DESC_385.fm Page 452 Monday, October 7, 2002 4:25 PM
Developmental Science 5:4 (2002), pp 452–466
PAPER
Blackwell Publishers Ltd
From sharing to dividing: young children’s understanding
of division
From sharing to dividing
Sarah Squire and Peter Bryant
Department of Experimental Psychology, University of Oxford, UK
Abstract
Children have particular difficulty with division problems, as compared to sharing problems. An inability to discriminate between
the dividend, divisor and quotient might contribute to their difficulty with division. This study investigates whether young children
(5–9 years) were able to discriminate between the divisor and quotient in simple division problems that were modeled for them.
Children were presented with partitive and quotitive division problems in which the dividend was grouped either by the divisor
or by the quotient. The children showed a very different pattern of results in the partitive and quotitive problems; they found it
easier to identify the answer (quotient) when the dividend was grouped by the divisor in partitive problems and by the quotient
in quotitive problems. It is argued that children rely on a schema of action of creating ‘portions’ when they first learn about
division, and that the ‘portions’ produced by sharing are different in partitive and quotitive problems. We discuss this finding in
terms of the importance of problem representation, children’s schemas of action and mental models.
In the study of children’s mathematical reasoning, there
is increasing interest in the way in which children initially approach mathematical problems. This includes
the way in which young children may use their informal
experiences in solving mathematical problems (e.g. Carpenter & Lehrer, 1999), the way in which children may
initially model mathematical problems using concrete
materials or drawings (e.g. Carpenter, Ansell, Franke,
Fennema & Weisbeck, 1993; Carpenter & Moser, 1982;
Murray, Olivier & Human, 1992) and the mental models
that young children may have for solving mathematical
problems (e.g. Huttenlocher, Jordan & Levine, 1994).
There are several reasons for the interest in children’s
initial knowledge and informal strategies. One is that
informal knowledge may be a key basis for learning
formal knowledge (Baroody, Gannon, Berent & Ginsburg,
1984; Carpenter, Fennema & Franke, 1996; Hughes,
1986) and it is important that children make connections
between informal and formal reasoning (e.g. Nunes,
Schliemann & Carraher, 1993; Schoenfeld, 1991). By
studying children’s informal strategies, we may establish
important links between conceptual and procedural
knowledge (e.g. Carpenter, Franke, Jacobs, Fennema &
Empson, 1997; Hiebert, 1986; Hiebert & Carpenter,
1992; Rittle-Johnson & Siegler, 1998) and between
children’s informal knowledge and the meaning of mathematical symbols (e.g. Mack, 1995). This research is also
of central importance to the development of teaching
strategies. It has been argued that children who use
invented strategies based on their informal knowledge,
and children who are encouraged to provide explanations for mathematical problems prior to learning standard algorithms are generally better at extending their
knowledge to new situations than students who initially
learn standard algorithms (Carpenter et al., 1997). Consequently, an important part of mathematics learning
and teaching is to build on children’s intuitive strategies
(e.g. De Abreu, Bishop & Pompeu, 1997; De Corte,
Greer & Verschaffel, 1996; Fennema, Carpenter, Franke,
Levi, Jacobs & Empson, 1996) and to mediate between
children’s concrete representations and abstract mathematical problems (Gravemeijer, 1997). In general, very
little is known about the connection between children’s
early informal experiences in mathematics and their learning of mathematics in school (Bryant, 1997), whereas we
know much more about the connections between children’s
early experiences and their success in learning to read
(e.g. Bradley & Bryant, 1985).
Most of the research in young children’s mathematics
has concentrated on children’s approaches to addition
Address for correspondence: Sarah Squire, Department of Experimental Psychology, University of Oxford, South Parks Road, Oxford OX1 3UD,
UK; e-mail: [email protected]
© Blackwell Publishers Ltd. 2002, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.
DESC_385.fm Page 453 Monday, October 7, 2002 4:25 PM
From sharing to dividing
and subtraction problems. There is evidence that children
can construct methods for some addition and subtraction
problems without explicit instruction (e.g. Carpenter &
Fennema, 1992; Carpenter & Moser, 1982; Carraher,
Carraher & Schliemann, 1987) but that they are quite
sensitive to the context in which these problems are posed
(e.g. Hughes, 1986; Riley, Greeno & Heller, 1983). Such
research has led to the proposal that children develop
mental models of situations which allow them to solve
non-verbal addition and subtraction tasks before they
acquire the conventional symbols for arithmetic and
mathematical problems (Huttenlocher et al., 1994; Levine,
Jordan & Huttenlocher, 1992). In all of this research,
children demonstrate more understanding than one might
observe if one were merely to present them with formal
additive problems in which a calculation was required.
However, despite this progress in understanding
children’s additive reasoning, far less attention has been
paid to children’s initial understanding of multiplicative
reasoning, especially division. Although multiplicative
reasoning might require a qualitative change in thinking
from additive reasoning (Confrey, 1994; Piaget, 2001/
1977; Steffe, 1994), it would not be surprising if children
also had some informal understanding of multiplicative reasoning and demonstrated informal knowledge
and effects of context similar to those that have been
observed in addition and subtraction.
We have some evidence that children as young as 4
years of age might have some understanding of fraction
concepts if the use of conventional symbols is avoided
(Goswami, 1989) and if they are given non-verbal tasks
(Mix, Levine & Huttenlocher, 1999). This research is
clearly relevant to children’s understanding of division
because concepts such as multiplication, division, fractions and ratios all fall within the Multiplicative Conceptual Field – ‘a set of problems and situations for the
treatment of which concepts, procedures and representations of different but narrowly interconnected types
are necessary’ (Vergnaud, 1983, p. 127). However, we
have much more information about older children’s and
adults’ understanding of division problems (e.g. Lago,
Rodriguez, Zamora & Madrono, 1999; Silver, Shapiro &
Deutsch, 1993; Simon, 1993; Tirosh, 2000) than we have
about the first stages of the problem-solving processes in
multiplication and division word problems (Vershaffel &
de Corte, 1997).
Children’s early understanding of division is potentially a very interesting topic for two reasons. The first is
that there is an informal, everyday activity – sharing – that
may be important in children’s initial understanding of
division. Sharing is dividing, in the sense that to share a
quantity successfully is to divide a dividend into equal
quotients. Teachers usually introduce division as a form
© Blackwell Publishers Ltd. 2002
453
of sharing and this approach to teaching division can be
justified because children are able to share quite proficiently. Even children in the Reception class (5-year-olds)
are quite familiar with the phrase ‘share out’ (although,
in the UK, the phrases ‘divide’, ‘divided by’ and ‘divided
into’ are not introduced until Year 2, i.e. around 7 years;
see Department for Education and Employment, 1999).
A number of studies have demonstrated that most 4and 5-year-old children know how to share out quantities in a distributive (one for A, one for B) manner, and
that by 5 years they tend to understand quite a lot about
the basis of this procedure (e.g. Desforges & Desforges,
1980; Frydman & Bryant, 1988; Miller, 1984). In addition
to children’s ability to share, it has been shown that young
children are able to model division problems using concrete materials well before they have been formally taught
about division (Carpenter et al., 1993; Correa, 1994) and
that children’s initial strategies tend to reflect the action
described in the problem (e.g. Marton & Neuman, 1996).
Children’s proficiency with sharing means that this activity is a good candidate for the ‘schema of action’ (Piaget,
1972/1947) from which an understanding of division might
develop.
The second reason why children’s understanding of
division is interesting is that the division problems that
children encounter at school are very difficult for them.
They find division much harder than addition or subtraction, even when quite small numbers are involved
(e.g. Brown, 1981). The reason for the apparent contrast
between children’s success in sharing and their difficulties
with division problems at school must lie in differences
between what children have to do when they share and
when they divide. The purpose of a division problem is
to work out the quotient; somebody trying to solve a
division problem must understand that s/he has to
divide the dividend by the divisor to produce the quotient. This does not, of course, have to imply an understanding of the terms dividend, divisor and quotient.
Correa, Nunes and Bryant (1998) have argued that in
order to have some understanding of division, children
should demonstrate an understanding of the inverse
relation between the divisor and the quotient. They
investigated whether children aged between 5 and 7
years had any understanding of this relation by presenting them with problems in which food had to be shared
at two parties – a red party and a blue party. There was
the same given quantity of food at each party but there
was a different number of recipients in each case (e.g. 12
sweets at each party, but three recipients at the blue
party compared with four recipients at the red party).
The children had to make a judgment about the relative
size of the portions in each case. In order to be successful in these tasks, children had to understand that the
DESC_385.fm Page 454 Monday, October 7, 2002 4:25 PM
454
Sarah Squire and Peter Bryant
more recipients there were, the less food each recipient
would receive. Correct answers thus reflected some
understanding of the inverse relation between the divisor
and the quotient. It was found that 5-year-olds had an
imperfect understanding of the inverse relation between
divisor and quotient (only 30% performed better than
chance) but that the 7-year-olds performed reasonably
well. Other studies, using slightly different tasks, have also
documented similar improvements with age (Kornilaki,
1999; Sophian, Garyantes & Chang, 1997).
Although these studies provide important information
about children’s understanding of the inverse relation,
they also raise other questions that require further
research. One question is: Are young children able to
distinguish between, and recognize the role of, different
terms in division problems? This is a key aspect of
understanding the meaning of an arithmetical operation:
‘It is important that students understand what each
number in a multiplication or division expression represents’ (National Council for Teachers of Mathematics
(NCTM), 2000). Children may be successful in distinguishing between the dividend, divisor and quotient and
recognizing the role of each in the division problem even
though (like the children in both Correa et al.’s (1998)
and Sophian et al.’s (1997) studies) they are unfamiliar
with the terms dividend, divisor and quotient and are
unable to compute the answer to the division problem
when it is posed in a conventional way. We suggest that
the ability to distinguish these three terms (dividend,
divisor and quotient) from each other, and to recognize
the role of each in the division problem, is an important
starting point in understanding division.
It is quite easy to test children’s awareness of the
distinction between these terms. The most direct way to
do this is to model a division problem with concrete
materials and to ask the child to identify the quotient.
For example, one could model the division problem 12 ÷ 4
by having 12 sweets and 4 recipients (e.g. 4 children). Here
the dividend is the 12 sweets and the divisor is the
number of recipients of the sweets. This dividend (12)
could then be broken up into groups of sweets in one of
two ways. Either the sweets could be grouped by the
divisor (the recipients) so that there is one group of
sweets for each recipient, or they could be grouped by
the quotient so that in each group there is one sweet for
each recipient. In the first case, the sweets are formed
into 4 groups of 3 (grouping by divisor); the fact that
there is one group of sweets for each recipient means
that it is similar to the end-point of sharing. In the second case (grouping by quotient), the sweets are formed
into 3 groups of 4; there is one sweet in each group for
each recipient and each group of sweets is like a round
of sharing. In both cases the quotient is represented in
© Blackwell Publishers Ltd. 2002
the model; the quotient is the number of sweets in each
group in the first case (grouping by divisor) and the
number of groups of sweets in the second case (grouping
by quotient). If a child understands the operation of
division s/he should be able, when asked about the quotient, to identify it correctly in both cases.1
The need to identify the quotient and to discriminate
it from the divisor becomes clear when division problems
are set in a specific context. This context can take one of
two forms. The context for the problem that we have
just described (and the context of Correa et al.’s problem described earlier) is partitive. In partitive problems,
a dividend is shared equally among a certain number of
recipients and the size of the portion (the quotient)
depends on the number of recipients (the divisor). For
example, if there are 12 sweets and there are 4 recipients
(divisor) then the size of each portion (quotient) must
be 3 sweets. The other context is quotitive. Here the
dividend is divided into fixed portions (divisor) and the
number of recipients (quotient) depends on the size of
the portion. Here, if there are 12 sweets and they are
distributed in bags of 4 (the divisor), then there can only
be 3 recipients (the quotient).
Of course, the problems do not have to be about sharing sweets; the same problem (12 ÷ 4) could be presented
as both a partitive and a quotitive division problem
concerning children sitting around tables. The partitive
problem (for 12 ÷ 4) would be: ‘There are 12 girls and 4
tables. If the girls have to sit around the tables so that
there is the same number of girls around each table, how
many girls will there be on each table?’ The quotitive
problem for 12 ÷ 4 would be: ‘There are 12 girls and they
have to sit so that there are 4 girls on each table. If all of
the children sit down, how many tables will we need?’ In
the partitive problem, the divisor is the number of recipients (tables) and the quotient is the size of the portion
(number of children on each table). The groups coincide
with the ‘final equal portions’ (the end-point of sharing)
when the dividend is grouped by the divisor (i.e. in the
example above, when there are 4 equal sized groups of
girls, one for each table). By contrast, in the quotitive
problem, the divisor is the size of the fixed ‘portion’
(number of children on each table), and the quotient is
the number of recipients (tables). In this case, the groups
coincide with the final portions when they are grouped by
the quotient (i.e. in the example above, when the girls are
arranged into 3 groups, with 4 girls in each group).
This leads to a clear prediction about a young child
who can share perfectly but has an imperfect grasp of
1
Again, it should be emphasized that we are not interested in whether
the child actually knows the terms ‘quotient’, ‘divisor’ and ‘dividend’
(indeed, we do not anticipate that young children would understand
these terms).
DESC_385.fm Page 455 Monday, October 7, 2002 4:25 PM
From sharing to dividing
the role of the divisor and quotient in a division problem. If one were to give such a child the kind of concrete
model of a division problem that we have described, and
ask him/her to identify the quotient, that child would
manage much better when the objects were grouped
into the final equal portions (the end-point of sharing)
than when they were not. It should be apparent from
the examples described that the role of the size of the
portion and of the number of recipients actually reverses
in partitive and quotitive problems. The child has to
recognize that the significance of these two variables
changes with the context in order to be able to identify
the quotient in the concrete model.
The change in context is one reason why the distinction
between partitive and quotitive division is of interest.
Although the underlying mathematical structure is the
same for partitive and quotitive problems, it is possible
that partitive and quotitive division problems may differ
in difficulty (the evidence for this to date is very mixed,
e.g. Bourgeois & Nelson, 1977; Correa et al., 1998;
Fischbein, Deri, Nello & Marino, 1985; Gunderson, 1955;
Zweng, 1964). The idea that the same problem may vary
in difficulty depending on the context in which it is
posed ties in to extensive research in cognitive psychology on isomorphic problems and the role of problem
representation in accounting for problem difficulty and
expert–novice differences (e.g. Kotovsky, Hayes &
Simon, 1985; Kotovsky & Simon, 1990; Novick, 1988;
Reed, Ernst & Banerji, 1974). The distinction between
partitive and quotitive problems is also of interest because it is possible that the two types of division relate
differently to sharing. We will return to these issues in
the Discussion.
To our knowledge, there is no evidence on children’s
ability to detect the quotient and to distinguish it from
the divisor. It is an interesting way to study children’s
understanding at an age where they are able to model
problems by manipulating concrete materials (Carpenter
et al., 1993) and are beginning to understand something
about the inverse relation between divisor and quotient
(Correa et al., 1998; Sophian et al., 1997), but are as yet
unable to compute the answers to the problems, because
it provides insight into the conceptual structures that
children might be using without requiring them to have
knowledge of formal algorithms. Although the children
are not required to carry out an arithmetical computation
in these tasks, such problems are mathematical in the
sense that they require the child to manipulate numbers,
recognize the relevant variable and think about the link
between the division word problem and the concrete
material with which they are presented. Fennema, Sowder
and Carpenter (1999) state that ‘included in the definition
of school mathematics is also the way students think
© Blackwell Publishers Ltd. 2002
455
about it, understand it, and manifest their understanding’
(p. 186) and we would argue that the types of modeling
problems that we have described begin to investigate
these aspects of division.
The aim of our experiment was therefore to use
models of division problems to investigate children’s
understanding of division and to test the prediction that
young children find it easier to identify the quotient
when the portions are grouped by the divisor in partitive
problems and by the quotient in quotitive problems.
Method
Participants
One hundred and eighteen children from two state
schools (with mixed socioeconomic status) took part in
this study. There were 29 children (13 male and 16
female) in Year 1, 31 (15 male and 16 female) in Year 2,
30 (18 male and 12 female) in Year 3 and 28 children (11
male and 17 female) in Year 4. The mean age of the
children in Year 1 was 5 years 7 months (ages ranged
from 5 years 2 months to 6 years 2 months), the mean
age of the children in Year 2 was 6 years 8 months (ages
ranged from 6 years 2 months to 7 years 1 month), the
mean age of the children in Year 3 was 7 years 9 months
(ages ranged from 7 years 3 months to 8 years 2 months)
and the mean age of the children in Year 4 was 8 years
11 months (ages ranged from 8 years 3 months to 9
years 7 months). A further child, who failed the pre-test,
was excluded from the study.
Design
Each child was seen individually in two testing sessions;
the second session took place within a week of the first
session. In the first testing session, the child was given a
sharing pre-test, which s/he had to pass in order to take
part in the main study. The child was then given eight
baseline trials, which included both partitive and quotitive division problems. The scores in these baseline trials
were used to match children into two experimental
groups. In the second session, one experimental group
received partitive division problems and the other group
received quotitive division problems.
Materials and procedure
Pre-test
All children received a pre-test. This pre-test was loosely
based on that used by Correa et al. (1998) and was
DESC_385.fm Page 456 Monday, October 7, 2002 4:25 PM
456
Sarah Squire and Peter Bryant
designed to ensure that the children who participated in
the study were able to share and had some understanding of numerical equivalence. An A4 sized card was presented, on which there was a picture of four identical
girls (7 cm in height) in a line at the top of the page. The
child was also given eight red circular pieces of card,
measuring 11 mm in diameter, which were the pretend
‘sweets’ to be shared. The child was asked to share the
sweets out fairly between the girls. The pre-test was
the only task in which the child was able to manipulate
the materials. The experimenter asked the child how
many sweets had been given to the first girl, then covered
and removed the sweets and asked how many had been
given to the other girls (numerical equivalence).
Baseline trials
Children who passed the pre-test were then given eight
baseline trials in the same session. Figure 1a shows the
type of picture that the child was presented with in a
partitive baseline trial for the problem 12 ÷ 4. An A4
sized card was presented with pictures of girls (in blue)
positioned randomly on the page and pictures of tables
at the top of the page. Each picture of a girl measured
2.4 × 1 cm and each picture of a table measured 4.3 × 2.5
cm. The number of girls depended on the dividend and the
number of tables depended on the divisor in the problem.
Round tables were used to avoid children’s answers being
influenced by the number of sides that the table had. In
each partitive baseline trial, the child was presented with
the picture for a particular problem and told how many
girls and how many tables there were in the picture. The
child was then asked how many girls s/he thought there
would be on each table, if the girls sat down at the tables
so that there was the same number on each table.
Figure 1b shows the type of picture that the child was
presented with in a quotitive baseline trial for the sum
12 ÷ 4. The pictures of the girls and of the tables were the
same size as in the partitive baseline trials. An A4 sized
card was presented with pictures of girls (in blue) positioned
randomly on the page and a picture of one table at the top
of the page, with a certain number of girls sitting around
it. The total number of blue girls depended on the dividend and the number of girls around the table depended
on the divisor in the problem. The girls around the table
were dressed in red rather than blue, so that they could
be distinguished from the pictures of girls waiting to sit
down (but note that the red and blue colors are not
shown in Figure 1b). It was explained to the child that
the blue class had to sit at the tables in the same way as
the red class (e.g. with three girls at each table). On each
quotitive trial, the child was presented with the picture
for a particular problem and told how many girls there
© Blackwell Publishers Ltd. 2002
Figure 1 Schematic representation of (a) a baseline partitive
division problem for the problem 12 ÷ 4. (b) and the equivalent
baseline quotitive division problem for the problem 12 ÷ 4.
Note that the girls around the table in (b) were a different color
from the other girls.
were, and how many girls had to be seated on each table.
The child was then asked how many tables would be
needed for all of the girls in the blue class to sit down.
The eight problems presented in the baseline trials
were: 12 ÷ 2, 12 ÷ 3, 12 ÷ 4, 12 ÷ 6, 15 ÷ 3, 15 ÷ 5, 20 ÷ 4
and 20 ÷ 5. Four of these problems (12 ÷ 2, 12 ÷ 4, 15 ÷ 5
and 20 ÷ 4) were partitive trials and the other four problems (12 ÷ 3, 12 ÷ 6, 15 ÷ 3 and 20 ÷ 5) were quotitive
trials. The problems were presented in the same random
order for each child. Half the children received the partitive tasks first and the remainder received the quotitive
tasks first.
A time limit of approximately 25 s was imposed on
each baseline trial. After this time, the child was
prompted, and if s/he still failed to answer the experimenter proceeded to the next trial. No feedback was
given about the correctness of the response. The scores
achieved in the baseline trials were used to match the
DESC_385.fm Page 457 Monday, October 7, 2002 4:25 PM
From sharing to dividing
(a) Grouping
-byDivisor
(b) Grouping
-byQuotient
(a) Grouping
-byDivisor
457
(b) Grouping
-byQuotient
Figure 2 Schematic representation of a partitive division
problem (12 ÷ 4) in (a) the Grouping-by-Divisor, and (b)
Grouping-by-Quotient condition.
Figure 3 Schematic representation of a quotitive division
problem (12 ÷ 4) in (a) the Grouping-by-Divisor, and (b)
Grouping-by-Quotient condition. The girls around the table in
(b) were a different color from the other girls.
children to each of the two types of division problems
(partitive and quotitive) in the experimental trials.
quotitive) and the within-subject factor was Condition
(Grouping-by-Divisor vs. Grouping-by-Quotient).
Figure 2 illustrates the type of picture presented in the
Grouping-by-Divisor and Grouping-by-Quotient condition
of a partitive experimental trial. Figure 3 illustrates the
type of picture presented in the Grouping-by-Divisor
and Grouping-by-Quotient condition of a quotitive
experimental trial. Note that, in a particular trial, there
was only ever one vertical line of girls, positioned down
the centre of the page. The pictures of the girls and the
tables were the same size as in the baseline trials but this
time the pictures of the girls were positioned in small
equal sized groups in a vertical line, with a 2 cm space
between each group. This required larger sheets of card,
measuring approximately 29 × 64 cm. In the Groupingby-Divisor condition, the number of groups was the same
as the value of the divisor and the number in each group
Experimental trials
The experimental trials were presented in the second
session. Each child received only one type of division
problem – partitive or quotitive – in the experimental
trials. The key difference between the baseline trials and
the experimental trials was that in the baseline trials
the girls were not grouped whereas in the experimental
trials the girls were grouped either by the divisor (the
Grouping-by-Divisor condition), or by the quotient (the
Grouping-by-Quotient condition). Each child received
four problems in each condition. In summary, the
between-subjects factors in the experimental tasks were
Year (1, 2, 3 or 4) and Type of division (partitive vs.
© Blackwell Publishers Ltd. 2002
DESC_385.fm Page 458 Monday, October 7, 2002 4:25 PM
458
Sarah Squire and Peter Bryant
was the same as the value of the quotient. In the Groupingby-Quotient condition, the number of groups was the
same as the quotient and the number of girls in each
group was the same as the value of the divisor.
The experimental tasks were conducted in one corner
of the classroom. The child sat beside the experimenter
and the materials were clipped onto a large board placed
in front of them. At the beginning of each session, the
experimenter reminded the child about the problems
that s/he had solved in the previous session concerning
children sitting around tables. The experimenter then
explained that the child was going to solve some similar
problems again, but that this time the girls were in
groups, ready to sit down for dinner.
The experimenter began a particular trial by placing
the relevant picture for that trial onto the board facing
the child. She told the child how many girls there were,
and either how many tables there were (partitive), or
how many girls had to sit around each table (quotitive).
The experimenter then asked the child two questions:
Table 1 Mean scores in the baseline trials according to year
(maximum score = 4)
Type of division
Year
N
Year 1
Mean
Standard Deviation
29
Year 2
Mean
Standard Deviation
31
Year 3
Mean
Standard Deviation
30
Year 4
Mean
Standard Deviation
28
Partitive
Quotitive
0.52
0.74
0.55
1.02
1.52
1.31
1.39
1.36
2.17
1.42
2.13
1.48
2.79
0.96
3.21
0.83
Results
(i) How many groups of girls are there?
(ii) How many girls are there in each group?
The order of asking these questions was counterbalanced
across children and across conditions.
The experimenter then summarized the information
by saying: ‘So, there are x groups and there are y girls in
each group’ [in the case of the order (i), (ii) ] and reminded
the child about how many tables there were (partitive) or
how many girls had to be placed on each table (quotitive).
The child was then asked the crucial question, which
was how many girls would be seated at each table (in the
partitive tasks) or how many tables were needed (in the
quotitive tasks). As in the baseline trials, the color cues
in the quotitive task helped children to realize that they
were being asked about how many tables were required
for the blue class to sit down. If the child seemed to be
including the table shown at the top of the page, s/he was
reminded that the experimenter only needed to know
how many more tables were required, for the blue class.
Each child received the same eight problems as had
been presented as baseline trials. Half the problems
were assigned to the Grouping-by-Divisor condition and
half to the Grouping-by-Quotient condition. The correct answer was equally often larger than the divisor as
it was smaller than the divisor. Counterbalancing occurred
across children such that the problems assigned to the
Grouping-by-Divisor condition in half of the children were
assigned to the Grouping-by-Quotient condition in the
other children and vice versa. The child was not given
any feedback regarding whether or not s/he had answered
correctly.
© Blackwell Publishers Ltd. 2002
Baseline trials
Each child was given a score of 1 for each baseline trial
that was correct and a score of 0 for each incorrect trial.
Table 1 presents the baseline scores. An analysis of variance
(ANOVA) was carried out on these scores; the two
between-subjects factors in the analysis were Type of
division in the experimental tasks (partitive vs. quotitive)
and Year (1, 2, 3 or 4) and there were repeated measures
on Baseline scores (partitive vs. quotitive). An alpha level
of 0.05 was used for all statistical tests. The only significant effect was that of Year [F(3, 110) = 29.4, p < 0.001],
indicating that the older children achieved higher baseline scores. The main effect of Type of division in the experimental tasks was not significant [F(1, 110) = 0.070],
which showed that it was not possible to ascribe any
differences between the groups in the experimental tasks
to initial differences between the two groups of children.
The ANOVA also demonstrated that the partitive and
quotitive problems were of comparable difficulty when
presented in the baseline tasks because there was no significant effect of Baseline scores [F(1, 110) = 0.437].
Experimental tasks
Each child was given a score of 1 for each experimental
trial that was correct and a score of 0 for each incorrect
trial. The maximum possible score in each condition was
4. Table 2 presents the mean scores.
The most interesting result was that differences were
found between the mean scores achieved in the Grouping-
DESC_385.fm Page 459 Monday, October 7, 2002 4:25 PM
From sharing to dividing
459
Table 2 Mean score in each condition of the experimental tasks (maximum score is 4)
Type of division
Partitive
Year
n
Year 1
Mean
Standard Deviation
14
Year 2
Mean
Standard Deviation
15
Year 3
Mean
Standard Deviation
15
Year 4
Mean
Standard Deviation
14
Grouping
-byDivisor
Grouping
-byQuotient
1.71
1.64
0.43
0.65
2.47
1.55
1.33
1.35
2.47
1.19
2.07
1.39
3.93
0.27
2.86
1.10
n
Grouping
-byDivisor
Grouping
-byQuotient
0.87
1.25
1.93
1.53
1.75
1.34
2.50
1.51
1.87
1.19
3.53
0.83
3.21
0.97
3.86
0.36
15
16
15
14
by-Divisor and Grouping-by-Quotient conditions, depending on the type of division problem. Figure 4 shows that
in the partitive tasks the Grouping-by-Divisor condition
was much easier than the Grouping-by-Quotient condition. In the quotitive tasks, the reverse pattern was
found; the Grouping-by-Quotient condition was much
easier than the Grouping-by-Divisor condition. This
pattern occurred in all years.
The between-subjects factors in the ANOVA of these
results were Year (1, 2, 3 or 4) and Type of division
Figure 4 Mean scores (+95% confidence intervals) obtained
in each condition in the partitive and quotitive division
problems.
© Blackwell Publishers Ltd. 2002
Quotitive
(partitive vs. quotitive), and there were repeated measures on Condition (Grouping-by-Divisor vs. Groupingby-Quotient). The only significant main effect was of
Year [F(3, 110) = 25.1, p < 0.001]. This confirmed that
the children’s performance in the experimental tasks
improved with age. There was no main effect of type of
division.
There was a significant Type of division × Condition
interaction [F(1, 110) = 62.1, p < 0.001]. Post-hoc analysis
(Bonferroni t-tests) revealed that there was a significant difference between the mean scores achieved in the
Grouping-by-Divisor and the Grouping-by-Quotient conditions in both types of division problem. However, in
the partitive tasks the higher score was achieved in the
Grouping-by-Divisor condition and in the quotitive tasks
the higher score was achieved in the Grouping-by-Quotient
condition. This confirmed that the Grouping-by-Divisor
condition was the ‘easy’ condition in the partitive tasks
whereas the ‘Grouping-by-Quotient’ condition was the
‘easy’ condition in the quotitive tasks.
The children’s age (the variable of Year) did not interact with any of the other variables. We will comment on
the implications of the absence of a significant Year ×
Type of Division × Condition interaction in the Discussion.
The general improvement with age, demonstrated by
the significant Year term in the analysis, could be
explained in two ways. One is that there is a growing
understanding with age of the mathematical relationships in division, and particularly of the relations
between divisor and quotient. The other is that older
children have more knowledge of the number facts (e.g.
are more likely to know that 12 ÷ 4 = 3) and are therefore more successful in all types of division problem.
DESC_385.fm Page 460 Monday, October 7, 2002 4:25 PM
460
Sarah Squire and Peter Bryant
Our hypothesis is that the first alternative is the correct
one, and in order to investigate this we carried out an
analysis of co-variance. In this analysis the main terms
were the same as in the analysis that we have just
described, and the co-variate was the children’s scores
in the baseline trials. We used the baseline scores as the
co-variate because, in our opinion, they provided a
reasonably good measure of the children’s ability to carry
out simple numerical divisions.2 If the improvement with
age in the experimental problems was merely a matter
of their knowledge of simple division facts, the year
difference should no longer be significant when the baseline problems are entered as a co-variate. In fact, the
results of the analysis of co-variance were very similar
to those of the previous analysis. The main term of
Year [F(3, 109) = 3.33, p < 0.05] and the Type of division
× condition interaction [F(1, 109) = 61.53, p < 0.001] were
again significant.
How did children’s performance in the experimental
tasks compare with their performance in the
baseline tasks?
Comparison of Tables 2 and 3 suggests that there is very
little difference between children’s mean scores in the
baseline tasks and their mean scores in the ‘difficult’
experimental tasks. However, it appears that their mean
scores in the ‘easy’ experimental tasks were much higher
than their mean scores in the baseline tasks. In order to
investigate this, four further ANOVAs were performed.
In these analyses, children’s scores in either the partitive
or quotitive baseline tasks were compared with the two
conditions in the experimental tasks. In other words, in
these ANOVAs Year was the between-subjects factor
and there were repeated measures on Task (baseline vs.
one experimental condition).
Two ANOVAs were carried out on the data from
children who received the partitive experimental tasks.
In the first ANOVA, the between-subjects factor was Year
and there were repeated measures on Task (partitive baseline vs. Grouping-by-Divisor [‘easy’] experimental). There
2
Ideally, we would have had a direct measure of children’s knowledge
of division facts. However, we wanted the control trials to be as similar
as possible to the experimental trials in terms of the materials presented and that is why we chose to give the children ‘baseline’ trials in
which the same problems and materials were presented and where the
only difference was that the items were not grouped. Although we
cannot be sure that children used knowledge of division facts in the
baseline trials, they were unable to manipulate the materials and so
could not have used a simple strategy such as sharing, or direct manipulation or grouping of the items to be shared. Also, there was an
improvement in the baseline scores with age and we think that the
most likely explanation for this is associated with an increase in knowledge of division facts or multiplication tables.
© Blackwell Publishers Ltd. 2002
were significant main effects of Year [F(3, 54) = 12.2,
p < 0.001] and Task [F(1, 54) = 26.8, p < 0.001], showing
that children’s scores in the ‘easy’ partitive experimental
trials were better than their scores in the partitive baseline trials. In the second ANOVA, the between-subjects
factor was Year and there were repeated measures on
Task (partitive baseline vs. Grouping-by-Quotient [‘difficult’] experimental). There was a significant main effect
of Year [F(3, 54) = 28.9, p < 0.001], but no significant
effect of Task [F = 0.734]. This shows that children’s scores
in the ‘difficult’ experimental condition in the partitive
tasks were no better than those achieved in the baseline
trials.
Two similar ANOVAs were carried out on the data
from children who received the quotitive experimental
tasks. In the first ANOVA, Year was the between-subjects
factor and there were repeated measures on Task (baseline
quotitive vs. Grouping-by-Quotient [‘easy’] experimental).
The ANOVA revealed significant main effects of Year
[F(3, 56) = 15.4, p < 0.001] and Task [F(1, 56) = 39.9,
p < 0.001], showing that children’s scores in the ‘easy’
quotitive experimental trials were better than their scores
in the quotitive baseline scores. The final ANOVA, with
the between-subjects factor of Year and repeated measures
on Task (baseline quotitive vs. Grouping-by-Divisor
[‘difficult’] experimental), revealed only a significant
effect of Year [F(3, 56) = 16.8, p < 0.001] and no main
effect of Task [F = 0.004]. None of the above analyses
revealed a significant Year × Task interaction.
Together, these analyses show that children’s scores in
the ‘difficult’ experimental condition were no better than
their scores in the baseline trials, but that their scores in
the ‘easy’ experimental trials were significantly better
than their scores in the baseline tasks.
Children’s errors
We also looked at the kind of errors that children made.
When the children got the quotient wrong they could
give one of the other two terms in the problem – the
dividend or the divisor – as the answer. Alternatively,
they could give an answer that represented none of these
three terms (‘other’).3 Table 3 shows the proportion of
these different types of error that children made.
It is apparent from Table 3 that divisor errors were not
proportionally more frequent in the ‘difficult’ conditions
3
Approximately 13% of the ‘other’ errors were null responses. Most
of the ‘other’ errors were arbitrary numbers, but these numbers were
almost always smaller then the dividend (i.e. the ‘other’ number that
the child gave as the answer tended to fall between 1 and 20). Quite
often the child gave a number that was close in size to either the
quotient or the divisor, but which did not correspond to either of these
numbers.
DESC_385.fm Page 461 Monday, October 7, 2002 4:25 PM
From sharing to dividing
Table 3 The proportion of different types of error made in
each type of division problem and in each condition
Type of error
Division problem and
condition
Divisor
Dividend
Other
Partitive
Grouping-by-Divisor
Grouping-by-Quotient
29.1
27.4
3.80
0.74
67.1
71.9
Quotitive
Grouping-by-Divisor
Grouping-by-Quotient
48.4
40.6
9.52
15.6
42.1
43.8
than the ‘easy’ conditions in either the partitive or the
quotitive tasks; therefore, the kind of error made does
not tell us much about the reasons for the pattern of
performance in the experimental trials. However, one
noticeable pattern observed from Table 3 is that divisor
errors and dividend errors accounted for a greater
proportion of the total errors in both conditions of the
quotitive tasks than in both conditions in the partitive
tasks. Sign tests4 showed that in both conditions in the
quotitive tasks, there was no difference between the
number of children who made more divisor errors than
other errors, and the number of children who showed
the reverse pattern. By contrast, a sign test showed that
the number of children who made more other errors
than divisor errors in the Grouping-by-Quotient condition of the partitive tasks was greater than the number
of children who showed the reverse pattern (z = 4.07,
p < 0.0001). In the Grouping-by-Divisor condition of the
partitive tasks the same pattern was observed, although
it did not reach significance.
Why might a greater proportion of divisor errors have
occurred in quotitive tasks than in the partitive tasks?
There are several possible explanations, all related to the
differences between the partitive and quotitive tasks.
In the partitive task, the children answered a question
about how many girls there would be around each table.
It is possible that the children were more inclined to
give arbitrary numbers in this task (e.g. based on their
knowledge of how many children sit on each table in
their classroom, at dinner or at home) than when asked
about the number of tables in the quotitive tasks. Thus,
children may have been more likely to give an ‘arbitrary’
number in the partitive tasks and a number that they
had heard in the question in the quotitive tasks.
4
Note that Chi-square tests could not be used because the data are
not independent; the same children made more than one type of error.
Also, it was of little use to consider the actual numbers of errors made
because it has already been established that, overall, more errors were
made in one condition than the other (depending on the type of division).
© Blackwell Publishers Ltd. 2002
461
Another possible explanation could be that in the
partitive tasks the child was able to see the number of
tables (divisor). In the quotitive tasks, the number of
girls on each table represented the divisor but there was
no explicit representation of the tables (as the number
of tables was the quotient in these problems). It may
therefore have been more difficult for children to have
imagined allocating groups of girls to each table. This
did not result in an overall difference in difficulty
between partitive and quotitive tasks, but it might explain why there were different patterns of errors. In the
partitive tasks, it might have been fairly easy to detect a
‘mismatch’ between the number of tables and the
number of groups of girls in the difficult condition; if
this happened, the child probably realized that the
correct answer could not be the number of girls in each
group (the divisor) and may instead have given an
arbitrary numerical response. In cases where children
were not sure about the answer in the quotitive tasks,
it may have been more difficult for them to have noticed
a ‘mismatch’ between the number of children in each
group and the number of children which had to be
put on each table. In other words, the size of the divisor
may have been less salient in the quotitive tasks (there
was always a picture of one table at the top of the page
and it was the number of girls around it that varied).
Children who made errors in the quotitive tasks may
have experienced difficulty in deciding whether the girls
in the picture were grouped by the divisor or by the
quotient. This could have caused confusion between the
divisor and the quotient and resulted in a greater proportion of ‘divisor’ errors overall in the quotitive tasks.
It is worth commenting on the fact that, in both types
of division, very few dividend errors were made. This suggests that children realized that the answer was smaller
than the dividend, and were more likely to confuse the
quotient with the divisor than with the dividend (this is
consistent with other research, e.g. Squire & Bryant, 2002).
Discussion
The main result from this experiment is that, in both
the partitive and the quotitive tasks, one condition was
much easier for children than the other. However, there
was a striking difference between the partitive and quotitive tasks in terms of which condition children found
easier. In the partitive task, the ‘easy’ condition was the
Grouping-by-Divisor condition and the ‘difficult’ one
was the Grouping-by-Quotient condition. In the quotitive task, the reverse was true; the Grouping by-Quotient
condition was easy and the Grouping-by-Divisor condition was difficult.
DESC_385.fm Page 462 Monday, October 7, 2002 4:25 PM
462
Sarah Squire and Peter Bryant
Non-mathematical factors must be involved because
in mathematical terms there is no difference between the
two conditions; the divisor and the quotient are represented in both conditions (via the number of sets and the
number of girls in each set). In our view, the most
convincing reason for this difference between the two
conditions in each type of division problem is a psychological one. The easy condition coincided with the
‘portions’ and therefore the child could use a strategy of
portion allocation, whereas this strategy would not be
successful in the difficult condition. The child may have
had a ‘mental model’ (Johnson-Laird, 1983) of the problem in which the dividend was grouped into portions
and where the easy model coincided with these portions
whereas the difficult condition did not.
Johnson-Laird argues that although some models may
be highly artificial and acquired only by cultural training
(e.g. in pure mathematics), others are ‘presumably acquired without explicit instruction, and used by everyone
in the course of such universal processes as inference and
language comprehension’ (Johnson-Laird, 1983, p. 11).
According to this approach, it is quite plausible that informal
experience could contribute to the formation of a ‘mental
model’ of a concept and hence that children could begin
to acquire a mental model of division through sharing,
which provides such experience. In other words, sharing
might be the ‘schema of action’5 (Nunes & Bryant, 1996;
Piaget, 1972/1947) from which an understanding of division develops. The pattern of results that we have described fits well with this notion because we already have
evidence that children might have different schemas of
action for partitive and quotitive division. When given
concrete materials, children often model partitive problems
by grouping by the divisor and quotitive problems by
grouping by the quotient (e.g. Carpenter et al., 1993;
Correa, 1994; Murray et al., 1992). The ‘easy’ condition
in both our partitive and quotitive problems was the
one in which the model presented to the children coincided with the model of the problem that they would
probably have created themselves. This suggests that
young children’s mental models of division may be the result
of their experience of sharing and of creating portions.
The theory of schemas of action is a developmental
one. It includes the claim that children start from these
schemas and progress, as they grow older, to a more
abstract understanding of the mathematical concept in
question. If correct, one would expect that the differences that we found between the easy and the difficult
5
‘Schemas of action’ are familiar actions that might provide a first
understanding of arithmetic operations, because the logical requirements and relationships that must be kept constant in arithmetical
operations also have to be invariant in the child’s schema of action
(Nunes & Bryant, 1996; Piaget, 1972/1947; Vergnaud, 1985).
© Blackwell Publishers Ltd. 2002
conditions in the two types of division would diminish
with age. Yet, this did not happen. In our study, older
children found the same tasks easy and difficult as the
younger ones did, and in our analysis of the results the
interaction which would have shown a diminution of
the pattern of differences (the Year × Type of division ×
Condition interaction) was not significant.
One possible explanation for this is that the older
children were more familiar with the division facts and
used these more often than the younger children did.
However, our analysis of co-variance suggests that it is
not this simple, because when the children’s baseline
scores were added as a co-variate in the analysis of our
results, there was still a significant effect of Year. This
means that the older children’s better performance is
unlikely to simply be due to their greater knowledge of
division facts. An alternative explanation for the age
differences is that children become more able to concentrate on the abstract relations between the mathematical
terms in these problems as they grow older. In other words,
one condition continues to coincide better with children’s
mental model of the problem (built up from their schema
of action) than the other, possibly explaining why the difference between the easy and the difficult conditions in
our study did not diminish with age. Older children may
achieve higher scores in both conditions because they
can also use their understanding of the relations between
the variables in division problems – such as the interchangeability between the divisor and quotient (e.g. if
12 ÷ 4 = 3, 12 ÷ 3 = 4) – to find solutions. This should help
them in all the problems, and not just in the more difficult ones. One reason why this explanation is a plausible
one is because of the evidence that children’s understanding of the inverse relation between the divisor and
the quotient improves significantly across the age range
that we studied (Correa et al., 1998; Sophian et al., 1997).
An interesting question for future research is whether
the difference between the two conditions would diminish if even older children or adults were studied. Older
children might be expected to have a greater understanding of the concept of interchangeability and to also have
better working memory, making it easier for them to
‘transform’ the model that they are presented with into
a model that coincides with their mental model (e.g. to
imagine re-grouping three groups of four into four
groups of three). Alternatively, it is possible that the difference in difficulty between the two conditions might
also exist with older participants; the ‘easy’ condition
may still coincide better with their mental model of the
problem and therefore remain easier to solve. Also,
an improvement in the understanding of the relations
between terms in division problems might continue
to improve performance in both conditions. Further
DESC_385.fm Page 463 Monday, October 7, 2002 4:25 PM
From sharing to dividing
research (possibly using more difficult division problems,
or response times) would be required to distinguish
between these two possibilities.
Presenting children with models of division problems
in which the objects are grouped in the two different
ways – Grouping-by-Divisor and Grouping-by-Quotient
– should help them to go beyond sharing (and a simple
strategy of portion-allocation) to consider the structure
of the problem with respect to the dividend, divisor and
quotient. It may also help children to mentally reorganize the model that they are presented with (e.g. to reorganize four groups of three into three groups of four)6
and this could help to improve children’s understanding
of multiplicative relations. Since the difficult condition
in the partitive problems (Grouping-by-Quotient) is the
easy condition in the quotitive problems and vice versa,
the different conditions also provide the opportunity for
children to learn something about the relation between
partitive and quotitive division.
In the educational and the psychological literature
considerable emphasis is placed on providing children
with different problem representations. The NCTM
(2000) suggests that ‘understanding of the meanings of
these operations [multiplication and division] should
grow deeper as they [children] encounter a range of representations and problem situations’ (p. 148). This view
is also reflected in the psychology literature about
children’s mathematics (e.g. Gravemeijer, 1997; Murray
et al., 1992; Nunes, 1999) and in other fields of psychology.
For example, Spiro, Feltovich, Coulson and Anderson
(1989) recommend that multiple analogies be used in
learning about complex concepts. Our study suggests
that children may have two different schemas for the
operation of ‘division’ (sharing in partitive division
versus forming quotas in quotitive division). In order to
bridge the gap between their informal and formal understanding, children need to anchor their understanding of
the mathematical operation of division with the schemas
of action for both partitive and quotitive division and,
eventually, these actions schemas must also be related to
the conventional sign(s) for the operation of division.
Our study also highlights the importance of presentation in children’s problem solving; although the underly6
We have some anecdotal evidence that, in the difficult conditions,
some children tried to restructure the model. For example, in one partitive problem (20 girls who had to be seated fairly around 4 tables) in
the Grouping-by-Quotient condition (i.e. five groups of four girls), a
Year 3 child explained that he was solving the problem by putting one
group of girls on each table and with last group, he was putting one
girl on the 1st table, one on the 2nd table, one on the 3rd table and so
on. In this way, the child mentally reorganized the model into a more
meaningful context, which was one that coincided with the end-point
of sharing (four equal groups of girls for the four tables). This strategy
resulted in him giving the correct answer.
© Blackwell Publishers Ltd. 2002
463
ing mathematical structure was the same for partitive
and quotitive problems, how the problem was presented
had an impact on problem difficulty. Extensive research
in cognitive psychology has shown that the manner in
which problems are represented produces substantial
differences in difficulty (Kotovsky & Simon, 1990). For
example, Hayes and Simon (1974, 1977) showed that
there were great differences in difficulty between sets of
problems that were isomorphic (possessed the same structure) but which were described by different cover stories
and therefore engendered different representations.
Differences in problem representation have also been
used to explain some of the differences between experts
and novices (Adelson, 1984; Chi, Feltovich & Glaser,
1981). It has been argued (e.g. Novick, 1988) that
although surface information may be included in the
representations of both experts and novices, experts’ representations include abstract, structural features of the
problem (e.g. how the quantities in the problem are
related to each other). For this reason, when two analogical problems share surface, but not structural similarity,
one might expect spontaneous negative transfer to be
stronger for novices (focusing on surface similarities)
than for experts (Novick, 1988). However, this prediction has not always been confirmed in the literature;
Novick (1988) found that experts were just as likely as
novices to think that the two complex arithmetic word
problems should be solved in a similar manner because
they had similar surface features. This could be because
surface similarities between problems are very salient
and readily learned (Lewis & Anderson, 1985; Medin &
Ortony, 1989). Our findings tie in well with this research
because we found that even the older children (who
might be considered more ‘expert’) showed a difference
in difficulty between the two conditions. The older
children continued to find one condition easier than the
other and this may have been because they were not
always examining the underlying structure of the problem, but were instead relying on surface features (e.g.
both problems are about sitting girls around tables).
Interestingly, Novick found that the only response measure which was sensitive to experts’ superior performance
was the type of solution procedures used, not the time
taken to complete the problem, or accuracy of solutions
(which was the response measure in our experiment).
This raises an interesting question for future research,
which is whether any developmental changes might have
occurred if strategies used in the two conditions (and
two types of division problem) were considered as the
response measure.
To summarize, our results are consistent with the idea
that children begin to solve division problems by relying
on their mental model of the problem, which is built up
DESC_385.fm Page 464 Monday, October 7, 2002 4:25 PM
464
Sarah Squire and Peter Bryant
from a schema of action that depends on the context.
Our claim is that the experience of sharing may influence
children’s mental models of division. When sharing, children allocate equal portions and we claim that children
found it easy to answer the problems that we gave them
when the sets in the model corresponded to the portions.
More generally, we have also argued that it may be
important to expose children to different problem representations and problem contexts in order to improve
their ability to recognize the important variables in a
problem, to develop their conceptual understanding of
multiplicative relations and encourage them to think
flexibly in particular and diverse contexts. Further
research, including longitudinal studies, is required to
establish whether there is a direct link between children’s
ability to share and their understanding of division.
Acknowledgements
We thank the Medical Research Council (UK) for providing a graduate studentship for the first author. We
are also grateful to the teachers and children of North
Kidlington Primary School and Wolvercote County First
School for taking part in this project.
References
Adelson, B. (1984). When novices surpass experts: the difficulty of a task may increase with expertise. Journal of Experimental Psychology: Learning, Memory and Cognition, 10,
483 – 495.
Baroody, A.J., Gannon, K.E., Berent, R., & Ginsburg, H.P.
(1984). The development of basic formal mathematics abilities. Acta Paedologica, 1, 133 –150.
Bourgeois, R., & Nelson, D. (1977). Young children’s behavior
in solving division problems. Alberta Journal of Educational
Research, 23, 178 –185.
Bradley, L., & Bryant, P. (1985). Rhyme and reason in reading
and spelling. IARLD Monographs. Ann Arbor: University
of Michigan Press.
Brown, M. (1981). Number operations. In K.M. Hart (Ed.),
Children’s understanding of mathematics (pp. 11–16). Oxford:
John Murray.
Bryant, P. (1997). Mathematical understanding in the nursery
school years. In T. Nunes & P. Bryant (Eds.), Learning and
teaching mathematics: An international perspective (pp. 53–
67). Hove: Psychology Press.
Carpenter, T.P., Ansell, E., Franke, M.L., Fennema, E., &
Weisbeck, L. (1993). Models of problem solving: a study of
kindergarten children’s problem-solving processes. Journal
for Research in Mathematics Education, 24, 428–441.
Carpenter, T.P., & Fennema, E. (1992). Cognitively guided
instruction: building on the knowledge of students and
© Blackwell Publishers Ltd. 2002
teachers. International Journal of Research in Education, 17,
457–470.
Carpenter, T.P., Fennema, E., & Franke, M.L. (1996). Cognitively guided instruction: a knowledge base for reform in
primary mathematics instruction. Elementary School Journal,
97, 3–20.
Carpenter, T.P., Franke, M.L., Jacobs, V.R., Fennema, E., &
Empson, S.B. (1997). A longitudinal study of invention and
understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3–20.
Carpenter, T.P., & Lehrer, R. (1999). Teaching and learning
mathematics with understanding. In E. Fennema & T.A.
Romberg (Eds.), Mathematics classrooms that promote
understanding (pp. 19–32). Studies in mathematical thinking and learning series. Mahwah, NJ: Lawrence Erlbaum
Associates.
Carpenter, T.P., & Moser, J.M. (1982). The development of
addition and subtraction problem solving skills. In T.P.
Carpenter, J.M. Moser & T. Romberg (Eds.), Addition and
subtraction: A cognitive perspective (pp. 10–24). Hillsdale,
NJ: Lawrence Erlbaum Associates.
Carraher, T.N., Carraher, D.W., & Schliemann, A.D. (1987).
Written and oral mathematics. Journal for Research in
Mathematics Education, 18, 83–97.
Chi, M.T.H., Feltovich, P.J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and
novices. Cognitive Science, 5, 121–152.
Confrey, J. (1994). Splitting, similarity, and rate of change: a
new approach to multiplication and exponential functions.
In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291–
330). Albany, NY: State University of New York Press.
Correa, J. (1994). Young children’s understanding of the
division concept. University of Oxford: Unpublished PhD
thesis.
Correa, J., Nunes, T., & Bryant, P. (1998). Young children’s
understanding of division: the relationship between division
terms in a noncomputational task. Journal of Educational
Psychology, 90, 321–329.
De Abreu, G., Bishop, A.J., & Pompeu, G. Jr. (1997). What
children and teachers count as mathematics. In T. Nunes &
P. Bryant (Eds.), Learning and teaching mathematics: An
international perspective (pp. 233–265). Hove, UK: Psychology
Press.
De Corte, E., Greer, B., & Verschaffel, L. (1996). Mathematics.
In D. Berliner & R. Calfee (Eds.), Handbook of educational
psychology (pp. 491–549). New York: Macmillan.
Department for Education and Employment (1999). The
national numeracy strategy: Mathematical vocabulary.
Watford: GPS Printworks.
Desforges, A., & Desforges, G. (1980). Number-based strategies of sharing in young children. Educational Studies, 6,
97–109.
Fennema, E., Carpenter, T.P., Franke, M.L., Levi, L.,
Jacobs, V.R., & Empson, S.B. (1996). A longitudinal study
of learning to use children’s thinking in mathematics
instruction. Journal for Research in Mathematics Education,
27, 403–434.
DESC_385.fm Page 465 Monday, October 7, 2002 4:25 PM
From sharing to dividing
Fennema, E., Sowder, J., & Carpenter, T.P. (1999). Creating
classrooms that promote understanding. In E. Fennema &
T.A. Romberg (Eds.), Mathematics classrooms that promote
understanding ( pp. 185–199). Mahwah, NJ: Lawrence Erlbaum
Associates.
Fischbein, R., Deri, M., Nello, M., & Marino, M. (1985). The
role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics
Education, 16, 3 – 17.
Frydman, O., & Bryant, P.E. (1988). Sharing and the understanding of number equivalence by young children. Cognitive
Development, 3, 323–339.
Goswami, U. (1989). Relational complexity and the development
of analogical reasoning. Cognitive Development, 4, 251–268.
Gravemeijer, K. (1997). Mediating between concrete and
abstract. In T. Nunes & P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 315–345).
Hove, UK: Psychology Press.
Gunderson, A.G. (1955). Thought-patterns of young children
in learning multiplication and division. Elementary School
Journal, 55, 453 – 461.
Hayes, J.R., & Simon, H.A. (1974). Understanding written
problem instructions. In L.W. Gregg (Eds.), Knowledge and
cognition (pp. 167–200). Hillsdale, NJ: Erlbaum.
Hayes, J.R., & Simon, H.A. (1977). Psychological differences
among problem isomorphs. In N.J. Castellan, D.B. Pisoni &
G. Potts (Eds.), Cognitive theory (pp. 21–44). Hillsdale, NJ:
Erlbaum.
Hiebert, J. (Ed.) (1986). Conceptual and procedural knowledge:
The case of mathematics. Hillsdale, NJ: Erlbaum.
Hiebert, J., & Carpenter, T.P. (1992). Learning and teaching
with understanding. In D.A. Grouws (Eds.), Handbook of
research on mathematics teaching and learning (pp. 65–97).
New York: Macmillan.
Hughes, M. (1986). Children and number. Oxford: Blackwell.
Huttenlocher, J., Jordan, N.C., & Levine, S.C. (1994). A mental
model for early arithmetic. Journal of Experimental Psychology:
General, 123, 284 –296.
Johnson-Laird, P.N. (1983). Mental models. Cambridge: Cambridge University Press.
Kornilaki, E. (1999). Young children’s understanding of multiplicative concepts: a psychological approach. University of
London: Unpublished PhD thesis.
Kotovsky, K., Hayes, J.R., & Simon, H.A. (1985). Why are
some problems hard? Evidence from Tower of Hanoi.
Cognitive Psychology, 17, 248–294.
Kotovsky, K., & Simon, H.A. (1990). What makes some
problems really hard: explorations in the problem space of
difficulty. Cognitive Psychology, 22, 143–183.
Lago, M.O., Rodriguez, P., Zamora, A., & Madrono, L.
(1999). Influencia de los modelos intuitivos en la comprehension de la multiplication y la division [The influence of
intuitive models on the comprehension of multiplication
and division]. Anuario de Psicologia, 30, 71–89.
Levine, S.C., Jordan, N.C., & Huttenlocher, J. (1992). Development of calculation abilities in young children. Journal of
Experimental Child Psychology, 53, 72–103.
Lewis, M.W., & Anderson, J.R. (1985). Discrimination of
© Blackwell Publishers Ltd. 2002
465
operator schemata in problem solving. Learning from examples. Cognitive Psychology, 17, 26–65.
Mack, N.K. (1995). Confounding whole-number and fraction
concepts when building on informal knowledge. Journal for
Research in Mathematics Education, 26, 422–441.
Marton, F., & Neuman, D. (1996). Phenomenography and
children’s experience of division. In L.R. Steffe, P. Nosher,
P. Cobb, G.A. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 315 –333). Mahwah, NJ: Lawrence
Erlbaum Associates.
Medin, D.L., & Ortony, A. (1989). Psychological essentialism.
In S. Vosniaou & A. Ortony (Eds.), Similarity and analogical
reasoning (pp. 179–195). NewYork: Cambridge University
Press.
Miller, K. (1984). The child as the measurer of all things:
measurement procedures and the development of quantitative concepts. In C. Sophian (Ed.), Origins of cognitive
skills (pp. 193–228). Hillsdale, NJ: Erlbaum.
Mix, K.S., Levine, S., & Huttenlocher, J. (1999). Early fraction
calculation ability. Developmental Psychology, 35, 164–174.
Murray, H., Olivier, A., & Human, P. (1992). The development
of young students’ division strategies. Proceedings of the
Sixteenth International Conference for the Psychology of
Mathematics Education, 2, 152–159.
National Council for Teachers of Mathematics (2000).
Principles and standards for school mathematics [http://
standards.nctm.org/document/prepost/cover.htm]
Novick, L.R (1988). Analogical transfer, problem similarity
and expertise. Journal of Experimental Psychology: Learning,
Memory and Cognition, 14, 510–520.
Nunes, T. (1999). Mathematics learning as the socialisation of
the mind. Mind, Culture, and Activity, 6, 33–52.
Nunes, T., & Bryant, P. (1996). Children doing mathematics.
Oxford: Blackwell.
Nunes, T., Schliemann, A.L., & Carraher, D. (1993). Street
mathematics and school mathematics. New York: Cambridge
University Press.
Piaget, J. (1972). The psychology of intelligence (M. Piercy &
D.E. Berltyne, trans.). Totowa, NJ: Littlefield, Adams & Co.
(Original work published in 1947.)
Piaget, J. (2001). Studies in reflecting abstraction (R.L. Campbell,
trans.). Hove: Psychology Press. (Original work published in
1977.)
Reed, S.K., Ernst, G.W., & Banerji, R. (1974). The role of
analogy in transfer between similar problem states. Cognitive
Psychology, 6, 436–450.
Riley, M.S., Greeno, J.G., & Heller, J.I. (1983). Development
of children’s problem-solving ability in arithmetic. In H.
Ginsburg (Ed.), The development of mathematical thinking
(pp. 153–196). New York: Academic Press.
Rittle-Johnson, B., & Siegler, R.S. (1998). The relation
between conceptual and procedural knowledge in learning
mathematics: a review. In C. Donlan (Ed.), The development
of mathematical skills (pp. 75 –110). Studies in developmental
psychology. Hove: Psychology Press.
Schoenfeld, A.H. (1991). On mathematics and sense-making:
an informal attack on the unfortunate divorce of formal and
informal mathematics. In J.F. Voss & D.N. Perkins (Eds.),
DESC_385.fm Page 466 Monday, October 7, 2002 4:25 PM
466
Sarah Squire and Peter Bryant
Informal reasoning and education (pp. 311–343). Hillsdale,
NJ: Lawrence Erlbaum Associates.
Silver, E.A., Shapiro, L.J., & Deutsch, A. (1993). Sense making
and the solution of division problems involving remainders:
an examination of middle school students’ solution processes
and their interpretations of solutions. Journal for Research in
Mathematics Education, 24, 117–135.
Simon, M.A. (1993). Prospective elementary teachers’
knowledge of division. Journal for Research in Mathematics
Education, 24, 233 –259.
Sophian, C., Garyantes, D., & Chang, C. (1997). When three
is less than two: early developments in children’s understanding of fractional quantities. Developmental Psychology,
33, 731–744.
Spiro, R.J., Feltovich, P.J., Coulson, R.L., & Anderson, D.K.
(1989). Multiple analogies for complex concepts: antidotes
for analogy-induced misconception in advanced knowledge
acquisition. In S. Vosniadou & A. Ortony (Eds.), Similarity
and analogical reasoning (pp. 498 – 531). New York: Cambridge University Press.
Squire, S., & Bryant, P. (2002). The influence of sharing on
children’s initial concept of division. Journal of Experimental
Child Psychology, 81, 1– 43.
Steffe, L. (1994). Children’s multiplying schemes. In G. Harel
© Blackwell Publishers Ltd. 2002
& J. Confrey (Eds.), The development of multiplicative reasoning
in the learning of mathematics (pp. 3–40). Albany, NY: State
University of New York Press.
Tirosh, D. (2000). Enhancing prospective teachers’ knowledge
of children’s conceptions: the case of division of fractions.
Journal for Research in Mathematics Education, 31, 5–25.
Vergnaud, G. (1983). Multiplicative structures. In R. Lesh &
M. Landau (Eds.), Acquisitions of mathematics concepts and
procedures (pp. 127–174). New York: Academic Press.
Vergnaud, G. (1985). Conceptes et schèmes dans une théorie
opératoire de la représentation [Concepts and schemes in an
operational theory of representation]. Psychologie Francaise,
30, 246–252.
Verschaffel, L., & de Corte, E. (1997). Word problems: a
vehicle for promoting authentic mathematical understanding
and problem solving in the primary school? In T. Nunes &
P. Bryant (Eds.), Learning and teaching mathematics: An
international perspective ( pp. 69 –97). Hove: Psychology Press.
Zweng, M.J. (1964). Division problems and the concept of
rate. Arithmetic Teacher, 11, 547–556.
Received: 2 February 2001
Accepted: 15 August 2001