searching for something useful and applicable tothe problem. This shift appears to be self-invoked; that is to say, there has been nodeliberate, external intervention to cause her to decide to search her textbook-although obviously the question and therefore the teacher have contributed to thisoccurring. t may be that her history ofworking with this teacher encourages her inthe belief that she does possess the elements within her primitive knowing that willsupport her growth of understanding.Initially Rosemary believes that, unlike Jasmin in Figure 2, she cannotimmediately tackle the problem that the teacher expects her to be able to solve,andshe clearly sees a need to fold back to her primitive knowing and to use previousunderstandings in this new topic. She seems, however, unsure which aspects of herprimitive knowing to actually draw upon and her thinking is unfocused in itsnature. After a pause, she tells Kerry that she is looking for the area section . Shehas decided that she needs to calculate the area of a circle. She finds this section inthe book, intending to search for the required formula, confident that she alreadyknows t and that having re-collected it she can return to image making for theproblem in hand. She expects to be able to use her primitive knowing to continueworking, n a similar manner to Paulo n Figure 2. n the later stage of the extract,we see that Rosemary does find the information she is searching for (bothinternally in terms of her own understanding and externally and physically in the

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Case 2: Simon and AnnThe remaining extracts are taken from a teaching session with tWo Year 12

students (about 17 years old), Simon S and Ann A . They are working oncalculus and within this topic, on the concept of differentiation from first.principles. In order to do this the teacher has invoked them to fold back to work onthe necessary primitive knowing, in this case on making an image for the conceptof limits. They have already answered a number of straightforward questions and3are now trying to find lim h ? Their initial attempt is simply to replace h by

~ h 2h

textbook). She collects the area of a circle formula, taking it back to the level ofimage making where she attempts to continue working. In fact, though, she findsthat she cannot immediately use her formalised rule to f ind a numerical answer,because the problem the teacher has posed gives no dimensions for the circle. Inthis, her collecting differs from that of Paulo in that, having collected, she still hasto determine how she will use her understanding. Nonetheless, her response toKerry s final statement, Just to make it easy , is evidence that she is now thinkingabout the question of finding the area of a sector. She is seeing it as a portion of thewhole circle-that is to say, she is constructing an image for the notion of sector aspart of a circle-and is intending to use her recollected understandings.

closer inspection, we see that the images Kerry and Rosemary initially formfor a sector are interestingly different from one another-at least partly because thetwo students draw upon different primitive knowing. Rosemary s mention of thearea section has a marked effect on the thinking of Kerry and, probably as aconsequence of this student intervention, Kerry too folds back to her primitiveknowing. However, her shift appears more intentional: She goes directly to theconcept of the area of a triangle. Later conversation with Rosemary reveals that,prior to folding back, she saw the sector as a triangular shape and attempted tocollect her understanding of triangular area in order to make it possible to workwith her image for the problem. She suggested working on a quarter circle to createan easy right-angled, isosceles triangle. But for her too, the formula she neededwas not immediately applicable.Both students collected inner understanding which they attempted to use toincrease outer understanding. But the knowledge and understandings theycollected resulted in differing images of a sector. Both girls folded back to collect onthe occasion of the given problem. Both then acted to reformulate previousunderstandings into an understanding of a sector. But their collecting led them todifferent understanding actions, just as their perceptions of the problem invokeddifferent collectings.

135he Role of ollecting in the Growth ofMathematical Understanding

zero.A: It s nought divided by nought... she writes 0 =.Q

O+2xO 0 Yeah,. but you re saying what s nought d iv ided by nought? Is it nothingor is it infinity? How many nothings in nothing? Is there none or is therean infinite number?


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The difficulty here has been caused by the fact that the h in the denominator ofthe rational expression leads to a division by zero. With their present image forfinding limits they are led to replace h by zero, and they are left with a situationthat they cannot solve. Their difficulty here has two aspects. Firstly, their existingunderstanding of limiting values is insufficient to allow them to solve the question;and secondly, they do not use the necessary algebraic primitive knowing to allowthem to modify their image making. Both students are seen folding back to theirprimitive knowing of arithmetic.The way in which the two students then work, though, differs. Simon appearsto be trying to retrieve a fact at this inner level: Nought divided by nought is eithernought or infinity, he has forgotten which. (This inference of ours is supported bylater dialogue between the students). From careful scrutiny of all the video data(and not relying solely on a transcript), we also infer that, in fact, Ann and Simonhave different images for zero. Ann calls zero nought and regards i t as a number,like any other number. Simon calls zero nothing (he says nought only whenrepeating Ann s comments) and regards it as emptiness . His face, gestures, andtone of voice all imply that he is looking to see how much emptiness is insomething empty. His image is, quite reasonably but probably unconsciously,influenced by the common language meaning of nothing. When Ann refers to thenumber two, Simon responds, Nothing s totally different . Although he can callhis image to mind and he attempts to state what he has recalled, he is unable toapply it.With Ann the situation is different. She folds back to an understanding of aproperty of division a 1) and moves out of the topic of limits to work withthis property. In fact, the actions of Ann suggest that her existing understanding isnot sufficiently developed or complete enough to allow her to collect i t and use itanew. Instead, she seems to have a need to work on her primitive knowing, to havea greater understanding of division, which she can then use in the new context ofnought divided by nought . Here, therefore, we do not see her actually engage inan act of collecting. Simon is aware of the inadequacy of her notion, but cannotoffer an alternative idea and both students are effectively unable to procee,d. At thisstage the teacher T intervenes.

T: Right, you can t actually give me an answer to it as it stands? In fact, canyou do something to that pointing to original expression in h ? I mean,what s the problem out of here is the zero on the bottom, isn t it? Cosyou don t know how to divide by zero, you don t know, as you say,howmany nothings there are in something, O Can you do something tothis the original expression ? Can you simplify that in some way? .

A: You can knock them off you see ... that s what we can do, can t we? Youcan do that, you can make it hover h plus ... argh, we ve got two h

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A:A:

A:

Pirie Martin

Is it one? Is there one nothing in nothing? .It s no t one, there s not one nothing in nothing...No, but if you go two by two pause over two, it s one:It s one. That s different though, nothing s nothing, nothing s totallydifferent.I suppose pause . It s n o t i n ~ or in inity or one, we haven t decideIt s no t one...


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squared, knock off one h squared. Get h, h plus h squared. She writes x x h th d hcrosses out x e numerator, an wrztes 2 .h 2 hx h h

S: Right, so h plus he writes as they workA h times h, so that s inaudible times h times h Knock off hmm ....S: See, I ve got h plus h squared over h, that s still not right . . ..A: It s exactly the same as it used to be, hmm. Well surely there s something

we can do with these, can t we? So it s still nothing divided bysomething, you divide i tby nothing, no it s nothing ....T: Are you actually happy with what was going on here pointing to

h x h x h ? Cos I wasn t quite clear what was going on.h 2 h xA OK, we ve got, on the top we ve got h times h times hT: Yeah.A On the bottom we ve got h plus h times h ... times h times h. She now has

written: h h + h x h ~T: OK, I m a bit unhappy about what s going on here. Why were you able

to cross that out with that pointing to the earlier writing ?A: Because tha t s what we did in maths a couple of days ago, what was it,

was it? Factorials? over minus factorial factorial something likethat.

The teacher here has recognised the problem the students are having andinitially validates this by saying cos you don t know how to divide by zero. Shemakes an intentionally invocative intervention to get the students to fold backagain to their primitive knowing, but this time the teacher is able to give theintervention a more explicit focus than the printed question had provided. Sheasks, Can you do something to this? Can you simplify that in some way? Thelanguage here is a prompt to particular algebraic techniques. The word simplifyseems to provide the invocative trigger for Ann and Simon. They fold back to theirprimi tive knowing and collect from this inner layer their method andunderstandings of algebraic manipulation, which they proceed to work with whilet ry ing to construct an image for the notion of limits. Unfortunately, it is evidentthat their algebraic understanding which they collect is either incomplete orinappropriate to the task at hand. Hence they will need further image making,including other re-collecting, in order to proceed.The teacher then makes two interventions which appear to have the aim ofprompting the students to do just this. She first asks, Are you actually happy withwhat was going on here? This intervention has the potential to be invocativealthough it is somewhat non-directional in nature, but it is not taken as invocativeand the students continue to work in the same way. The second question, againwith invocative intent, is more explicit: Why were you able to cross that out withthat? Had the students been able to answer this question, it would have acted as avalidating question. Where the students are incorrect, it is reasonable to suggest

The Role of Collecting in the Growth ofMathematical Understanding 137

The notion of validating is used here to describe an intervention which confirms the level ofunderstanding currently employed by the person.


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3 For obvious reasons, we do not have an y d at a o n her collecting acts when sh e is workingalone and therefore silently.

that the aim was to get them to change their understandings in some way.This time the teacher's intervention triggers something for Ann, who foldsback to he r primitive knowing, aiming to collect understanding that will providean explanation. The nature of the question asked by the teacher results in arelatively unfocused shift in thinking by Ann and sh e recalls some notion offactorials. She seems to be searching he r primitive knowing again for somethingt hat s he thinks will help her, although she does not appear to know why factorialsma y do this or even how she might apply them-she merely recalls a similar.situation in a previous lesson. She is unable to deliberately collect the requiredmathematics; instead, she first locates where it occurred for he r an d then utilisesthis flag to find the knowledge she needs. She justifies he r working by sayingBecause that s what we did in maths a couple of days ago and is thisreferencing that then seems to trigger the actual collecting of the notion offactorials, n over n minus r factorial factorial something like that .

The way in which Ann sets about finding what she wants to collect isespecially interesting. t is a common feature in her working, as is the fact that theactual mathematics recalled does no t initially seem to be particularly clear in herown thinking. Her typical pattern of action seems to be to reference her thinking bythe time when sh e worked on th e collectable concept and by the events in theclassroom that surrounded it. She needs to re-si tuate herself in her previousunderstanding activity. She relatively easily collects the mathematical label for aconcept-in this case, factorials-but she then frequently relies on Simon to supplythe actual piece of useable mathematics3. Indeed she takes on an almost teacherlyrole when calling on Simon t o supply or recall the actual mathematics, as thefollowing episode illustrates.

At one poi nt Simon and nn think that they have to deal with one divided bynought.

Which is ? expectant pause) You know Sime.S Do I? Yeah, you re bi g on these sorts of things. Come on.S Why am I big? Why am I bi g on these s or ts of things?A: C os we were doing it th e other day.S W er e we?A Proving God and all that. Or whatever you were doing.S That was more t ha n t wo years ago

Yeah, well.S That was th e Hitch-Hiker s Guide the Galaxy. It's infinite.Once Simon ha s identified the mathematics, Ann is usually able to recollect it

and apply i t with understanding. This, and many similar examples, suggests thatknowledge of the need for collecting plays a frequent and vital role in Ann sgrowth of understanding. She seems to have developed he r ow n internal labellingand referencing strategies to allow he r to make the collecting process easier andmore efficient. She h as c re ate d and uses a tw o stage mental referencing system,

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is probably referring to how they were introduced to integration by drawingrectangles to approximate the area under a curve. The diagram has the appearance of a setof stairs.5 The textbook from which they have been working.6 She is probably trying to recall the expansion of x h n.

which links events to mathematical labels to mathematical understanding.Later in the same session, the students return to differentiation and can be seenusing their primitive knowing from the concept of limits to make a new image fordifferentiation. They are working with the equation y x2 and using the notion ofthe difference in y divided by difference in x for a small increment At theh

Ann realises that she cannot immediately expand the expression and she foldsback to her primitive knowing. Although the shift is again primarily caused by thematerial and the question, there is a definite element of Ann also choosing to foldback herself. She is aware tha t somewhere in her primitive knowing she has thenecessary techniques and that she needs to fold back and recall or re-collect them.The shift itself is unfocused-she seems to be combing her primitive knowing to tryand find what she knows that may help. There is a very real sense of her trying tofind some appropriate understanding. She is not folding back to develop theseideas but to pull them out and collect them ready to use in the new situation.Once again Ann s referencing strategy is seen in operation. She says, we justdid it, we did on, hmm, on those stairs Mathematical Methods and This isthe great long thing we had an exam on . She has not only labelled her thinking bytime and event but also by a physical reference point; and she uses this to try andfacilitate the collection of the attached mathematics. Despite an apparently welldeveloped strategy, however, Ann is still unable to precisely or fully collect whatshe needs. Atypically, although Simon knows she wants to multiply out thebrackets, he fails to come to her aid. She continues to struggle to recall the relevantknowledge, with comments such as It s the great big thing and we se t them all outin class and I said No miss, that s wrong, I did that wrong and I s it the one withthe Pascal s triangle?

Throughout the process of hunting through primitive knowing, Ann is not

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and are tryingh

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7 The General Certificate of Secondary Education examination, which the students sateighteen months previously.

actually talking directly about the mathematics that she is trying to fold back to butrather about the events of the lesson that surrounded that work and the physicallocation of it. This referencing and collecting is again taking place in two stages.Ann first describes the events of the lesson, then collects and states the labelPascal's triangle . When, after searching her memory she is still unable to findexactly what she knows she needs, she tries, as Rosemary, Kerry, and Paulo did inthe earlier examples, to actually physically locate where she may find what sheknows she needs. She begins looking in her bag and her file for her notes from thelesson to help her collect what she knows she needs. Although she can access.thegeneral topic area she is aware that she is unable to precisely collect what isrequired and calls on Simon s assistance, saying Have you got your book withyou? and later You've got your folder, you might have your book and lateragain review sheet it might help you sic to work it out .

Throughout the rest of this episode, there is a sense of Ann moving in and outof her primitive knowing as she retrieves more small pieces of mathematics to helpher. She seems to be t rying to reconstruct a jigsaw puzzle of her understandingspiece by piece-in effect repeatedly saying to herself i f I knew such-and-such Icould probably act with understanding -until the whole is ready to be used.

What Ann is gradually collecting is a relevant, i f overly sophisticated, piece ofprimitive knowing-namely, the expansion of binomial expressions. Final ly theteacher intervenes.T: Right, I know what you re going for, and when you find it you d be

right. But you re going an awfully difficult way round.S: Yeah, complicated.T: What you re going for is useful if you want x plus h to the eleventh,

rather a waste of time if you want x plus squared. What does x plus squared mean?

S: It means x plus A x plus h, argh This is GCSE7 very excited and laughing). So x squared plus

xh, plus hx plus h squared ....S and So its x squared plus two xh plus h squared ....A triumphantly) There you go.T: Now, do you remember why youwanted that?Even s the teacher speaks, nn and Simon return immediately to the sheet o paper onwhich they had been working.The first intervention of the teacher here is in response to the difficulties the

students are having. Her comment is explicit and intentional, she wants them tonow fold back further, to their specific formalised understanding for quadratics.The effect on Ann is qui te startling, she suddenly folds even further back to theimage she has for the expansion of brackets term by term, and is able to quicklyand easily collect it. Again, though, she firstly describes it in terms of the labelattached to i t n event, in this case GeSE . Having collected the image from theinner layer, she is able to move back out with it to their image making fordifferentiation from first principles and immediately and successfully use it to

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expand the expression. The final comment of the teacher is important as it isdesigned to validate that the students are in fact working back at the image makinglevel and using their algebraic understandings in the context of differentiationrather than merely expanding a quadratic without purpose. The end result here iseventually one of successful folding back collecting and using earlierunderstanding although it was not achieved without a struggle

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Limits

Algebra

Other Understanding Differentiationigure Ann s path of growth of understanding of differentiation.


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Ann s cognitive path through the various levels of understanding is illustratedin Figure 3 This diagram appears complex and hard to interpret, and we haveincluded it here for this very reason. tgives a good illustration of the difficulty offinding ways to represent dynamic understanding, itself an extremely complexprocess. the past ,many readers have taken the eight nested circles toosimplistically and as literally the model of the theory . In fact, they are merelyoffered as a visual aid to our humanly inadequate, verbal description of one set ofthe features of the enactive process of coming to understand. Those who, however,can start at the point marked a in the Differentiation circles diagram and follow theline as i t weaves back and forth, in and out of the understanding diagrams forLimits, Arithmetic, Algebra and Other-all of which form part of the PrimitiveKnowing for differentiation-back to the point b in the Differentiation diagram areaided in their understanding by the visual representation. The zigzag line is whatwe call Ann s path of growth of mathematical understanding during the lessondescribed above.

Implications for Learners and TeachersThe examples discussed above have been selected to illustrate students foldingback not to a reconstructive inner level activity, but to select and read anew for

current use knowledge and understandings which they did not have available inalgorithmic or definitional form. We call such folding back actions collecting. As isobvious from the examples, the usefulness of collecting in on-going understandingis dependent on what is collected and how it is read into the new situation.So what then are the implications of this for teachers and their students? twould certainly appear that students differ widely in their ability and their methodof collection. Most of the s tudents we have studied have, like Jasmin in Figure 2some of their inner layer understandings formalised into an instantly accessible

and automated process. Indeed this is likely to be the case for many students inmuch of their mathematics. For example, students working on advanced calculuswould not be expected to stop, fold back, and re-collect their earlierunderstandings every time they need to multiply or add two numbers together.Instead, such facilities have become an unconscious tool, used when requiredwithout thought as to their origin or meaning.

However, as the examples show, collecting from an inner layer is a vital part ofthe growth of understanding and we need to provide the students with the abilityto facilitate this growth. Paulo, Rosemary, and Ann are all very aware that theypossess the understanding and the knowledge they need to be able to continueworking, and this is the key feature of the phenomenon of collecting. What we areinterrogating is the interaction between the person and the problem at hand. Themathematics they are working on is not in itself problematic, they do not have alack of understanding which inhibits their attempting to work on the new problem(as John did in Figure 2), nor, we believe, do they need to fold back to enhance theirexisting knowledge. Instead, for these students the initial difficulty lies in theirbeing unable to automatically access an earlier understanding.For Rosemary, overcoming this difficulty is a matter of retrieving the requiredformula from the textbook. Thus, the folding back is no more than a momentary

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shift in her cognitive state which empowers her to continue to develop her outerlevel understanding. Nonetheless, the importance of this shift in state, thiscollecting, should not be minimised. Even her collecting involves recollectingprevious mathematical experiences and understanding, and creating acontemporary use for them in a current act of understanding. This act obviouslyshapes the understanding being currently built (the idea of a sector of a circle) butalso reshapes her previous understanding of circles.With Ann, the situation is more complex. She shares the awareness ofRosemary that she has the understanding she needs, but for her the process offinding and collecting this understanding is problematic. The internal two stagefiling system she has created for her mathematical concepts certainly helps her todo this, and at times the system is highly successful-although she relies quiteheavily on Simon to provide the precise mathematics. Ann s invoked collecting ofbinomial manipulation shapes her ability to understand limits of functions, but italso clearly changes and adds to her understanding of the binomial form of thequadratic function. In contrast, Simon appears better able to fold back and ~ o l l tmathematical understanding, most particularly when prompted by Ann sinvocative location system. Simon does not appear to search his existingunderstandings in the way that Ann does, but waits for her to provide him with aprompt. Ann and Simon both have a good understanding of what they are doingand what they need to do to be able to continue. Their problem lies in an inabilityto effectively find, collect, and use the earlier concepts-the primitive knowingthat they need. would, of course, have been interesting to have followed thesestudents working with different partners, but unfortunately this was not possible.

How widespread is this technique of Ann s? Could teachers help students todevelop such reference systems? Certainly an important action when initiallyworking on any problem is to ask oneself Have I seen a problem like this before?Although it might initially be hard for teachers to do this (since the mathematicsthat they teach is likely to be, for them, at least at the formalising stage of automaticrecall), a conscious, frequent, overt modelling of collecting when workingexamples in front of students would demonstrate the need and the power of suchmathematical activity.The examples in this paper also clearly illustrate the fact that studentsfrequently return to textbooks or written notes to find what they know they need.Although such texts do not constitute the understanding needed, they are certainlya valuable aid to the cognitive processes involved in collecting. Thus a teacher whopromotes writing about one s understanding, careful reading of texts, and studentdiscussion indirectly provides the ground for such collecting.This leads us to consider the effect and importance of teacher interventions inhelping students to fold back and collect when needed. The dilemma lies indeciding the extent and explicitness of the interaction. In the case of Ann it was thefinal invocative intervention of the teacher, explicitly directing her to fold back tothe precise piece of mathematics that she needed to collect, that enabled her to beable to do this and then continue working. How specific should one be? How soonshould one intervene? ,Clearly, neither question can be answered in the form ofgeneral advice and the teacher s motives for the students undertaking particularmathematical activities need to be considered in every case individually.

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situations. Educational Studies in Mathematics, 23, 99-122.Davis, B 1996 . Teaching mathematics: Toward a sound alternative. New York, NY: Garland.Gray, E M., Tall, D. O. 1994 . Duality, ambiguity and flexibility: A proceptual view ofsimple arithmetic. Journal for Research n Mathematics Education, 25, 83-94.Hiebert, J., Carpenter, T P 1992 . Learning and teaching with understanding. In D A.Grouws Ed. , Handbook o research on mathematics teaching and learning pp. 65-97 . NewYork, NY: Macmillan.Kieren, T E., Pirie, S E B 1991 . Recursion and the mathematical experience. In L PSteffe Ed. , Epistemological foundations o mathematical experience pp. 78-101 . New York,NY: Springer-Verlag.Kieren, T E., Pirie, S E B 1992 . The answer determines the question-Interventionsand the growth of mathematical understanding. In W. Geeslin K Graham Eds. ,Proceedings o the 16th annual meet ing o the International Group for the Psychology oMathematics Education Vol. 2, pp . 1-8 . Durham, NH: Program Committee.Martin, L. C 1999 . The nature o the folding back phenomenon within the Pirie-Kieren Theory forthe growth o mathematical understanding and the associated implications for teachers andlearners o mathematics. Unpublished doctoral dissertation, University of Oxford, Oxford,England.

Maturana, H. R, Tomm, K 1986 . Languaging and the emotion flow. Paper presented at aconference of the Department of Psychiatry, University of Calgary, Alberta, Canada.Maturana, H. R, Varela, F J 1987 . The tree of knowledge: The biological roots of human

understanding. Boston, MA: Shambhala.Pirie, S E B Martin, L., Kieren, T E 1996 . Folding back to collect: Knowing you knowwhat you need to know. In L. Puig A. Gutierrez Eds. , Proceedings o the 20th annualconference o the International Group for the Psychology o Mathematics Education Vol. 4 pp.147-154 . Valencia, Spain: Program Committee.

Pirie, S E B Kieren, T E 1989 . A recursive theory of mathematical understanding. orthe Learning o Mathematics, 9 3 , 7-II.Pirie, S E B Kieren, T E 1991a, April . The characteristics o the growth o mathematicalunderstanding. Paper presented at the annual meet ing of the American EducationalResearch Association, Chicago, IL.Pirie, S E B Kieren, T E 1991b . Folding back: Dynamics in the growth ofmathematical understanding. In F. Furinghetti Ed. , Proceedings o the 15th annualmeeting o the International Group for the Psychology o Mathematics Education Vol. 3, pp.169-176 . Assisi, Italy: Program Committee.

Pirie, S E B Kieren, T E 1992 . Watching Sandy s understanding grow. Journal oMathematical Behavior, 11, 243-257.Pirie, S E B Kieren, T E 1994 . Growth in mathematical understanding: How can wecharacterise it and how can we represent it? Educational Studies in Mathematics, 26, 165190.Sfard, A. 1991 . nthe dual nature of mathematical conceptions: Reflections on processesand objects as different sides of the same coin. Educational Studies in Mathematics, 22, 136.Sierpinska, A. 1990 . Some remarks on understanding in mathematics. or the Learning oMathematics, 10 3 ,24-36.

Sierpinska, A. 1994 . Understanding in mathematics. London: Falmer.Skemp, R R 1976 . Relational understanding and instrumentaltmderstanding. MathematicsTeaching, 77, 20-26.Tall, D. 1978 . The dynamics of Ull:derstanding mathematics. Mathematics Teaching, 84,5052.Von Glasersfeld, E 1987 . Learning as a constructive activity. nC. Janvier Ed. , Problems o

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representation in the teaching and learning mathematics pp. 3-17 . Hillsdale NLawrence Erlbaum.

Walkerdine, V 1988 . The mastery re son London England: Routledge.

AuthorsSusan E Pirie, University of British Columbia, Department of Curriculum Studies 2125Main Mall, Vancouver, V6T 124 Canada E-mail: .Lyndon C. Martin University of British Columbia Department of Curriculum Studies 2125Main Mall, Vancouver, V6T 124, Canada E-mail: .

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