Teaching Children About
the Inverse Relation Between Addition and Subtraction
Department of
Education, University of Oxford
Terezinha Nunes, Peter
Bryant, Darcy Hallett, Daniel Bell, and Deborah Evans
Two intervention
studies are described. Both were designed to study the effects of
teaching children about the inverse relation between addition and
subtraction. The interventions were successful with 8-year-old
children in Study 1 and to a limited extent with 5-year-old children
in Study 2. In Study 1
teaching children about
inversion increased their success not just in Inverse problems (a + b
– b = ?)
but also in Transfer
complement problems (a + b = c; c – b = ?).
The issue of children’s
understanding of the inverse relation between addition and
subtraction
plays an important part
in psychological theories about cognitive development, and it is also
relevant to research on
the teaching of mathematics (Nunes et al., 2007). Theoretical
discussions
about the inverse
relation go back to Piaget (Piaget, 1952; Piaget & Moreau, 2001)
who linked
this form of
understanding to his idea of reversibility, which is the ability to
cancel the effect of
a transformation to an
object or a set of objects by imagining the opposite transformation
to the
material at hand.
Piaget (1952) thought that young children lack reversibility, and in
one of his
last publications, he
(Piaget & Moreau, 1977, 2001) argued that this prevented them
from under-
standing the inverse
relations between addition and subtraction and between multiplication
and
division.
Piaget and Moreau
adapted a well-known guessing game to study children’s
understanding of
inversion. They asked
children aged from 6 to10 years to choose a number but not to tell
them
what this was. Then
they asked the child first to add 3 to this number, next to double
the sum,
and then to add 5 to
the result of the multiplication. They continued by asking the child
what the
final sum was and went
on to tell the child what was the number he or she chose to start
with.
Finally, they asked the
child to explain to them how he or she had managed to work out what
this initial number
was.
Piaget and Moreau
reported that this was a difficult task. The youngest children in the
sample
did not understand that
the experimenters had performed the inverse operation, subtracting
The authors are
thankful to the ESRC-Teaching and Learning Research Programme,
without whose generous
support (through grant
number L139251015) this work could not have been carried out. They
are also very grateful to all
the children who
participated in this project. Without their collaboration, and that
of the schools and teachers who
allowed them to disturb
their routines, no research about children’s learning would be
possible..
where the child had
added and dividing where he or she had multiplied. The older children
did
show some understanding
that this was how the experimenters reached the right number but did
not understand that the
order of the inverse operations was important.
This highly impressive
and original study has been largely ignored since its original
publication
in 1977, but in recent
years several researchers have re-discovered how hard it is for
children in
the 5- to 10-years age
range to solve problems that involve the inverse relation between
adding
and subtracting (Bisanz
& Lefevre, 1992; Siegler & Stern, 1998; Stern, 1992). This
body of
evidence raises an
obvious question: What exactly is the nature of children’s
difficulties with the
principle of inversion?
One possible answer
lies in a distinction made by Bryant, Christie, and Rendu (1999)
between two levels in
the understanding of the inverse relation between addition and
subtraction.
One is the level of
identity: when identical stuff is added and then subtracted to an
object, the
final state of this
object is the same as the initial state. Young children have many
informal expe-
riences of inverse
transformations at this level. A child gets his shirt dirty (mud is
added to it)
and then it is cleaned
(mud is subtracted) and the shirt is as it was before. At meal-times
various
objects (knives, forks,
etc.) are put on the dining room table and then subtracted, and the
table
top is as empty after
the meal as it was before. Note that understanding the inversion of
identity
may not involve
quantity. The child can understand that if the same (or identical)
stuff is added
and then removed the
status quo is restored without having to know anything about the
quantity
of the stuff.
The other possible
level is the understanding of the inversion of quantity. If I have 10
sweets
and someone gives me
three more and then I eat three, I have the same number left as at
the start
and it doesn’t matter
whether the three sweets that I ate are the same three sweets as were
given
to me or different
ones. Provided that I eat the same number as I was given, the
quantitative
status quo is now
restored.
Bryant, Christie, and
Rendu (1999), working with toy bricks, established that 5- and
6-year-
old children found
identity problems in which exactly the same bricks were added and
then
subtracted to the
initial set (or vice versa) easier than quantity problems in which
the same
number of bricks was
added and then subtracted (or vice versa) but the bricks added were
quite
different from the
bricks that were subtracted. Bryant et al. also found a greater
improvement
with age in children’s
performance in quantitative inversion problems than in identity
inversion
problems, in which the
procedures were exactly the same apart from the fact that identical
objects were added and
subtracted in the identity condition but not in the quantity
condition.
These results point to
a developmental hypothesis: children’s understanding of the
inversion of
identity precedes and
may provide the basis for their understanding of the inversion of
quan-
tity. First they
understand that adding and subtracting the same stuff restores the
physical status
quo. Then they extend
this knowledge to quantity, realizing now that adding and subtracting
the same quantity
restores the quantitative status quo whether the addend and
subtrahend are
the same stuff or not.
If this hypothesis is
correct, teaching children about the connection between these two
levels would be an
effective way to improve their understanding of quantitative
inversion.
This is one issue that
we tried to settle in the two intervention studies that we will
present
here. So far there have
been hardly any reports on research about teaching children the
inverse
addition-subtraction
relationship. We know of only two such studies. One was by
Lai, Baroody, and
Johnson (2008), but this produced no evidence that is relevant to the
identity-quantity
distinction that we have just made. The other study (Nunes et al.,
2008) does
touch on this
distinction.
This was a study of
deaf children. Some previous work (Nunes et al., 2008) established
that
deaf children find
inversion problems particularly difficult. This result was the spur
for an
intervention project in
which deaf children were taught about the inverse relation between
addition and
subtraction with the help of concrete objects. The children were
taught about
inversion by first
solving inversion of identity problems (the same bricks added and
subtracted)
and then moving on to
the inversion of quantity (different bricks added and subtracted).
Color
cues were used to
facilitate the transition between the inversion of identity and of
quantity.
When different bricks
were added and subtracted, those added were of a different color so
that
the children could
compare those that had been subtracted with those that had been
added. This
helped them realise
that it was possible to answer the question without counting the
bricks or
doing the sums.
This intervention was
effective. In two posttests (one immediately after training and the
other
2 to 4 weeks later) the
children in the intervention group did significantly better than the
children in a control
group who had been taught other number skills. There was no
difference
between the
intervention and the control groups’ success with control items,
which did not
involve inversion. So
the effect of the intervention was specific to promoting the
understanding
of inversion and not a
general one that improved deaf children’s knowledge of addition and
subtraction facts. One
possible reason for this positive result may have been the emphasis
in the
intervention on the
link between the two kinds of inversion: identity and quantity.
Intervention studies
are valuable not just for assessing ways of teaching mathematical
principles but also for
settling theoretical issues about children’s mathematical
understanding.
One such issue is the
possible importance of understanding inversion in various familiar
math-
ematical tasks. For
example, the problem a + b = c; c – a = ? also causes children a
great deal
of difficulty. This was
termed the complement problem by Baroody (Baroody, 1999; Baroody,
Ginsburg, & Waxman,
1983; Baroody & Tiilikainen, 2003). Most commentaries on this
problem (Resnick, 1983;
Putnam, de Bettencourt, & Leinhardt, 1990) emphasise the fact
that c
consists of two parts,
a and b, and make the claim that the main reason for the children’s
failure
is their difficulty in
grasping the relationship between parts and wholes. Another
possibility is
that children fail to
solve the problem because solutions rest on understanding that
subtractions
cancel out additions.
They may not grasp that, because c is the result of adding b to a,
subtract-
ing b from c will
cancel out the effects of that addition and will therefore restore
the initial
quantity a.
If one reason for
children’s mistakes in the complement problem is the difficulty
that they
have in understanding
the inverse relation between addition and subtraction, an
intervention that
improves their
understanding of inversion should also increase their success with
complement
problems.
We designed two studies
to answer the questions that we have just raised. In the first
study we looked at the
effect of two kinds of teaching inversion on 7- to 8-year-old
children’s
understanding of inversion and also on their ability to solve
complement
problems. One of the
intervention methods stressed the connection between the inversion of
identity and of
quantity while the other concentrated just on quantity. In the second
study
we looked at the
effects of the first kind of intervention on a younger group of
children who
were five years old.
STUDY 1
Method
Participants
Sixty schoolchildren
(32 girls and 28 boys) from two state schools took part in this
study.
The children in the two
schools came from a wide range of socioeconomic backgrounds.
The participants were
all in their second or third year at school. They were randomly
selected
from their class list
and standard procedures for asking for permission for their
participation
were used. The mean age
of the sample was 8 years (SD = 6 months). There were no refusals.
Design
All the children were
given the same pretest to assess their understanding of inversion.
They
were then randomly
assigned to one of three intervention groups.
1. Visual Demonstration
group (N = 21; mean age 8 years 0 months; SD = 6.3 months);
2. Oral and Calculator
group (N = 19; mean age 8 years 0 months; SD = 6.4 months);
3. Control group (N =
20; mean age 8 years 0 months; SD = 5.5 months).
Then each child went
through two intervention sessions. Finally the children were given a
posttest that was
identical to the pretest. The children in the three intervention
groups were given
different experiences
in the intervention sessions.
Procedure
The pretest and
posttests contained three kinds of items, which were mixed and placed
in
random order in both
testing sessions. They were:
1. Inverse (a + b – b
= ?) items designed to assess understanding of the inverse relation
between addition and
subtraction (n = 21); in some of these items, the last term differed
from the second term by
1 or by 2.
2. Transfer (a + b = c;
c – a = ?) items, which involved the complement relation and which
might be solved more
easily by those who understand the inverse relation than by those
who do not (n = 12).
3. Control (a + a – b
= ?) items, which did not involve the inverse relation (n = 9).
The pre- and posttest
items were presented verbally.
The Inverse task
consisted of numerical questions of the type a + b – b = ? or a + b
– (b +/– 1) = ?.
The numbers were larger
than those in addition facts that the children had practised in
school
(for example, 18 + 7 –
7 = ?). In some items the addition came first and subtraction second
and
in others they came in
the opposite order.
In each of the transfer
task items the children were first told about an addition or
subtraction
and then asked to do
the inverse operation. So, they were told that a + b = c and then
asked to
solve the sum c – a =
? (or c – b = ?), or they were told that a – b = c and then asked
to solve the
sum c + b = ?. No item
of this form was included in the intervention.
The control task
involved a series of sums that could not be solved by using the
inverse
principle. They
contained a repeated number (a + a – b = ?) and had the same result
as one of
the inversion items
(e.g., 11 + 11 – 4 gives the same result as the item in the
preceding inversion
example).
In the intervention
sessions the children were randomly assigned to the intervention
groups: a
visual demonstration,
an oral-calculator, and a control group. There were two intervention
sessions for each
child, during which a trained experimenter worked with each child
individually
in a room close to the
classroom. All the children were given the same number of trials.
The intervention for
the visual demonstration group took the form of a series of trials
that
began with the
experimenter showing the child a row of joined up Unifix bricks and
asking him/
her to count them. The
experimenter then placed the bricks under a cloth with both ends
visible
so that the child could
see transformations made to the row. Other bricks were then added to
and
subtracted from the
initial row in each trial. There were six types of trials, which we
expected to
be of increasing
difficulty. The children solved six items of each type, but these
were not
presented as completely
independent blocks (the last two items of each block were mixed with
the first two items of
the subsequent block).
The first six items
used colour and identity cues: the bricks added to the row were of a
different
color from the initial
bricks and these new and distinctive bricks were added and then
subtracted, and the
children were asked how many items there were in the row. These were
termed identity items
and the bricks added were the same bricks that were subtracted. The
second type of trial
involved identity and colour cues, but the numbers added differed by
one
(+1 or –1) from the
number subtracted. The child had to monitor these transformations
carefully
because the answer was
not always the initial number of bricks in the row.
The next two types of
trial were inversion trials without the color cues; the bricks added
and
subtracted were
different (i.e., bricks were added to one end and subtracted from the
other). In
the third type of
trial, the number added and subtracted was the same; in the fourth
type of trial,
the number subtracted
differed by one (+1 or –1) from the number added.
In the fifth and sixth
types of trial the researcher subtracted the bricks before adding
some to
the other side. The
bricks added and subtracted were not the same ones: the experimenter
placed
those bricks that had
been subtracted into a container and took other bricks from the same
container. In the fifth
type of trials, the numbers subtracted and added were the same; in
the
sixth type, they
differed by one (+1 or –1).
After each trial, the
child was asked how many bricks were under the cloth. After providing
an answer, the child
could either compare the row of bricks that had been added to that
formed
by the bricks that had
been subtracted or could count the number of bricks under the cloth
to find
out whether he or she
had answered correctly. Thus the child progressed from experience
with
inversion of identity
problems to inversion of quantity problems.
The children in the
oral and calculator group solved the same questions as those in the
visual
demonstration group but
these were presented orally only. For example, the experimenter
asked:
Imagine that I have 9
bricks on the table, and then I add 7, and then I take 7 away. How
many
bricks will I have
left? The researcher used gestures while saying that bricks were
added or
taken away, but these
would not be specific enough to show whether the same bricks that
were
added were taken away.
After the children had answered, the experimenter suggested that they
check their answer by
entering the operations into a calculator (9 + 7 – 7) to find out
whether
they had answered
correctly. This gave them the opportunity to rehearse the trial
verbally
because they said “nine
plus seven minus seven” as they entered the operations into the
calculator.
If the child had
forgotten a number, the researcher helped the child to reconstruct
the trial.
The control group
children worked with the same experimenter for approximately as long
as
the children in the
other two intervention groups. They solved all the computation
problems
presented to the
intervention groups but in separate form: e.g., as a control for 18 +
7 – 7, the
control group solved,
at different times during the sessions, 18 + 7 and 25 – 7. The
questions
were presented orally
in the same way as they were presented to the oral-calculator group.
The
children were allowed
to check their answers by entering the operation in a calculator. It
was not
expected that this
would improve their performance in the control task, as most of the
specific
sums that they were
practising were not the same ones that were included in the pre- and
posttest.
The children were given
the pretest and the first half of the intervention on Day 1 in the
experiment. On Day 2,
they were given the second part of the intervention, followed by the
posttest.
None of the specific
Inverse items used in the pre- and posttest was included in the
training.
Results
Preliminary analyses
showed no differences between the boys and girls, and so their
results were
combined.
The main purpose of the
pretest was to establish a baseline for the children’s scores in
the
different tasks.
However it also gave us a measure of the children’s initial
understanding of the
inversion principle. We
were able to compare the children’s success in the a + b – b
inverse task
problems, which they
could solve either by invoking the inverse relation or by
computation,
with their success in
the a + a – b control task problems, which they could only solve
through
computation. A higher
mean score in the Inverse than in the control task would therefore
indi-
cate some knowledge in
the sample of children about the inverse relation between addition
and
subtraction. This is
the criterion for understanding inversion that has been adopted very
widely
(Bisanz & Lefevre,
1992; Bryant et al., 1999; Rasmussen, Ho, & Bisanz, 2003; Stern,
1992).
It could be argued that
there is an alternative explanation for such a result. The
alternative is
that children might do
better in the inverse than in the control task because they are
carrying out
a right-to-left
analysis (e.g., for 18 + 7 – 7 = ?, they work out first that 7 –
7 = 0 and then
that18 + 0 = 18). In
our view, however, children who solve the problem in this way do so
because they understand
inversion. They realize that an equal addend and subtrahend cancel
each other out and
conclude that there has been no change because of this cancellation.1
Table 1 gives the mean
proportion of correct scores for the three types of items: inverse,
control, and transfer.
(We present proportional scores here because there were different
numbers
of items in these three
tasks.) The children were more successful in the inverse than in the
control task. This
difference clearly establishes some knowledge of inversion in the
sample of
children in our study.
However, this knowledge
did not seem to be universal in our sample at the time of the
pretest.
We divided the
individual children into two categories: those whose proportional
scores were
higher with the inverse
than with control items and those whose Inverse scores were no better
then their control
scores (i.e., either the two scores were equal or the control score
was the
higher one). At the
time of the pretest, 39 of the 60 children (65%) were in the first
category:
1
One of the editors (AB)
of this special issue advised us to mention this alternative
hypothesis.
TABLE 1
Mean Proportion of
Correct Answers Per Task, Group, and Test (Study 1)
Inverse Task
Mean
SD
Complement Task
Mean
SD
Control Task
Mean
SD
Visual Demonstration
Group (N = 21)
Pretest
.34
.29
Posttest
.60
.20
Oral-Calculator Group
(N = 19)
Pretest
.29
.25
Posttest
.57
.22
.49
.97
.60
.97
.45
.66
.39
.04
.35
.04
.36
.38
.29
.39
.21
.34
.23
.41
.25
.32
.23
.25
.23
.30
Control Group (N = 20)
Pretest
.37
Posttest
.52
.25
.29
they had a higher
proportional score with the Inverse than with the control items. The
remaining
35% fell into the
second category. Thus, some children apparently knew about inversion
and
used this knowledge in
the pretest while others did not.
The mean proportional
scores presented in Table 1 also produced an unexpected result.
The children did
better in the transfer task than in the Inverse task. This was a
surprise because
the equal additions and
subtractions were juxtaposed directly in a single sum in the inverse
task
problems, while in the
transfer problems they were separate because they occurred in two
different
sums. For this reason,
we had predicted that the children would find it easier to take
advantage
of the inverse relation
with the Inverse than with the transfer problems.
We carried out a 3 x 3
mixed design ANOVA on the proportional correct scores in the
pretest, which
confirmed these differences. The main terms were group (visual
demonstration,
oral calculator,
control) and item (inverse, transfer, control) with repeated measures
on the
second variable. The
analysis produced a highly significant items difference with a large
effect
size (F(2, 114) =
20.14, p < .001, partial h2 = .261). Post-hoc tests of
within-subjects contrasts
confirmed that the
transfer items scores were significantly higher than the inverse task
scores
(F(1, 57) = 12.68, p =
.001, partial h2 = .182) and the inverse scores significantly higher
than the
control task scores
(F(1, 57) = 16.92, p < .001, partial h2 = .229).
There was neither a
significant group difference, nor a significant interaction, in this
analysis.
So, we can assume that
the performance of the children in the three different intervention
groups
was equivalent before
the interventions.
In our posttest
analyses we predicted that:
1. The intervention
given to the children in the visual-demonstration and oral-calculator
groups, which was about
the inverse relation, would improve their performance in the
inverse task and in the
transfer task as well.
2. The intervention
groups would not perform better than the control group in the control
task, which were about
calculating; the effect of the intervention was not expected
to result from a
general improvement in calculation ability but from the specific
under-
standing of the inverse
relation.
Table 1 shows that the
children’s scores were generally better after each intervention
than
before. It also shows
that as predicted the visual-demonstration and oral-calculator
groups, the
two groups who were
taught about inversion, improved more than the control group in the
inverse task and in the
transfer task which, in our view, involved an understanding of
inversion.
Also, as predicted, the
pattern of the groups’ scores was quite different in the control
task, in
which the groups taught
about inversion did no better (in fact they did slightly worse) than
the
control group.
The overall pattern of
posttest results, therefore, suggests that our inversion intervention
may
have had a positive and
specific effect. The children who were taught about inversion
benefited
from it with problems
that involved understanding inversion.
Our main purpose in
analysing these results was to establish the specific effects of each
intervention. In
particular, we wanted to know if the two types of teaching inversion
(visual
demonstration and oral
calculator) had significantly improved the children’s ability to
solve the
inverse items and also
the transfer items. We were not interested in directly comparing
these two
forms of teaching
inversion because in strict terms they were not comparable. There
were
several differences
between these two interventions.
The effect of
intervention type across the different types of items was analyzed
using three
difference scores,
which were calculated as the difference between each participant’s
posttest
score and pretest score
for each of the three tasks (inverse, transfer, control). Although it
is
usually preferable, in
an experimental design, to control for the effects of pretest using
an
ANCOVA (Tabachnick &
Fidell, 2001), difference scores were used instead because prelimi-
nary analyses
demonstrated a violation in the homogeneity of regression assumption
for both the
inverse and transfer
tasks.
The main analysis was a
group (visual demonstration, oral calculator, control) by task
(inverse, transfer,
control) mixed design ANOVA of the proportional correct difference
scores,
with repeated measures
on the last factor. Although there was a significant main effect for
task
(F(2, 114) = 13.73, p <
.0005, partial h2 = .194) and a near significant main effect for
group
(F(2, 114) = 2.58, p =
.084, partial h2 = .083), these results are mitigated by a
significant task by
group interaction (F(4,
114) = 3.21, p = .015, partial h2 = .101). This interaction was
expected
because it was
predicted that the interventions would not have the same effect on
the control
items as they would
have on the inversion and transfer items.
To confirm that this
difference was the source of the interaction and to understand the
inter-
action in general three
separate ANOVAs were conducted with the group as the between factor
and each of the tasks
as the three dependent variables. For both the inverse task and the
transfer
task, there was a
significant effect of group (F(2, 57) = 4.96, p = .010, partial h2 =
.148 and
F(2, 57) = 3.31, p =
.044, partial h2 = .104, respectively). Post-hoc Tukey Honestly
Significant
Difference (HSD) tests
established that in the inverse task both intervention groups demon-
strated more
improvement from pretest to posttest (34 percentage points for
vVisual demon-
stration, 31 percentage
points for oral calculator) compared with the control group (17
percentage points) but
did not differ from each other. In the transfer tasks, post-hoc tests
demon-
strated a significantly
greater improvement in the visual demonstration group (48 percentage
points) compared to the
control group (21 percentage points). The change in the oral
calculator
group (37 percentage
points) fell in between the other two groups but was not
significantly dif-
ferent from either of
them. Finally the ANOVA demonstrated no main effect of group on the
control items (F(2, 57)
= 0.62, p = .543), as predicted.
STUDY 2
The positive results of
Study 1 stimulated us to test whether it would be possible to improve
younger children’s
understanding and use of the inverse relation. In Study 2 we compared
the
effects of the visual
demonstration intervention with a control intervention that
concentrated on
the decimal structure.
Some changes were
introduced both in the pre- and posttest outcome measures and in the
intervention. First,
the numbers used in all the tasks were smaller than in Study 1: we
reasoned
that young children
might be discouraged if we presented them with the relatively large
numbers
that we used in tasks
designed for older children.
Second, we gave the
children two posttests: one immediately after the intervention and
the
second, delayed
posttest about three weeks later. The aim of the delayed posttest was
to assess
whether there was
forgetting, or whether the children maintained their level of
performance
across the two
posttests, or, perhaps (optimistically) whether there was any sign of
a sleeper-
effect which we have
obtained in other interventions with young children’s understanding
of
logical principles
(Nunes, Bryant, Pretzlik, & Hurry, 2006).
Third, Stern (1992)
observed that mixing inverse and control items has a deleterious
effect on
children’s use of the
inverse principle. We thus decided to use both inverse and control
items in
the pretest to assess
whether there was a difference in their performance across item type,
and in
the immediate posttest
in order to assess the results of the intervention but not in the
delayed
posttest.
Fourth, in view of the
children’s age, we administered fewer items in the pre- and
posttests in
order not to tax their
attention.
Finally, the items in
the pretest and posttests were presented in the context of concrete
objects
and drawings rather
than only orally, as in Study 1. It has been observed in different
studies
(e.g., Hughes, 1986;
Levine, Jordan, & Huttenlocher, 1992; Nunes, Schliemann, &
Carraher,
1993), and also
specifically in one previous study about inversion (Bryant et al.,
1999) that
children perform better
in problems presented in the context of a story than when they are
asked
to solve operations
presented orally.
We predicted that the
children in the experimental group would do better than children
in the control group in
the inverse task in the immediate and the delayed posttest but that
there would be no such
group difference in the control task at the time of the immediate
posttest. We did not
expect that the control group children would improve more than the
experimental in the
control task in the immediate posttest because the intervention given
to the control group
children was mainly about the decimal structure of the counting
system.
Method
Participants
The children (N = 39;
16 boys, 23 girls; mean age 5 years; SD = 3.5 months) were recruited
from two schools. They
were in their first year in school and were tested during their
second
term. Both schools
catered to a varied clientele in socio-economic terms.
Design
The children were
randomly assigned either to the experimental group (n = 20) or to the
con-
trol group (n = 19).
They were tested on three occasions. On the first, they were given a
pretest,
consisting of inverse
and control items, which was immediately followed by the first of two
intervention sessions.
On the second occasion they were given the second intervention
session,
which was followed by
the immediate posttest. On the third occasion, roughly three weeks
later,
they were given the
delayed posttest.
Procedure for the Pre-
and Posttests
In the pretest and the
immediate posttest the children were given six inverse problems and
six
control problems. These
items were not presented in separate blocks but mixed together in
random order. In half
of each of these problems we showed the children an initial row of
bricks,
which we then covered
with a cloth. Next we added and then subtracted (or vice versa) some
bricks to and from the
initial, and now covered, row. The bricks added and subtracted were
not
the same: addition was
performed to one end of the row and subtraction was applied to the
other
end of the row. In the
inverse items the number of bricks added and subtracted was equal (a
+ b – b).
In the control items
the numbers added and subtracted differed by at least 3 (a + a –
b), as in
Study 1.
The six remaining items
were story problems. These were presented with the support of
drawings, which
depicted adding and subtracting the same type of object (e.g.,
lollipops, books)
to a number of items
already inside a box. The number of objects in the box was presented
orally
to the children as part
of the story and was also written on the box; the numbers added and
subtracted were
represented as individual items. Again, the same number of objects
was added
and subtracted in each
inverse item, whereas different numbers (different by at least 3)
were
added and subtracted in
the control items. Figure 1 presents an example of a story problem.
The
pictures that we
presented to the children were in color; the color and spatial
arrangement of the
items added and
subtracted differed in all the items.
In the delayed
posttest, for the reason explained earlier on (see reference to
Stern, 1992), we
gave the children the
inverse task but not the control task. We started the block of
Inverse items
in this delayed
posttest with two practice trials with bricks in which the children
received
feedback about their
answers. This feedback took the form of allowing them to count the
number of bricks under
the cloth after they had given their answer. Children in both groups,
the
intervention and the
control group, received these two trials with feedback before the
test items
were presented. Then we
gave them the same six inverse items as in the pretest and the
immediate
posttest.
The children in this
study also formed the control group for another intervention study
anal-
ysing a different
aspect of children’s mathematical reasoning.
Procedure in the Two
Intervention Sessions
The children were
randomly assigned either to the experimental group or to the control
group. The children in
the control group participated in an intervention designed to attain
a
different aim. They
played a shop-game individually with the experimenter in which they
–5
+5
7
FIGURE 1 Example of an
inverse word problem used in the pre- and
posttests. The child
was presented with the card and told that there were
7 marbles in a bag. A
boy came and took 5 out and then another boy
came and put 5 in. The
child was then asked how many marbles were left
in the bag.
worked with coins of
different denominations and were asked to pay for small items in this
pretend shop. For
example, they could have 1 five-cent coin and 4 one-penny coins and
would
be asked to pay seven
cents for a toy in the pretend shop. The trials used different
combinations
of coins (including 1p,
2p, 5p, 10p, and 20p); in every trial, two different denominations
were
used. The children
could solve these problems by addition (e.g., 5 plus 2 is 7) or by
counting
(e.g., they could take
the 5-cent coin and count on, as they took one-penny coins). They
were
given feedback by
helping them count on from the larger value, when they could not
provide the
correct combination of
coins to pay the exact amount. Each child was given 20 shop-game
prob-
lems in each of the two
sessions.
The children in the
experimental group followed a procedure similar to that adopted in
Study
1 with the visual
demonstration group. However, there were some differences between the
two
studies. Because the
children in this study were younger, the value of the numbers used in
the
problems was always
under 10 and the form of presentation of the items differed. Two
types of
items were used: items
using bricks and items using objects.
1. The children first
were presented with a series of items using Unifix bricks, as those
described in Study 1,
starting with items that involved identity inversion and color cues,
and then proceeding to
items where these cues were not present. There were 15 items
using bricks in the
first session (six used color cues and nine did not) and six in the
second session (three
used color cues and three did not).
2. After the children
solved the problems with bricks, a series of items with other objects
(either marbles, or
pencils, or cards) was presented. The initial set of items was placed
into a box, and then
addition and subtraction transformations to the initial set were
carried out. These
children could not see inside the box, and so they could not tell
whether the items
subtracted were the same ones that had been added; the researcher let
go of the items when
they were placed in the box and shook the box around, giving no
indication that the
items were the same. The aim of using objects was to increase the
variety in the problems
in order to keep the children engaged in the task. There were six
trials using objects in
each of the two sessions.
As in Study 1, the
addition and subtraction performed in each problem given to the
experi-
mental group sometimes
cancelled each other out and sometimes involved values that differed
by 1 so that the
children had to monitor the operations carefully and could not answer
simply by
saying the initial
value of the set. The children received feedback about their
responses by being
allowed to count the
objects to check their answer.
The different
procedures that we adopted for the two groups both involved practice
in
addition (the control
group children had to add different amounts of money) but only the
experi-
mental group had to
carry out subtractions as well. Different material and procedures
were used
with each group. Thus
the intervention sessions differed for the two groups in more than
one
way and we have to be
cautious in interpreting positive results.
The kind of environment
where the interventions were carried out differed markedly between
the two schools. In
School 1, the experimenter and the child had a room at their
disposal.
In School 2 the
interventions were carried out in the library. This library opened
onto the play-
ground and was
therefore noisy. Other children often entered it during the
intervention sessions.
We were aware therefore
that school membership might influence the intervention effects.
Results
Preliminary analyses
showed no reliable differences between the boys’ and girls’
scores, whose
results we combined in
the subsequent analyses.
Pretest Analyses
Table 2 presents the
pretest scores for the two groups. Both groups did better in the
pretest in
the inverse than in the
control task and there was very little difference between the two
groups in
TABLE 2
Mean Number of Correct
Answers (Out of 6) Per Task, Group, and Test (Study 2)
Inverse Task
Mean
Experimental Group (N =
20)
Pretest
Immediate Posttest
Delayed Posttest
Control Group (N = 20)
Pretest
Immediate Posttest
Delayed Posttest
SD
Control Task
Mean
SD
1.05
1.95
3.25
1.65
2.10
2.84
0.95
1.64
2.10
1.53
1.62
2.19
0.05
0.05
–
0.15
0.10
–
0.22
0.22
–
0.37
0.31
–
pretest scores. Thus,
the two groups were on an equal footing in both tasks at the start of
the
experiment and at least
some of the children in the sample already knew about and made some
use of the inverse
relation.
A groups (experimental,
control) by tasks (control, inverse items) ANOVA was carried out
on these pretest scores
with repeated measures on the second factor.
In this analysis there
was no significant difference between groups and this variable did
not
interact significantly
with tasks. This means that the performance of the children in the
two
groups was at the same
level at the outset of the study before the two intervention
sessions.
The analysis produced a
highly significant tasks difference with a large effect size
(F(1, 37) = 40.25; p <
.001; partial h2 = .514). This confirmed that the children did much
better
in the Inverse task
than in the control task, which indicates some knowledge and use of
the
inverse relation at the
time of the pretest.
Since some of the
children definitely could use the inverse principle, we inspected the
distri-
bution of scores, which
showed that all the children scored below 2 items (out of 6) in the
control task; 36 gave
no correct answers and only 4 produced one correct answer. In
contrast, in
the inverse task 15
children scored between 2 and 5 (out of 6) items correctly; the
remaining 25
either scored 0 or 1.
None of the children got all 6 inverse items right. So, the
difference between
the levels of
difficulty of the two tasks is accounted for by 37% of the children.
We concluded that some,
but probably the minority, of the children in the sample started the
study with some
knowledge of inversion and also that the two groups were reasonably
well
matched in the pretest
before the intervention.
Posttest Analyses
We made two predictions
about the posttest results.
1. The children who had
been taught about inversion would improve more than the control
group children from
pretest to immediate posttest in the inverse task, but there would be
no such difference in
the control task.
2. In the delayed
posttest the experimental group would do better than the control
group in
the inverse task. (As
mentioned earlier, we did not administer the control task in the
delayed posttest).
Table 2 presents the
mean correct scores for the inverse and control tasks in the pretest
and
immediate posttest and
for the Inverse task only in the delayed posttest. There was no sign
of the
predicted group
differences in the immediate posttest. Both groups’ scores in the
inverse task
were better in the
immediate posttest than in the pretest, while there was no
improvement for
either group in the
control task. Since the control group improved as much as the
experimental
group in the immediate
inverse posttest, our first prediction was apparently not confirmed.
Both groups also
improved from the immediate to the delayed posttest, but the strength
of
this improvement
differed to some extent between the two groups. The experimental
group
children improved more
than the control group children, but the group difference was only a
moderate one.
As in the first
experiment, we analysed the difference scores for each posttest (the
difference
between each posttest
score and the appropriate pretest score). In the first analysis we
looked at
the difference scores
between the pretest and the immediate posttest for the Inverse and
control
tasks. This was a
groups (experimental and control) x tasks (inverse and control items)
ANOVA,
with repeated measures
on the second factor. There was one significant result in this
analysis,
which was between the
two tasks (F(1, 37) = 8.39, p = .006, partial h2 = .181). This
established
that the difference
between pre- and posttest was greater in the Inverse task than in the
control
task. However, since
the interaction between tasks and groups was not significant, the
greater
improvement from
pretest to posttest in the Inverse task had nothing to do with the
intervention.
Thus, the analysis did
not support our first prediction.
The aim of the next
analysis was to test our second prediction, which includes the scores
in
the delayed posttest.
This analysis was of the difference scores (the difference between
each
posttest score and the
pretest score) for the inverse task in the pretest and the immediate
and
delayed posttests. It
took the form of a group (experimental and control) by test
(immediate and
delayed posttests)
ANOVA with repeated measures on the second factor. A highly
significant
tests term (F(1, 37) =
16.28, p < .001, partial h2 = .306) established that the delayed
posttest
scores differed more
from the pretest scores than the immediate posttest scores did. As in
the
first analyses the
groups term was not significant and there was no interaction between
groups
and tests. This
analysis, therefore, did not support our second prediction either.
However, we noticed a
striking difference between the two schools, particularly in the
delayed posttest. The
experimental group children from one school improved in both
posttests
and quite markedly from
immediate to delayed posttest. This was not true of the children in
the
same group in the other
school. Table 3 illustrates this difference. The experimental group
children in school 1
did very well and much better than the control group children in this
school
in both posttests,
notably in the delayed posttest. There was no sign of this group
difference in
the delayed posttest
among the school 2 children.
We examined this
apparent discrepancy in an analysis of the two schools’ Inverse
task differ-
ence scores for both
posttests. This was a group x school x test (immediate posttest,
delayed
posttest) ANOVA with
repeated measures on the last factor. Its aim was to establish
whether the
apparent intervention
effect in school 1’s performance was reliable enough to produce a
group x
school x test
interaction.
This analysis produced
a significant test effect (F(1, 35) = 14.39, p = .001, partial h2 =
.291),
which was due to the
difference scores being higher for the delayed than for the immediate
TABLE 3
Mean Number of Correct
Answers (Out of 6) in the Inverse Task
Per School, Group, and
Test (Study 2)
School 1
Mean
Experimental Group
Pretest
Immediate Posttest
Delayed Posttest
Control Group
Pretest
Immediate Posttest
Delayed Posttest
N = 12
0.83
2.33
4.08
N=9
1.40
1.80
2.56
SD
Mean
N=8
1.38
1.38
2.00
N = 10
1.90
2.40
3.10
School 2
SD
0.84
1.88
1.68
1.58
1.81
1.81
0.22
0.22
2.14
1.52
1.43
2.56
posttest. There was
also a significant School effect (F(1, 35) = 4.40, p = .04, partial
h2 = .112),
which was due to the
difference scores being generally higher for school 1 than for school
2.
Finally, the
interaction between schools and groups narrowly missed significance
(F(1, 35) = 3.66,
p = .064, partial h2 =
.021).
The obvious discrepancy
in this analysis between schools allowed us to do separate groups
x tests analyses for
each of the two schools. This analysis produced no significant
effects with
school 2. The same
analysis with school 1, however, produced a significant difference
between
groups (F(1, 19) =
4.27, p = .05, partial h2 = .184), which established that the
improvement from
pretest to posttests
was greater for the experimental group than for the control group.
There was
also a significant
tests difference (F(1, 19) = 15.07, p < .001, partial h2 = .442),
which was due to
the difference scores
being generally higher for the delayed than for the immediate
posttest.
Finally, the
interaction between groups and tests was very nearly significant
(F(1, 19) = 4.04,
p < .06, partial h2
= .175). We explored the interaction in Bonferroni adjusted t-tests
(a = .0125).
These showed a
significant difference between the inverse and the control groups in
the delayed
posttest (t(19) = 2.84,
p = .01) but not in the immediate posttest (t(19) = 1.40, p = .18).
We can
conclude that by the
time of the delayed posttest there was a definite improvement in
school 1 as
a result of the
intervention. Thus, these analyses suggest that at the time of the
delayed posttest
the children from
school 1 had benefited from instruction about the inverse relation,
whereas the
children from school 2
had not.
We interpret this
result as an existence proof of the possibility of teaching young
children
about the inverse
principle. However, it is not yet certain when and under what
circumstances
the intervention will
work with young children.
DISCUSSION
We set out to answer
three questions in these studies. The first was whether it is
possible to
improve the
understanding and use of the inverse relation between addition and
subtraction in
children between the
ages of 5 and 8 years. The second was whether one needs to stress the
connection between the
inversion of identity and of quantity for such an intervention to
work.
The third was whether
direct teaching of the inverse relation will also help children to
solve
complement problems.
It is not always easy
to teach young children about logical principles (Inhelder, Sinclair,
&
Bovet, 1974). Yet, both
interventions in Study 1 were successful, even though they only
lasted
for two sessions. The
improvement in the children’s solutions to the inverse problems in
the
posttest was striking
in both cases, and much of this improvement must have been specific
to the
instruction given to
the children. The only difference between the interventions given to
the oral
and calculator group
and to the control group was that one group was taught about
inversion and
the other about
calculation. Both groups improved in the posttest, but in different
tasks. Those
taught about inversion
were better as a result in answering the Inverse problems. Those
taught to
calculate improved in
the control problems, which needed calculation.
After this promising
start, we need further studies to establish how durable the effects
are and
whether they can be
strengthened. For practical reasons we were only able to give the
children
an immediate posttest
in Study 1: we need to know whether the effects of intervention would
still be detected in a
later posttest after a delay of several months. Also, although it is
quite
impressive that our
two-session interventions were so effective, it would be useful to
know
whether there would be
even greater improvements in children’s understanding of inversion
after a longer
intervention.
The effect of the
intervention with younger children in Study 2 was less convincing.
The
intervention appeared
to work in one school, but not in the other. We have post-hoc
explanations
for this difference
between the two schools, but we cannot be sure that they are right.
At this
stage we can only
conclude that it is sometimes possible to improve 5-year-old
children’s under-
standing of inversion
but that it is also possible to fail to do so. We must also note that
for
practical reasons the
control group children were given some experience in addition during
the
intervention, but none
in subtraction, and we should therefore be cautious about this one
positive
result in Study 2.
The success that we did
have in Study 1 has a clear educational implication. It is that
instruc-
tion about this
important mathematical principle is needed in school and is
effective. The
amount of teaching that
children are given about inversion varies from syllabus to syllabus
and
country to country but
in the United Kingdom, at any rate, it has only a small part to play
in the
mathematics curriculum.
Some of the teaching offered to children in their second year in
school
clearly assumes that
they understand inversion: for example, they are taught that, instead
of
adding 9 to a number,
which they might find difficult, they could add 10 and then take 1
away.
In order to understand
why this works, it might be helpful for the children to know that 9 +
1 – 1
is 9. However, there
are no checks to see whether the children understand this or to help
them, if
they do not. Our
results suggest that more teaching about this vital arithmetical
principle could
be effective and
thoroughly profitable.
Our second question was
whether it is necessary to make a specific connection between the
inversion of identity
and of quantity in this teaching. Study 1’s results suggest that it
is not. One
intervention stressed
the identity-quantity connection and the other did not, but both
worked
well. Whether
re-capturing and incorporating children’s normal development in the
teaching
sequence is a good
teaching practice is always an interesting question. In this case,
our results
suggest that attention
to development in teaching is not essential but not damaging either.
The answer to the third
question was clear but it also contained a surprise. The surprise was
that the transfer
complement problems were actually easier than the Inverse problems.
We had
expected the opposite
because the need to use the inverse relation was more obvious in the
Inverse than in the
transfer problems, and also because the transfer problems also made
demands
on children’s
understanding of part-whole relations whereas the inverse problems
did not seem
to us to do so. One
possible explanation for the transfer task being the easier of the
two might lie
in the format of the
problems in these tasks. The transfer problems consisted of two
successive
questions (a + b = c
and c – b = ?), each with one operation. The inverse problems
contained two
operations in the same
question (a + b – b). Young children are more used to questions
with
one operation; two
operations in sequence, in a single item, are quite unusual, at any
rate in the
classroom.
Despite this surprise,
the results in Study 1 suggest that there may be a connection between
understanding the
inverse relation between addition and subtraction and solving
complement
problems. Complement
problems were not part of either intervention, and yet the visual
demon-
stration interventions
improved the children’s ability to solve these problems. Using
visual
methods to teach
children about inversion helps them to solve complement problems, and
this suggests that the
understanding of inversion may play a role in solving these problems.
Understanding
part-whole relations may also be a constraint in these problems, and
the relative
success here of the
visual demonstration intervention may have been due to this method
also
prompting the children
to think more clearly about parts and wholes than before.
The connection between
children’s understanding of inversion and their solutions to
comple-
ment problems is
another indication of the importance of the inverse relation, which
has been
rather neglected in the
past both in theories about children’s mathematical development and
in
mathematical education.
Yet, this understanding is a crucial part of children’s
mathematical
progress, and
eventually of their mathematical understanding in adulthood.
Convincing
evidence for this last
point was provided recently by Stern (2005) in a longitudinal study
which
established that
children’s performance in inversion tasks during their second year
at school
significantly predicted
their performance in an algebra assessment when they were 23 years
old
even when the effects
of IQ were partialled out. Our evidence that it is easy and
practicable to
teach children about
the inverse principle should now encourage teachers to include this
vital
principle in their
teaching when it is possible to do so.
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