MEMORY AND UNDERSTANDING IN MATHEMATICS
EDUCATION
Leo Jonker, Queen’s University
ABSTRACT
The most common metaphor for mathematics education, the constructivist metaphor, is
usually taken to imply that learning always proceeds from familiar situations to
abstracted concepts and that the process of discovery is to be valued above the
acquisition of competencies. In this paper, we suggest that the matter is not always that
straightforward – that there are situations where memory plays a greater role than is
often acknowledged, and that competencies are sometimes necessary before discovery
can take place. We examine some of these paradoxical situations and interpret them in
the light of work by Lakoff and Núñez (2000) on conceptual metaphors and in the light of
the APOS model of Dubinsky et al. (2001).
Three paradoxes
In the teaching of mathematics, understanding is given greater importance than
memorization, and autonomy is valued over competency. This is how most career
mathematicians think about mathematics learning; this is how most of them approached
their own. For others, understanding may not be the word that first comes to mind when
they think back to their mathematics classes. In fact, understanding may not have been
the main focus of school mathematics years ago, and the experience of learning
complicated mechanical procedures without understanding their meaning still haunts
many. Today this situation is changing. One of the six Principles enunciated by the
influential Principles and Standards of the National Council of Teachers of Mathematics
(NCTM 2000) is the “Principle of Learning”:
Students must learn mathematics with understanding, actively building new
knowledge from experience and prior knowledge.
An earlier (1991) NCTM document “Professional Standards for Teaching
Mathematics” indicated five shifts in the classroom to be engineered by teachers to help
students learn mathematics. Two of the shifts suggested are

Toward mathematical reasoning and away from mere memorizing procedures

Toward conjecturing, inventing, and problem solving and away from an emphasis
on the mechanistic finding of answers
I would certainly count these among the principles I hold high as a teacher, and
virtually all school mathematics curricula today reflect these shifts of emphasis.
However, education is fiendishly complicated, and every good idea can be countered with
an exception or at least a reservation. In this paper, I will look at some instances where
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the implications drawn from the mathematics reform movement initiated by the NCTM
may have to be modified, or at least nuanced. I will then use aspects of the work of
others, notably Lakoff & Núñez (2000) on concept formation, and Dubinsky et al. (2001)
on learning mathematics, to shed some light on the relationship between imagination,
understanding and memory, and what that might mean for mathematics teaching.
Discovery and competency
For more than twenty years, I have been teaching a weekly mathematics
enrichment class to students in grades 7 and 8, always in the same school. In this
enrichment program I try to introduce students to open-ended, varied, and imaginative
mathematics that is related to the curriculum but not necessarily included in it. I try to
introduce ideas that are strange and paradoxical – ideas that stretch their world and
engage their romantic impulses (Egan, 1997). Thus, I explore with the students why some
fractions produce repeating decimals while others produce terminating decimal
expressions. We explore why the Theorem of Pythagoras should be true. We investigate
why the trick for divisibility by 3 works the way it does. We discuss how you might
accommodate the passengers of a bus with infinitely many seats in a motel with infinitely
many rooms.
Over the last fifteen years, also at the school at which I do the enrichment
teaching, the reforms inaugurated by the NCTM have altered the way mathematics is
taught. Teaching methods rely more strongly on student discovery and less on the
mastery of technique. I approve of the change, for mastery of technique can be
deadening if students are not given the time and the opportunity to construct meaning of
the technical material they are asked to learn. Paradoxically, though, I find that the
students I see today are less ready to engage the enriched fare I present to them than were
students fifteen years ago.
The puzzling success of the JUMP program
In every school system some students fall by the wayside along the strongly
incremental mathematics curriculum. For many of these, the problem begins when they
try to learn fractions. Normally, to teach them fractions, we present students with
problems in which something must be divided: A pizza is divided into four wedges, and
you get one; twenty smarties are divides into five portions, of which you get two. Visual
aids and manipulatives are supposed to help students make the transition from enacted to
iconic to symbolic representations (Bruner, 1966). At some point, students are presented
with the problem of calculating the total amount of pizza contained in one-third of one
pizza and one-half of another. Students are urged to play with pictures of pizzas or cut
paper disks and to imagine dividing the pieces further and rearranging them until it
becomes clear that the same amount of pizza is obtained if you divide a pizza into six
equal wedges and take five of them. Only after much discovery of this sort is the student
encouraged to identify equivalent fractions in their symbolic representations and to
multiply one fraction (top and bottom) by 2 and the other by 3 to produce two fractions
whose denominators are both 6. This is the right way to teach this, I feel. The
understanding leads to algorithms. This is how understanding is constructed logically. If
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there are students who do not learn how to handle fractions (and there are), we may tell
ourselves that we should have gone through the steps of the process more carefully, or
perhaps we can agree that these students just will not learn to do fractions and that with
calculators to help them it does not really matter.
The “Junior Undiscovered Mathematics Prodigies” (JUMP) program was started
by the Toronto Governor-General’s Award winning playwright and mathematician John
Mighton. Some years ago, when he was between jobs, Mighton responded to a request
for a volunteer mathematics tutor for students who were at serious academic risk. His
very positive experience as a volunteer resulted in the creation of a mathematics tutoring
program, now supported by a website, a research department, and a small army of JUMP
tutors in Canada and abroad (http://www.jumpmath.org/).
The methods of the JUMP tutoring program run counter to many of the principles
of mathematics education. The focus of the program (say for grade 4 students) is on
fractions. However, there are few pictures and no manipulatives. Instead, the program
involves a great deal of technical drill of procedures that are broken down into minute
steps. While it is clear from the things John Mighton has written that he is as concerned
about understanding as any of us, there is very little in the JUMP workbooks that focuses
on helping students understand the meanings of the operations. For example, to teach the
addition
1 1
 ,
3 2
students are first taught, in a series of nearly identical exercises, to place multiplication
signs beside the fractions:
1 1

3 2
→
1 1

3 2
Once this skill has been learned, they are taught, again through repetition, to place copies
of the denominators of the opposite fractions next to the multiplication signs, like this:
1 1

→
3 2
2 1 1 3

.
23 23
Following this, they are given a series of questions that ask them to do the
multiplications:
2 1 1 3
2 3

→  ;
23 23
6 6
after that, a set of questions in which they are allowed to supply both the multiplication
signs and the multipliers at once. Finally, they are given questions in which they are
asked to complete the problem by adding the resulting fractions, which will now have
identical denominators. Of course, before this all takes place, students have completed
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several pages of exercises in which they add fractions with like denominators. Almost no
explanation is in sight anywhere.
This ought not to work. Or at least it ought not to lead to understanding. Or at
least it ought not to be enjoyable to the students involved! At this point, the evidence is
partly anecdotal, and not complete; but it suggests otherwise. Studies at OISE, focusing
on student attitudes to mathematics (Hughes, 2004), and at Brock University, focusing
on the change in mathematics test scores (Luckiw, Shuve, Miggiani, and Mellamphy,
2005) indicate a significant and positive outcome. Hughes found that
There is a very significant relationship between students’ math confidence and the
JUMP program. Furthermore, the qualitative data compiled from short answer
questions on the survey has revealed that students both enjoy the program and feel
more confident about their math education after receiving the program. (Hughes,
2004, p. 2).
Luckiw et al (2005) state in a similar manner:
The preliminary results of this pilot study show that the fractions unit of the
JUMP Math program has significant impact on student math scores on the
fractions pre/post test. (p. 6).
The paradox of the Chinese mathematics student
The third in my set of paradoxes concerns the East Asian mathematics student.
Partly in response to the mediocre performance of U.S., British, and to some extent
Canadian students on international comparative studies of student understanding (TIMSS
and PISA), a very strong and growing interest in cross-cultural comparisons of education
has emerged. East Asian students do especially well on the comparative studies, even
though some of the teaching in these regions would be dismissed out of hand by North
American observers as a form of rote learning. Several early studies (Oxford and
Anderson, 1995; Dunbar, 1988) came to this conclusion.
Kember and Gow (1991) write:
As newly arrived academics in Hong Kong, we were greeted with numerous
anecdotal descriptions of the students we were about to teach. The anecdotes
suggested that Hong Kong students relied heavily on rote-learning and
memorisation, but interestingly most of our new colleagues maintained that they
were good, or even very good, students. It was moreover suggested that they were
more passive and less interactive in class than typical western students. Yet they
were also described as very keen and competitive. (p. 117).
Meaning or competency?
All three paradoxical examples suggest that there may be more to be said about
the role of memory, practice and competency than is sometimes acknowledged in
discussions of mathematics education.
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The case of my enrichment students is the weakest example, since it is purely
anecdotal. Even so, it might cause one to wonder whether we can be confident that
education dominated by discovery and gradual construction of meaning out of familiar
material will automatically lead to the level of competency achieved by more traditional
methods. If the appropriate mathematical ideas do not become part of students’ toolkits of
competencies, it makes no sense to expect students to use them playfully and
imaginatively.
The success of the JUMP tutoring program suggests that for some students
mechanical competency is necessary before meaning can be put in place - that holding
both meaning and mechanical rules in mind together is too difficult for them.
The success of the East Asian mathematics student suggests that repeated practice
not only leads to the mastery of mechanical process but can even facilitate the
construction of meaning.
A more nuanced assessment of the Chinese learner.
Numerous recent studies (Kennedy 2002; Kember 1996; Leung 2001; Leung and
Park 2002; Wong; Wen and Marton 2002; Cooper 2004; Leung 2005; Marton, Wen and
Wong, 2005) have indicated that the situation is much more complex than suggested by
the earlier quotes about Chinese learners. Many of them have come to the conclusion
that the distinction between deep and surface learning, which originated with Marton and
Saljo (1976a, 1976b), is misused when surface learning is automatically associated with
rote. Leung and Park (2002) write:
The process of learning very often starts with gaining competence in the
procedure, and then through repeated practice, students begin to understand the
concepts behind the procedures. (p. 127).
Kennedy (2002) describes the Chinese student’s understanding of memorization as
follows:
Memorization has never been seen as an end in itself but as a prelude to deeper
understanding—mentally ‘photocopying’ texts, committing them to memory,
enabled the ‘learner’ to savour and reflect on them later, and, finally, to integrate
them with his/her prior learning and experience. (p. 433).
This is reminiscent of the reasons some of us will occasionally commit good
poetry to memory, just so we can savour the words again and again, and see new
significance in them each time we do. The poem becomes an intellectual tool on which
our imagination can create variations and which helps us interpret the world around us.
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Memory, metaphor, and meaning
For a deeper exploration of the three paradoxical situations with which we started
the paper, we will examine them in the context of two concepts found in the literature on
mathematics education. Each of these sheds useful light on the way mathematics is
learned. The first of these is the conceptual metaphor described in (Lakoff and Núñez,
2000). The second is the concept of encapsulation in the work of Dubinsky and his
colleagues (Asalia et al., 2004; Dubinsky, 2000; Dubinsky, 2005a; Dubinsky, 2005b).
Lakoff and Núñez (2000) claim that Mathematics operates on the world by
metaphor. Here metaphor is to be understood as a mental tool, a cognitive mechanism,
that takes the form of a mapping between two domains, a source domain and a target
domain. In most cases, particularly in the metaphors we use to give meaning to
arithmetic, the source domain is relatively concrete, grounded in fundamental schemas;
and the target domain is more abstract, the set of rules by which we manipulate numbers.
In each metaphor, the direction from source to target seems to indicate the order in which
things are learned. For example, the “Arithmetic as Object Collection” metaphor (one of
their “grounding metaphors”) enables us to understand addition and subtraction because
of our experience collecting and sorting objects. The metaphor relating arithmetic to
measuring sticks (another of the four grounding metaphors) enables us to give meaning to
the manipulation of fractions. Strangely, in their discussion of algebra, the order from
familiar situation to theoretical or symbolic description is reversed. In that case, the
authors posit the existence of a fundamental metaphor “algebra as essence” which they
relate back to the concept of essence in Greek philosophy. They suggest that somehow
our tendency to see structure (“essence”) in our world presents itself as a desire to
construct an axiom-based abstract theory. In this case, one does not get the same sense
that the more concrete instances give meaning to the symbols of the abstract theory, or
that there is a causal relationship in the construction of the metaphor:
The cognitive structure of an algebraic entity (e.g., a group) is not an essence that
inheres in other cognitive structures of mathematical entities (e.g., a collection of
rotations). Rotations are conceptualized independently of groups, and groups are
conceptualized independently of rotations. [italics mine] (Lakoff and Núñez,
2000, p. 119)
I am not clear as to why the conceptualization of groups takes place
independently of rotations, when the authors take pains to stress that the concept of
fraction derives its meaning only from metaphors based in object gathering, object
construction, measurement, and motion. I suspect it signals a truth we know instinctively,
and one suggested by the paradoxes described at the start of this paper: there is more than
one way to learn something.
If the direction of a conceptual metaphor in Lakoff and Núñez (2000) indicates
the way the cognitive structure comes about (that is, the source domain, together with a
capacity for metaphor construction leads to an understanding of the target domain), their
discussion suggests that algebra could be taught completely in the abstract and then
mapped to (more) concrete examples. Algebra is sometimes taught that way, but in most
6
cases it would not be considered good pedagogy. At the opposite extreme, it is possible
to introduce so many disparate examples of groups before introducing the group concept
that the students get confused. Generally, a middle road that combines a judicious
provision of examples and a building of theoretical structure and language is the best
choice. Both are possible: It is possible to construct the theory in purely symbolic form
and then to build the metaphor that connects it to a specific example; it is also possible to
look at examples and see bits of structure in them and then use those to build a formal
theory.
If the flow of meaning from source domain to target domain is ambiguous in the
case of the metaphors involved in abstract algebraic structures, one should also be able to
imagine the construction of the metaphor reversed in the case of fractions. Nothing in the
description of mathematical understanding in terms of conceptual metaphors prevents our
going from the formal manipulation of fractions to applications involving the
constructing and dividing of objects. It is difficult to imagine that the desire to impose or
discover “structured essence” does not also play a role in the creation of the theory of
fractions.
The success of the JUMP program suggests that if motivational issues are lined
up, fractions can be learned as a formal system and that for some students fractions can
only be learned that way. The formal theory of fractions is quite difficult for a young
child, and visual or physical explanations for, say, the equivalence of fractions are quite
complicated. Some students don’t find it possible to hold both of these in mind at once;
for them, it may in fact be better to suspend the need to attach meaning to the symbols.
This need not imply that meaning is neglected. Mighton (2003/2007, 2007) claims it
follows later, rather easily. It also does not mean that this way of learning fractions is not
embodied or based on experience. After all, a routine of doing worksheets in which the
formal addition and subtraction of fractions is broken up into steps small enough for even
a weak student to see the patterns and learn the method is itself a physical experience that
by its repetition creates an (embodied) change in the brain. All of us, when we do a hand
calculation involving fractions, focus only on the patterns of numbers, multiplication
symbols and fraction lines to do the calculation. Until the calculation has been
completed, we ignore the conceptual metaphor in the background that connects our
calculation to the problem we are trying to solve.
Teaching fractions mechanically, without first supplying all the elements of
meaning, also need not be seen as a violation of the constructivist metaphor for
mathematics education, though it does violate the way it is normally understood. When
carpenters build a house, they quite commonly assemble one of the walls separately,
before putting it in place on the foundation. In the same way, a theory may be prefabricated, as it were, before it is connected to the ideas that will eventually give it
meaning.
In the case of the Chinese learner, the same reality seems to hold. In a classroom
that emphasizes rote learning, the focus is on the creation of the target domain without
first attending to the metaphor. Once that target domain is a secure part of the student’s
understanding, an innate tendency to think metaphorically takes over. The structure of
7
the target domain is held against experience as a template for understanding. The
connections that result give meaning both to the target domain and to the more concrete
objects that are mapped to it.
Encapsulation
In a series of papers, Dubinsky and his colleagues developed what they refer to as
the APOS theory of mathematics education (Dubinsky, 2000; Dubinsky and McDonald,
2001; Dubinsky, 2005a; Dubinsky, 2005b). The acronym stands for Actions, Processes,
Objects and Schemas. Dubinsky and his colleagues see these four as the (repeated) steps
by which mathematical understanding proceeds. The process, as they describe it, is a
process of repeated abstraction. They see abstraction as essential to mathematics and key
to many kinds of understanding in the modern world (Dubinsky, 2000).
When students begin the process of learning a mathematical concept, they can do
little more than manipulate objects in very concrete ways:
Understanding a mathematical concept begins with manipulating previously
constructed mental or physical objects to form actions. (Dubinsky, 2000, p. 293)
This first stage is an exploration stage. Actions are transformations on objects that are
perceived by the learner as external and indicated by external cues. When the student
begins to reflect on the action and to describe it without actually carrying it out, the
student is forming a process conception of the action. The process conception is
perceived as internal to the learner and under his control. Dubinsky and his co-authors
(Asiala et al., 2004) refer to this movement from action to process as “interiorization.”
As the student continues this reflection, he or she begins to see a process as a totality, to
combine several processes, and to act on them. When this happens, the process has
become an object for the student, and Dubinsky and his colleagues say that the process
has been “encapsulated” as a cognitive object. The cyclic process in which actions
become processes and then objects, which in turn can be acted on, is indicated by Figure
1, which is based on Figure 2 in Asiala et al. (2004).
Figure 1: Constructions for Mathematical Knowledge
8
According to the authors, the learning of mathematics proceeds by cycling
through these stages in clockwise fashion. A mathematical topic involves many actions,
and processes, which have to be linked and organized into a coherent framework, called a
schema. Once the relevant mathematics has been learned, it seems that, to solve a
problem, the student moves back and forth between object and process, encapsulating and
de-encapsulating as the situation demands.
In APOS theory, the way in which mathematics is learned seems more
unambiguously unidirectional than in the analysis presented by Lakoff and Núñez (2000).
Conceptual metaphors seem to allow more flexibility in the way they are formed.
Dubinsky is an experienced research mathematician whose goal is to describe the process
whereby mathematics creates successive levels of abstraction and the difficulties implicit
in teaching this, especially at the university level. In the analysis presented in Asiala et
al. (2004), the process of abstraction seems to be driven by tendencies extracted from the
perceived objects, and less thought is given to the possibility of pre-existing cognitive
tendencies or potentialities in the human mind. Dubinsky puts it this way:
Abstraction, in general, is the determination in a given situation, which may be a
mathematical object, a procedure, or a combination of the two, of what is essential
in a component of the situation. In mathematical abstraction, one generally
expresses this essence in some systematic manner, such as formal language or a
set of axioms. (Dubinsky, 2000, p. 291)
Their description of the process whereby abstraction proceeds is insightful.
Especially instructive is their emphasis on the importance (and difficulty) of
encapsulating a process to create a new object that can then be perceived as external by
the student and acted on in turn. In particular, mathematics cannot grow if the student’s
activity is always in the nature of exploration; competency is an essential step to further
levels of abstraction. The absence of appropriate encapsulation is the key, I believe,
when grade 7 or 8 students in my enrichment mathematics program find it difficult to
play with the ideas presented to them. For example, if students cannot think of the long
division algorithm as an object, or if they have not encapsulated the idea of decimal
expansion, they cannot benefit from discussions about the relationship between fractions
and repeating or terminating decimals.
The part of the analysis of Dubinsky and his colleagues that is not well developed
is the construction of schemas. It may be that a closer examination of what the
construction of schemas entails will also reveal a good deal of freedom in the order in
which that is done. I suspect that to hold the understandings of Lakoff and Núñez and
that of Dubinsky and his colleagues together, we should look for conceptual metaphors to
play a role in the construction of these schemas.
Thinking specifically of the way in which we learn fractions, two separate actionprocess-object sequences can be considered quite independently of each other. On the
one hand, when the objects are paper circles representing pizzas, a student can act on
9
these as objects by cutting them in order to divide one into two, into three, etc. A student
can also be given two paper pizzas, one divided into two equal pieces and the other into
three, and a third into six, and then be asked how many pieces of the third pizza would
contain the same amount of pizza as one piece of the first, or one of the third. After a
number of exercises of this sort, students will begin to see and anticipate the pattern, and
interiorize the action. It is now a process. This process can then be acted on when the
student is asked how many pieces of the third pizza will equal one piece of the first plus
one of the second. The transformation from a fraction to an equivalent fraction has
become a cognitive object for the student. In effect, the student is learning about
equivalent fractions and how to add two fractions with different denominators, without
ever using the usual words and symbols. Of course, one would not normally take it this
far without introducing symbols.
Alternatively, the student could learn rules about adding such fractions formally,
as in the JUMP program, with all the meaning removed. At first, students follow the
very precise steps outlined in the JUMP manual: write in the multiplication signs, put in
the correct multipliers, and so on. This is action. The student is doing something
unfamiliar prompted by external cues (those of the tutor or the work book). After a page
or so of this, the student sees how to do all the steps and can reflect on it: “first I have to
do this, and then that …”. Eventually, when the student has learned to add fractions with
unlike denominators, as well as those with like denominators, and the special case when
one denominator divides the other, the student can reflect on the various possibilities:
“When one denominator divides the other, then I have to …”. The processes have
become objects. It is possible to imagine these processes taking place without the
students’ attaching meaning to any of them.
Thus, (potentially) two separate sets of objects (formal fractions and divided
paper disks) each form a piece of theory in the student’s mind. At some point (and in
most cases much sooner than in my description), these two sets of objects should be
assembled into a schema. This is where the symbols receive meaning. It would be noted
that, if properly mapped, the symbols behave just like the divided and re-assembled
pieces of paper. Lakoff and Núñez do not say much about the role of written symbols in
their discussion of metaphor but connecting concepts to symbols seems also to involve a
kind of conceptual metaphor. In most teaching situations, we would build this metaphor
in, along with the interiorization and encapsulation process, but in some cases we may
decide, for pedagogical reasons, not to.
The understanding of mathematics as successive abstractions made possible by
interiorization and encapsulation of actions helps to clarify why the students in my
mathematics enrichment class may have more difficulty engaging enrichment
mathematics than students who were taught in more traditional fashion. Playful,
imaginative engagement with mathematics requires well-established cognitive objects on
which to operate. These objects have to be created by a process of interiorization and
encapsulation in which memory plays a large role and from which rote is not necessarily
excluded as a device. In the learning situation, the teacher has to decide whether
appropriate motivation is in place for whatever method is used, but the goal has to be
competency as well as understanding.
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Writers in other fields, such as chess (Ross, 2006) and language instruction
(Myles, Hooper and Mitchell, 1998) have similarly recognized that memory and a
deliberate effort to commit material to memory play an important role in understanding.
Memory and Imagination
The word imagination is used in a variety of ways. In general, we associate it
with novelty, creative power, and ability to form mental images. I believe that the most
useful way to think about imagination is as the ability to make surprising and fruitful
connections between mental constructs and experience. This facility enables us to
interpret experience and is fundamental to interpretation. The philosopher Nigel Thomas
defines imagination this way:
Imagination is what makes our sensory experience meaningful, enabling us to
interpret and make sense of it, ... It is what makes perception more than the mere
physical stimulation of sense organs. (Thomas)
Gibbs sees the connections made in imaginative thinking as metaphors:
…human imagination is also an unconscious process that uses metaphor to map
aspects of long-term memory onto immediate experience. (Gibbs, 2005, p. 66).
Others, noting the importance of mental structures (cognitive objects in the
language of Dubinsky and cohorts) to make mapping possible, prefer to think of these
cognitive structures themselves as the images:
Some recent authors have recommended that the term ‘imagery’ should not be
understood as referring to a form of subjective experience, but, rather, to a certain
type of "underlying representation" (Block, 1981a, Introduction; Block, 1983a;
Kosslyn, 1983; Wraga & Kosslyn, 2003; and see also, Dennett, 1978). Such
representations are "mental" in the sense now commonplace in cognitive science:
i.e., they are conceived of as being embodied as brain states, but as individuated
by their functional (and computational) role in cognition. (Thomas)
Memory is not inimical to imaginative thinking; it is essential to it. Without
competency, even mechanical competency, it is not possible to think imaginatively.
While the long term memory required for imaginative thinking in mathematics must
include many inter-connecting conceptual metaphors, the way these metaphors are
constructed in a classroom situation depends as much on the context, especially
motivational elements, as on the nature of the metaphor. In particular, no unique way to
construct them exists.
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