terça-feira, 2 de abril de 2013

Teaching Children About the Inverse Relation Between Addition and Subtraction


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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