Mathematics Education Research Journal
2000i Vol. 12, No.2, 127-146
The Role of Collecting in the Growth of
Mathematical Understanding
Susan Pirie and Lyndon Martin
University of British Columbia
Folding back is one of the key components of the Pirie-Kieren Dynamical Theory
for the Growth of Mathematical Understanding. This paper looks at one aspect of
folding back, that of collecting. Collecting occurs when students know what is
needed to solve a problem, and yet their understanding is not sufficient for the
automatic recall of useable knowledge. They need to recollect some inner layer
understanding and consolidate it through use at an outer layer in the light of their
. now more sophisticated understanding of the concept in question. The collecting
phenomenon is described and distinguished through exemplars of classroom
discourse, and implications for teachers and learners are discussed.
In recent years, there has been much interest in exploring the nature of
mathematical understanding. For examples, see the work of Bergeron and
Herscovics (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 these
views of mathematical understanding is presented in Martin (1999). This paper is
concerned exclusively with a theory advanced over the past ten years or so by
Susan Pirie and Tom Kieren.
The Pirie-Kieren Theory
The Pirie-Kieren Dynamical Theory for the Growth of Mathematical
Understanding differs from other views of mathematical understanding in that it
characterises growth as a "whole, dynamic, levelled but non-linear, transcendently
recursive process" (Pirie & Kieren, 1991a, p. 1). This theory is compatible with the
constructivist view outlined by Von Glasersfeld (1987), according to which
individuals must reflect on and reorganise their own personal constructs in order
to build up new conceptual structures. However, the Pirie-Kieren theory views
understanding as something different from an internalised and mental process in
which a static notion is acquired and then applied. Instead, understanding is
characterised as occurring in action and not as a product resulting from such
actions. In particular, mathematical interactions (with others and the environment)
co-determine-that is, fully determine and are determined by-the mathematical
understanding 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 to
suggest that understanding can be observed as complex yet levelled and that "each
level is in some way defined in terms of itself (self-referenced, self-similar), yet
each 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 an
observer" and "the human knower of mathematics as self-referencing and self
maintaining in a particular niche of behavioural possibilities" (p. 8).
The Pirie-Kieren theory provides a way of considering understanding which
recognises and emphasises the interdependence of all the participants in an
environment. It shares the enactivist view of learning and understanding as an
interactive process. The location of understanding in the "realm of interaction
rather than subjective interpretation" and a recognition that "understandings are
enacted 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 be
achieved but as a dynamic and continuously unfolding phenomenon. Hence, it
becomes appropriate not to talk about understanding as such, but about the
process of coming to understand and about the ways that mathematical
understanding shifts, develops, and grows as a learner moves within the world.
Enactivism recognises the Piagetian or radical constructivist view that what a
learner learns is determined by his or her individual structure, and acknowledges
the work of Von Glasersfeld (1987) in moving away from a definition of
understanding as an acquisition to that of a continuing process of organising and
re-organising. However, enactivism departs from constructivism in that
understanding is seen not only as subjective and individually unique but also as
something that can be shared through interaction. Developing the enactivist notion
of 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 a
particular situation, one's understandings, as revealed in one's actions, may change
dramatically. And so, while understandings might be shared during moments of
interactive unity, they inevitably diverge as the participants come back to their
selves" (Davis, 1996, p. 200).
Enactivism's other major departure from constructivism is a move to
acknowledge the actions of the learner and to see understanding in terms of
effective actions. This notion of effective actions also allows for both formulated
and unformulated understandings. A learner who cannot state or verbalise their
understanding 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' the
difference between a single or a pair of raised fingers before it can count, an acting
that 'understands' that a sequence of two perpendicular cuts produces four pieces
before it realizes the process is multiplicative. These are understandings that are
actions of the body's doing" (p. 201). Within the Pirie-Kieren theory, cognition and
understanding are more than merely a process of reflective abstraction on mental
objectifications of experiences. Instead, experiences and actions such as Davis
describes themselves form part of the understanding and are enfolded and
enclosed within the more formal. In the Pirie-Kieren theory, growth in
understanding is seen as a dynamic and active process involving the building of
and acting in a mathematical world.
It is important to briefly consider the nature, purpose, and use of the Pirie-
The Role of Collecting in the Growth of Mathematical Understanding
129
Kieren theory. Von Glasersfeld (1995) talks of a "theoretical model" (p. 190) and
this is perhaps the most useful description of the Pirie-Kieren theory. It has grown
and evolved into a theory that can be used by a teacher or a researcher as a tool for
listening and observing in the context of mathematical activity. It offers a
theoretical way of looking at growing understanding as it is happening. It, is a
system by which an observer (a teacher or a researcher) can observe understanding
not in terms of a personal acquisition or an acquired state but as an on-going
process. (We prefer to use the word "knowledge" for static acquisition). Hence, it
allows a person to observe understanding in action and prompts the looking for
relationships between less and more formal understanding actions. It is a
theoretical thinking tool for a person who is observing mathematical
understanding and who might be interacting with students who are engaging in
understanding actions.
Levels for Understanding
The Pirie-Kieren theory contains, for a specific person and a specified topic,
eight potential levels for understanding. These are named Primitive Knowing,
Image Making, Image Having, Property Noticing, Formalising, Observing,
Structuring, and Inventising. A diagrammatic representation or model is provided
by the eight nested circles in Figure 1. Each layer contains all previous layers and is
included in all subsequent layers. This set of unfolding layers suggests that any
more formal or abstract layer of understanding action enfolds, unfolds from, and is
connected to inner, less formal, less sophisticated, less abstract, and more loca,l
ways of acting. Although the rings of the model grow outward toward the more
abstract and general, growth in understanding is not seen to happen that way.
Instead, growth occurs through a continual movement back and forth through the
levels of knowing, as the individual reflects on and reconstructs their current and
previous knowledge. Pathways of growth drawn across this model illustrate the
fact that growth in understanding need be neither linear nor unidirectional.
Of particular relevance to this article are the levels of Primitive Knowing and
Image Making. The term primitive is used not in the sense of low level or trivial,
but in the sense of "prime"-as in both "important" and "previous". Primitive
Knowing is all the previously constructed knowledge, outside of the topic, that
students bring to the learning of a topic. Much of this knowledge will, of course, be
irrelevant to the task in hand, but it is only on existing understanding that new
learning can be built. When approaching the teaching of any topic, the teacher will,
consciously or unconsciously, assume that learners possess certain prior
understandings. For example, the teaching of fractions assumes a certain level of
understanding of the concept of number and an understanding of the four rules of
arithmetic. It may also assume knowledge of the way pizzas and other circular
objects are frequently partitioned. It is, however, unlikely to call upon the nature of
the weather on that particular morning. Yet all these understandings-the
sequencing of numbers, addition, halving a pizza by cutting through the diameter
in a straight line (as opposed to, say, cutting out a circle of radius equal to 70.7% of
the original radius), and the nature of sun and rain-are likely to form part of each
learner's Primitive Knowing. From this Primitive Knowing, appropriate knowledge
must be selected and used as a basis for growth of understanding.
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Figure 1. 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 or
physical, that are intended to foster some initial or extended conceptions for the
topic to be explored. In the case of fractions, Image Making activities would
perhaps lead to the learner saying, "Ah, fractions are what you get when you cut
things up". At this point, the theory would claim that the learner has an image (is
acting within the Image Having level) for fractions-although one would hope that
this would later become refined to the image that "fractions involve the cutting of
items in equal pieces".
The above illustration of a possible path of understanding from Primitive
Knowing through Image Making to Image Having is not meant to indicate that the
growth of understanding moves smootWy outwards through the layers. We
contend that growth in understanding takes place through a continual movement
back and forth through the layers of knowing, as individuals reflect on and
reconstruct their current knowledge. The metaphor of recursion higWights the fact
that the dynamical understanding notions of a person involve states which differ in
character but are self-similar (Kieren & Pirie, 1991). A person's current
understanding action in some way acts to elaborate previous states and integrates
them in the sense that they are called into current knowing actions.
For a more complete description of the model, see Pirie and Kieren (1994).
Folding Back
A 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 word
invocative (Kieren & Pirie, 1992) is used to describe a cognitive shift to an inner level
The RQle of Collecting in the Growth ofMathematical Understanding
131
of understanding, and an invocative intervention is one which promotes such a
shift. An invocative shift is termed folding back when the person makes use of
current outer layer knowing to inform inner understanding acts, which in turn
enable further outer layer understanding (Pirie & Kieren, 1991b).
1
When faced with a problem that is not immediately solvable at any level, an
individual needs to return to an inner layer of understanding. The result of this
folding back is that the individual is able to extend their current inadequate and
incomplete understanding by reflecting on and then reorganising their earlier
constructs for the concept-or even to generate and create new images, should
their existing constructs be insufficient to work on the problem. However, the
person now possesses a degree of self-awareness about his or her understanding,
informed by the operations at the higher level. Thus, the inner layer activity cannot
be identical to that originally performed, and the person is effectively building a
thicker understanding at the inner layer to support and extend their understanding
at the outer layer that they subsequently return to. It is the fact that the outer layer
understandings are available to support and inform the inner layer actions which
gives rise to the metaphor of folding and thickening.
Although a learner may well fold back and act in a less formal, more specific
way, 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 thicker
piece 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 changing
understanding of the concept, and this extended understanding acts to inform
subsequent inner layer actions. Folding back, then, is a metaphor for one of the
processes of actions through which understanding is observed to grow and
through which the learner builds and acts in an ever-changing mathematical
world. Folding back accounts for and legitimates a return to localised and
unformulated actions and understandings in response to and as a cause of this
changing world. The Pirie-Kieren theory suggests that folding back is an intrinsic
and necessary part of the process by which understanding grows and develops.
Collecting
It is the purpose of this paper to distinguish what we see to be a particularly
important form of folding back which we call collecting. The process of folding back
to collect entails retrieving 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 folding
back. In what follows we give examples of collecting, distinguish it from other
forms of understanding actions, and discuss how teachers might act to occasion
1 We use the words "problem" and "solve" frequently in this article, but at no point are we
refering to the limited, specific activity that has come to be called problem solving in the
mathematics education literature. To enable people to solve problems throughout their
lives, not just those contrived problems set in mathematics lessons, is the reason that all
children are taught mathematics. For us, problem solving is simply working mathematically
when the route to the solution is not direct and immediately clear.
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such folding back to collect.
Of particular significance in the data relating to folding back is the occurrence
of a number of cases where, following a shift by the learner to an inner layer of
understanding, there has been neither any observable learning activity (in the sense
of any visible reorganisation or reconstruction of existing constructs) nor any
generation of wholly new understandings. Instead of working on existing ideas,
the inner layer activity has been more a process of finding and collecting an earlier
construct or understanding and then consciously using or re-reading it as useful in
a new situation.
Before we tum to actual classroom dialogue, we ask you to consider the three
examples of students tackling the question 93 - 47 = ? shown in Figure 2. The
vignettes are based on classroom events and have been deliberately constructed to
clearly illustrate and deliberately differentiate various ways of thinking about the
same problem.
Vignette 1
Jasmin:
So, three take seven, can't do (pause) nine becomes eight, thirteen
take seven is six (pause) and eight take four is four, gives forty six.
Vignette 2
John:
Hmm, three take seven ... (pause) Hang on, seven is bigger than
three, I can't do it, if it was seven take three it would be OK. (He
puts his hand up and the teacher comes over.) I can't do this 'cos seven
is bigger than three so you can't take it away.
Teacher: Could you do something to the nine and the three?
Hmm, no, I dunno, I can't do it.
John:
Teacher: OK then, I'll get the rods and blocks and we'll make ninety three
and forty seven.
They then work with the Cuisenaire rods and use these to solve the problem.
Vignette 3
Paulo:
Three take seven, can't do (pause) no, you can do something to the
nine and the three and borrow or tens it or something, lemme look.
(He opens his workbook and flicks through it.) Yeah, that's it, make the
nine an eight (pause) borrow ten so we get thirteen take seven is six.
Now the other bit is eight take four is four, forty six.
Figure 2. Three classroom vignettes.
Jasmin has no difficulty in solving the question at all. She has the necessary
understanding instantly accessible and the process she uses is essentially
automatic. There is no necessity for her to fold back.
John cannot deal with this problem at all. It is not clear whether he has met a
question like this before but cannot now solve it, or whether subtraction questions
of this type are new to him. What is clear, however, is that either he does not have
the necessary understanding or that his understanding is not well enough
The Role of Collecting in the Growth of Mathematical Understanding
133
developed to allow him to use it. Instead, prompted by the teacher, he folds back to
perform more image making, either to build a new image or to enhance an existing
one, perhaps by working on his image for subtractions where the unit subtrahend
is larger than the unit minuend. John needs to do more mathematical work at an
inner layer before he will be able to build for himself an algorithm to answer the
question, an algorithm that he can use with understanding.
We see something very different in Paulo's thinking about this problem. He too
cannot immediately solve the question-he does not have the understanding to use
an automated process in the way that Jasmin does. Neither, however, does he fold
back in the same way as John, to construct or modify an image. Paulo has an image
involving the reconceptualising of the numbers that he believes will allow him to
solve the question,..but he needs to fold back to the level of Image Having in order
to retrieve this image, to re-view its properties in terms of the specific task at hand,
and then to use it. There is a sense of him having, and being aware that he has, the
necessary understandings but that they are just not immediately accessible. Thus,
he needs to fold back to his more basic understanding and in some way recollectthat is to say, re-collect-it for use in his current thinking.
It is important to note that the process of collecting is a mental one. Although
here it is accompanied by Paulo searching his workbook, this is not essential to the
idea: It can equally be performed simply through the conscious searching of one's
thoughts. The workbook here is an aide-memoire-it is not in itself his
understandings. Although initially it may appear that he has a lack of
understanding of subtraction, this is not actually the case; he was not looking for a
new idea to help him. After successfully re-collecting the image he needs, he is able
to correctly complete the question using his existing understanding of the concept.
His language allows us to assume that he is not blindly applying, by rote copying,
a given algorithm. He has recollected the understanding process which legitimates
his subsequent, algorithmic action of subtraction. The major difference between
this and the folding back of John is that the inner level activity of Paulo does not
involve a modification of his earlier understandings. His working involves him,
instead, in finding and recalling what he knows he needs to solve the problem. He
is consciously aware that this knowledge exists. He collects his inner
understanding and consolidates it through intentional use.
Collecting in the Classroom
The rest of this paper is concerned with illustrating the phenomenon of
collecting as it happens in the classroom. Our intention is to show that even brief
fragments of dialogue are sufficient to alert us to the shifts in thinking that take
place. The examples are chosen to demonstrate some of the key features of
collecting, and to indicate the varied ways in which students carry out the process
and the various teacher actions which can facilitate it.
In the first of the following two case studies, students are seen successfully
collecting inner layer understanding and using this to continue working. In the
second case, the two students are initially less successfuL Their interaction provides
a valuable insight into their ways of thinking as they struggle to find and collect
what they know they need.
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Case 1: Rosemary and Kerry
The first extract is taken from a lesson with Year 9 students (about 14 years
old). The students, Rosemary (R) and Kerry (K) are of average ability and have
been set the task of finding out the area of icing on a slice of a circular birthday
cake. 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 is
from when the students begin working.
R:
K:
R:
K:
R:
K:
R:
K:
R:
There must be something on it in here. (Pause as she flicks through her
textbook.) I dunno (in doubtful tone), I'm looking for the area section.
(laughs) Area is page a hundred and thirteen. (She turns to this pageheaded "Area ofa Triangle"-in her book.)
Got it.
There, it's half the base times the height.
No, (pause) we need .... (pause) It's pi r squared isn't it and hrnm... (Pause
as 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. Circumference
equals two pi r squared. No, no, no, that's wrong, two pi r. Then area
equals pi r squared.
No, but we don't want ....
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 given
dimensions.) No, that's wrong.
Let's cut a quarter and make it easy ....
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. Before
Rosemary begins to work at making an image for the sector of a circle, she folds
back to her primitive knowing, searching for something useful and applicable to
the problem. This shift appears to be self-invoked; that is to say, there has been no
deliberate, external intervention to cause her to decide to search her textbookalthough obviously the question and therefore the teacher have contributed to this
occurring. It may be that her history of working with this teacher encourages her in
the belief that she does possess the elements within her primitive knowing that will
support her growth of understanding.
Initially Rosemary believes that, unlike Jasmin in Figure 2, she cannot
immediately tackle the problem that the teacher expects her to be able to solve,and
she clearly sees a need to fold back to her primitive knowing and to use previous
understandings in this new topic. She seems, however, unsure which aspects of her
primitive knowing to actually draw upon and her thinking is unfocused in its
nature. After a pause, she tells Kerry that she is looking for the "area section". She
has decided that she needs to calculate the area of a circle. She finds this section in
the book, intending to search for the required formula, confident that she already
knows it and that having re-collected it she can return to image making for the
problem in hand. She expects to be able to use her primitive knowing to continue
working, in a similar manner to Paulo in Figure 2. In the later stage of the extract,
we see that Rosemary does find the information she is searching for (both
internally in terms of her own understanding and externally and physically in the
The Role of Collecting in the Growth ofMathematical Understanding
135
textbook). She collects the area of a circle formula, taking it back to the level of
image making where she attempts to continue working. In fact, though, she finds
that she cannot immediately use her formalised rule to find a numerical answer,
because the problem the teacher has posed gives no dimensions for the circle. In
this, her collecting differs from that of Paulo in that, having collected, she still has
to determine how she will use her understanding. Nonetheless, her response to
Kerry's final statement, "Just to make it easy", is evidence that she is now thinking
about the question of finding the area of a sector. She is seeing it as a portion of the
whole circle-that is to say, she is constructing an image for the notion of sector as
part of a circle-and is intending to use her recollected understandings.
On closer inspection, we see that the images Kerry and Rosemary initially form
for a sector are interestingly different from one another-at least partly because the
two students draw upon different primitive knowing. Rosemary's mention of "the
area section" has a marked effect on the thinking of Kerry and, probably as a
consequence of this student intervention, Kerry too folds back to her primitive
knowing. However, her shift appears more intentional: She goes directly to the
concept 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 to
collect her understanding of triangular area in order to make it possible to work
with her image for the problem. She suggested working on a quarter circle to create
an "easy" right-angled, isosceles triangle. But for her too, the formula she needed
was not immediately applicable.
Both students collected inner understanding which they attempted to use to
increase outer understanding. But the knowledge and understandings they
collected resulted in differing images of a sector. Both girls folded back to collect on
the occasion of the given problem. Both then acted to reformulate previous
understandings into an understanding of a sector. But their collecting led them to
different understanding actions, just as their perceptions of the problem invoked
different collectings.
Case 2: Simon and Ann
The 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 on
calculus 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 on
the necessary primitive knowing, in this case on making an image for the concept·
of limits. They have already answered a number of straightforward questions and
3
are now trying to find lim
h ?
h~O h+2h-
Their initial attempt is simply to replace h by
zero.
A:
s:
It's nought divided by nought... (she writes:
0
=.Q)
O+2xO
0
Yeah,. but you're saying what's nought divided by nought? Is it nothing
or is it infinity? How many nothings in nothing? Is there none or is there
an infinite number?
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Pirie & Martin
A:
s:
A:
s:
A:
s:
Is it one? Is there one nothing in nothing? .
It's not 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 totally
different.
I suppose (pause). It's nothing~ or infinity, or one, we haven't decide
It's not one...
The difficulty here has been caused by the fact that the h in the denominator of
the rational expression leads to a division by zero. With their present image for
finding limits they are led to replace h by zero, and they are left with a situation
that they cannot solve. Their difficulty here has two aspects. Firstly, their existing
understanding of limiting values is insufficient to allow them to solve the question;
and secondly, they do not use the necessary algebraic primitive knowing to allow
them to modify their image making. Both students are seen folding back to their
primitive knowing of arithmetic.
The way in which the two students then work, though, differs. Simon appears
to be trying to retrieve a fact at this inner level: Nought divided by nought is either
nought or infinity, he has forgotten which. (This inference of ours is supported by
later 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 Simon
have different images for zero. Ann calls zero "nought" and regards it as a number,
like any other number. Simon calls zero "nothing" (he says "nought" only when
repeating Ann's comments) and regards it as "emptiness". His face, gestures, and
tone of voice all imply that he is looking to see how much emptiness is in
something empty. His image is, quite reasonably but probably unconsciously,
influenced by the common language meaning of nothing. When Ann refers to the
number two, Simon responds, "Nothing's totally different". Although he can call
his image to mind and he attempts to state what he has recalled, he is unable to
apply it.
With Ann the situation is different. She folds back to an understanding of a
property of division (a -;- a = 1) and moves out of the topic of limits to work with
this property. In fact, the actions of Ann suggest that her existing understanding is
not sufficiently developed or complete enough to allow her to collect it and use it
anew. Instead, she seems to have a need to work on her primitive knowing, to have
a greater understanding of division, which she can then use in the new context of
"nought divided by nought". Here, therefore, we do not see her actually engage in
an act of collecting. Simon is aware of the inadequacy of her notion, but cannot
offer an alternative idea and both students are effectively unable to procee,d. At this
stage the teacher (T) intervenes.
T:
A:
Right, you can't actually give me an answer to it as it stands? In fact, can
you 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? 'Cos
you don't know how to divide by zero, you don't know, as you say,how
many nothings there are in something, OK? Can you do something to
this (the original expression)? Can you simplify that in some way? .
You can knock them off you see ... that's what we can do, can't we? You
can do that, you can make it hover h plus ... argh, we've got two h
The Role of Collecting in the Growth ofMathematical Understanding
137
squared, knock off one h squared. Get h, h plus h squared. (She writes
T:
hxhxh
'
h2 )
- - , crosses out h x h'In th e numerator, and
wrztes
.
h+2(hx
h+h
Right, so h plus ... (he writes as they work)
h times h, so that's (inaudible) times h times h. Knock off hmm ....
See, I've got h plus h squared over h, that's still not right ....
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 by
something, you divide it by nothing, no it's nothing ....
Are you actually happy with what was going on here (pointing to
A:
T:
A:
h x h x h )? 'Cos I wasn't quite clear what was going on.
h+2(h x
OK, we've got, on the top we've got h times h times h.
Yeah.
On the bottom we've got h plus h times h ... times h times h. (She now has
S:
A:
S:
A:
written:
h x~
.)
h+(hxh)~
T:
A:
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)?
Because that's what we did in maths a couple of days ago, what was it,
was it? Factorials? ... n over n minus r factorial factorial ... something like
that.
The teacher here has recognised the problem the students are having and
initially validates2 this by saying "'cos you don't know how to divide by zero." She
makes an intentionally invocative intervention to get the students to fold back
again to their primitive knowing, but this time the teacher is able to give the
intervention a more explicit focus than the printed question had provided. She
asks, "Can you do something to this? Can you simplify that in some way?" The
language here is a prompt to particular algebraic techniques. The word "simplify"
seems to provide the invocative trigger for Ann and Simon. They fold back to their
primitive knowing and collect from this inner layer their method and
understandings of algebraic manipulation, which they proceed to work with while
trying to construct an image for the notion of limits. Unfortunately, it is evident
that their algebraic understanding which they collect is either incomplete or
inappropriate 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 of
prompting the students to do just this. She first asks, "Are you actually happy with
what was going on here?" This intervention has the potential to be invocative
although it is somewhat non-directional in nature, but it is not taken as invocative
and the students continue to work in the same way. The second question, again
with invocative intent, is more explicit: "Why were you able to cross that out with
that?" Had the students been able to answer this question, it would have acted as a
validating question. Where the students are incorrect, it is reasonable to suggest
The notion of validating is used here to describe an intervention which confirms the level of
understanding currently employed by the person.
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Pirie & Martin
138
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 folds
back to her primitive knowing, aiming to collect understanding that will provide
an explanation. The nature of the question asked by the teacher results in a
relatively unfocused shift in thinking by Ann and she recalls some notion of
factorials. She seems to be searching her primitive knowing again for something
that she thinks will help her, although she does not appear to know why factorials
may 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 required
mathematics; instead, she first locates where it occurred for her and then utilises
this "flag" to find the knowledge she needs. She justifies her working by saying
"Because that's what we did in maths a couple of days ago" and it is this
referencing that then seems to trigger the actual collecting of the notion of
factorials, lin over n minus r factorial factorial ... something like that".
The way in which Ann sets about finding what she wants to collect is
especially interesting. It is a common feature in her working, as is the fact that the
actual mathematics recalled does not initially seem to be particularly clear in her
own thinking. Her typical pattern of action seems to be to reference her thinking by
the time when she worked on the "collectable" concept and by the events in the
classroom that surrounded it. She needs to re-situate herself in her previous
understanding activity. She relatively easily collects the mathematical label for a
concept-in this case, factorials-but she then frequently relies on Simon to supply
the actual piece of useable mathematics3 . Indeed, she takes on an almost teacherly
role when calling on Simon to supply or recall the actual mathematics, as the
following episode illustrates.
At one point Simon and Ann think that they have to deal with one divided by
nought.
A:
S:
A:
S:
A:
S:
A:
S:
A:
S:
Which is ...? (expectant pause) You know Sime.
Do I?
Yeah, you're big on these sorts of things. Come on.
Why am I big? Why am I big on these sorts of things?
'Cos we were doing it the other day.
Were we?
Proving God and all that. Or whatever you were doing.
That was more than two years ago!
Yeah, well.
That was the Hitch-Hiker's Guide to the Galaxy. It's infinite.
Once Simon has identified the mathematics, Ann is usually able to recollect it
and apply it with understanding. This, and many similar examples, suggests that
knowledge of the need for collecting plays a frequent and vital role in Ann's
growth of understanding. She seems to have developed her own internal labelling
and referencing strategies to allow her to make the collecting process easier and
more efficient. She has created and uses a two stage mental referencing system,
3 For obvious reasons, we do not have any data on her collecting acts when she is working
alone and therefore silently.
The Role of Collecting in the Growth ofMathematical Understanding
139
which links events to mathematical labels to mathematical understanding.
Later in the same session, the students return to differentiation and can be seen
using their primitive knowing from the concept of limits to make a new image for
differentiation. They are working with the equation y = x 2 and using the notion of
the difference in y divided by difference in x for a small increment h. At the
h
h 2
and are trying
Ann realises that she cannot immediately expand the expression and she folds
back to her primitive knowing. Although the shift is again primarily caused by the
material and the question, there is a definite element of Ann also choosing to fold
back herself. She is aware that somewhere in her primitive knowing she has the
necessary 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 try
and find what she knows that may help. There is a very real sense of her trying to
find some appropriate understanding. She is not folding back to develop these
ideas 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 just
did it, we did it on, hmm, on those stairs ... Mathematical Methods" and "This is
the great long thing we had an exam on". She has not only labelled her thinking by
time and event but also by a physical reference point; and she uses this to try and
facilitate the collection of the attached mathematics. Despite an apparently welldeveloped strategy, however, Ann is still unable to precisely or fully collect what
she needs. Atypically, although Simon knows she wants to multiply out the
brackets, he fails to come to her aid. She continues to struggle to recall the relevant
knowledge, with comments such as "It's the great big thing and we set them all out
in class and I said 'No miss, that's wrong, I did that wrong'" and "Is it the one with
the Pascal's triangle?"
Throughout the process of hunting through primitive knowing, Ann is not
~ Ann is probably referring to how they were introduced to integration by drawing
rectangles to approximate the area under a curve. The diagram has the appearance of a set
of stairs.
5
The textbook from which they have been working.
6
She is probably trying to recall the expansion of (x + h)n.
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actually talking directly about the mathematics that she is trying to fold back to but
rather about the events of the lesson that surrounded that work and the physical
location 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 label
"Pascal's triangle". When, after searching her memory she is still unable to find
exactly what she knows she needs, she tries, as Rosemary, Kerry, and Paulo did in
the earlier examples, to actually physically locate where she may find what she
knows she needs. She begins looking in her bag and her file for her notes from the
lesson to help her collect what she knows she needs. Although she can access. the
general topic area she is aware that she is unable to precisely collect what is
required and calls on Simon's assistance, saying "Have you got your book with
you?" and later "You've got your folder, you might have your book" and later
again "... 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 out
of her primitive knowing as she retrieves more small pieces of mathematics to help
her. She seems to be trying to reconstruct a jigsaw puzzle of her understandings
piece by piece-in effect repeatedly saying to herself "if I knew such-and-such I
could probably act with understanding"-until the whole is ready to be used.
What Ann is gradually collecting is a relevant, if overly sophisticated, piece of
primitive knowing-namely, the expansion of binomial expressions. Finally the
teacher 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 h squared. What does x plus h
squared mean?
S:
It means x plus h ....
A:
x plus h, argh! This is GCSE7 (very excited and laughing). So x squared plus
xh, plus hx plus h squared ....
S and A: So its x squared plus two xh plus h squared ....
A:
(triumphantly) There you go.
T:
Now, do you remember why you wanted that?
Even as the teacher speaks, Ann and Simon return immediately to the sheet of paper on
which 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 to
now fold back further, to their specific formalised understanding for quadratics.
The effect on Ann is quite startling, she suddenly folds even further back to the
image she has for the expansion of brackets term by term, and is able to quickly
and easily collect it. Again, though, she firstly describes it in terms of the label
attached to it-an event, in this case "GeSE". Having collected the image from the
inner layer, she is able to move back out with it to their image making for
differentiation from first principles and immediately and successfully use it to
7 The General Certificate of Secondary Education examination, which the students sat
eighteen months previously.
The Role of Collecting in the Growth ofMathematical Understanding
141
expand the expression. The final comment of the teacher is important as it is
designed to validate that the students are in fact working back at the image making
level and using their algebraic understandings in the context of differentiation
rather than merely expanding a quadratic without purpose. The end result here is
eventually one of successful folding back, collecting and using earlier
understanding-although it was not achieved without a struggle!
Limits
Algebra
Other Understanding
Differentiation
Figure 3. Ann's path of growth of understanding of differentiation.
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Pirie & Mnrtin
Ann's cognitive path through the various levels of understanding is illustrated
in Figure 3. This diagram appears complex and hard to interpret, and we have
included it here for this very reason. It gives a good illustration of the difficulty of
finding ways to represent dynamic understanding, itself an extremely complex
process. In the past,many readers have taken the eight nested circles too
simplistically and as literally "the model of the theory". In fact, they are merely
offered as a visual aid to our humanly inadequate, verbal description of one set of
the 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 the
line as it weaves back and forth, in and out of the understanding diagrams for
Limits, Arithmetic, Algebra and Other-all of which form part of the Primitive
Knowing for differentiation-back to the point b in the Differentiation diagram are
aided in their understanding by the visual representation. The zigzag line is what
we call Ann's path of growth of mathematical understanding during the lesson
described above.
Implications for Learners and Teachers
The examples discussed above have been selected to illustrate students folding
back 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 in
algorithmic or definitional form. We call such folding back actions collecting. As is
obvious from the examples, the usefulness of collecting in on-going understanding
is 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? It
would certainly appear that students differ widely in their ability and their method
of collection. Most of the students we have studied have, like Jasmin in Figure 2,
some of their inner layer understandings formalised into an instantly accessible
and automated process. Indeed this is likely to be the case for many students in
much of their mathematics. For example, students working on advanced calculus
would not be expected to stop, fold back, and re-collect their earlier
understandings every time they need to multiply or add two numbers together.
Instead, such facilities have become an unconscious tool, used when required
without thought as to their origin or meaning.
However, as the examples show, collecting from an inner layer is a vital part of
the growth of understanding and we need to provide the students with the ability
to facilitate this growth. Paulo, Rosemary, and Ann are all very aware that they
possess the understanding and the knowledge they need to be able to continue
working, and this is the key feature of the phenomenon of collecting. What we are
interrogating is the interaction between the person and the problem at hand. The
mathematics they are working on is not in itself problematic, they do not have a
lack 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 their
existing knowledge. Instead, for these students the initial difficulty lies in their
being unable to automatically access an earlier understanding.
For Rosemary, overcoming this difficulty is a matter of retrieving the required
formula from the textbook. Thus, the folding back is no more than a momentary
The Role of Collecting in the Growth ofMathematical Understanding
143
shift in her cognitive state which empowers her to continue to develop her outer
level understanding. Nonetheless, the importance of this shift in state, this
collecting, should not be minimised. Even her collecting involves recollecting
previous mathematical experiences and understanding, and creating a
contemporary use for them in a current act of understanding. This act obviously
shapes the understanding being currently built (the idea of a sector of a circle) but
also reshapes her previous understanding of circles.
With Ann, the situation is more complex. She shares the awareness of
Rosemary that she has the understanding she needs, but for her the process of
finding and collecting this understanding is problematic. The internal two stage
"filing system" she has created for her mathematical concepts certainly helps her to
do this, and at times the system is highly successful-although she relies quite
heavily on Simon to provide the precise mathematics. Ann's invoked collecting of
binomial manipulation shapes her ability to understand limits of functions, but it
also clearly changes and adds to her understanding of the binomial form of the
quadratic function. In contrast, Simon appears better able to fold back and ~ollect
mathematical understanding, most particularly when prompted by Ann's
invocative location system. Simon does not appear to search his existing
understandings in the way that Ann does, but waits for her to provide him with a
prompt. Ann and Simon both have a good understanding of what they are doing
and what they need to do to be able to continue. Their problem lies in an inability
to effectively find, collect, and use the earlier concepts-the primitive knowingthat they need. It would, of course, have been interesting to have followed these
students working with different partners, but unfortunately this was not possible.
How widespread is this technique of Ann's? Could teachers help students to
develop such reference systems? Certainly an important action when initially
working 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 mathematics
that they teach is likely to be, for them, at least at the formalising stage of automatic
recall), a conscious, frequent, overt modelling of "collecting" when working
examples in front of students would demonstrate the need and the power of such
mathematical activity.
The examples in this paper also clearly illustrate the fact that students
frequently return to textbooks or written notes to find what they know they need.
Although such texts do not constitute the understanding needed, they are certainly
a valuable aid to the cognitive processes involved in collecting. Thus a teacher who
promotes writing about one's understanding, careful reading of texts, and student
discussion indirectly provides the ground for such collecting.
This leads us to consider the effect and importance of teacher interventions in
helping students to fold back and collect when needed. The dilemma lies in
deciding the extent and explicitness of the interaction. In the case of Ann it was the
final invocative intervention of the teacher, explicitly directing her to fold back to
the precise piece of mathematics that she needed to collect, that enabled her to be
able to do this and then continue working. How specific should one be? How soon
should one intervene? ,Clearly, neither question can be answered in the form of
general advice and the teacher's· motives for the students' undertaking particular
mathematical activities need to be considered in every case individually.
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Pirie & Martin
One solution is to direct the students explicitly to what they need to do when
they are stuck. This is not the totally inappropriate intervention that it is sometimes
painted to be. If, as in the case of Ann, the student needs to be reminded of a
particular technique (in her case, multiplying out quadratics) then in order to allow
that student to progress in the building of a new concept, the provision of direction
as to what to collect may be beneficial. This parallels the way in which we allow
students to use calculators to enable them to focus on methods of analysis by
avoiding heavy arithmetic. What is important is that the students know that they
do have the necessary primitive knowing somewhere. In the long run, however,
although this would have the effect that the students could continue with the
specific problem, it would not lead to their developing personal strategies for
collecting-which is, we believe, an essential tool for solving any mathematical
problem that one might encounter. For those students who do not naturally
construct a well-developed facility to collect their earlier understandings, teachers
can provide valuable assistance through careful, sensitive interventionsparticularly pertinent questioning. If we are to enable students to become
autonomous problem solvers, then we have an obligation to make explicit to them
the need for and the skills involved in both folding back and in collecting.
Conclusion
Although mathematical understanding remains a complex and intensely
individual phenomenon, there are common key features underlying the cognitive
processes involved. Both the general practices of the teacher and the students, and
specific interventions by the teacher act to invoke collecting. A cognisance of the
fact that an apparent lack of understanding of the current mathematics may, in fact,
be due to something very different holds for teachers and students of mathematics
both the challenge and the opportunity to enhance the growth of mathematical
understanding through their actions. A teacher who is not only aware of the role
that collecting plays in the growth of understanding, but also realises that students
have different strategies and abilities for finding, collecting, and using their
previous understandings, is in a position to consider the sort of actions that may
best enable the process to take place.
Acknowledgment
The conference presentation by Pirie, Martin, and Kieran (1996) contained
some of the ideas in this paper.
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Authors
Susan E. B. Pirie, University of British Columbia, Department of Curriculum Studies, 2125
Main Mall, Vancouver, V6T 124, Canada. E-mail: <[email protected]>.
Lyndon C. Martin, University of British Columbia, Department of Curriculum Studies, 2125
Main Mall, Vancouver, V6T 124, Canada. E-mail: <[email protected]>.