Teaching children how to include the inversion principle in their reasoning about quantitative relations

Teaching children how to include the inversion principle in their reasoning about quantitative relations

Terezinha Nunes & Peter Bryant & Deborah Evans & Daniel Bell & Rossana Barros

# Springer Science+Business Media B.V. 2011

Abstract

The basis of this intervention study is a distinction between numerical calculus

and relational calculus. The former refers to numerical calculations and the latter to the

analysis of the quantitative relations in mathematical problems. The inverse relation

between addition and subtraction is relevant to both kinds of calculus, but so far research on

improving children’s understanding and use of the principle of inversion through

interventions has only been applied to the solving of a+b−b=? sums. The main aim of

the intervention described in this article was to study the effects of teaching children about

the explicit use of inversion as part of the relational calculus needed to solve inverse

addition and subtraction problems using a calculator. The study showed that children taught

about relational calculus differed significantly from those who were taught numerical

procedures, and also that effects of the intervention were stronger when children were

taught about relational calculus with mixtures of indirect and direct word problems than

when these two types of problem were given to them in separate blocks.

Keywords Inverse relation between addition and subtraction . Relational calculus .

Numerical calculus . Teaching the inverse relation

Our aim in this study is to evaluate two different approaches to teaching children about the

inverse relation between addition and subtraction in the context of story problems. The

teaching of arithmetic must consider two different questions about the types of cognitive

demand that children face in solving problems. One is how they can efficiently and

accurately carry out arithmetical calculations. The second is how they know the right

arithmetical operation to use in order to solve a particular problem.

Vergnaud (1979) distinguishes these two abilities by referring to the first as “numerical

calculus” and to the second as “relational calculus”. By numerical calculus, Vergnaud

means the operations on the numbers to find the sum, in an addition, or the rest, in a

subtraction. By relational calculus, he means “the transformation and composition of

relationships given in [a] situation” (Vergnaud, 1979, p. 264). He further explains this idea

by contrasting two situations that require the same numerical calculus (24−6) but different

relational calculus. In the first, “I had 24 francs and spent 6 francs. How much do I have

now?”, little relational calculus is demanded because the situation is very close to children’s

initial conception of subtraction as “take away”. In the second, “My grandmother has just

given me 6 francs and I now have 24 francs. How much did I have before?”, the situation is

close to children’s initial conception of addition; in order to conceive of this problem as a

subtraction problem, the children must think of the transformation that did take place (my

grandmother gave me 6 francs) and consider what transformation would bring the amount

of money back to its initial state—i.e., what is the inverse of this transformation. We think

that the inverse relation between addition and subtraction is an important aspect of both

numerical and relational calculus.

1 The role of the inverse addition–subtraction relation in numerical calculus

In the domain of smaller numbers, there is evidence that children’s knowledge of addition

and subtraction number facts and their understanding of the inverse relation are independent

of each other (Bryant, Christie, & Rendu, 1999; Gilmore & Papadatou-Pastou, 2009). There

is also evidence that teaching children about the inverse relation improves their

performance in problems of the form a+b–b=? but not in other computation problems

that can be solved by memorization of number bonds (Nunes, Bryant, Hallett, Bell, &

Evans, 2009). However, there are suggestions in the literature that there may be a role for

children’s understanding of the inverse relation in the development of their computation

skills if the computations are challenging. There are three different kinds of reason for this

possible connection.

The first relates to understanding the exchanges in written addition and subtraction of

multi-digit numbers algorithms. Bryant and Nunes (2009), Fuson (1990), Gilmore (2006)

and Nunes and Bryant (1996) have argued that understanding carrying and borrowing

requires understanding the inverse relation between addition and subtraction. Fuson, for

example, argued that, when children are adding 7 tens and 6 tens, in order to understand the

ten-for-one to the left exchange, they need to realize that they are taking 100 away from the

tens place and adding 100 to the hundreds place; so the value of the total is not changed. A

similar reasoning is required in a subtraction such as 2,107−72; the conservation of the

minuend cannot be understood unless one understands that taking away 100 from the

hundreds place and adding 100 in the form of 10 tens to the tens place does not change the

quantity.

A second reason for an effect of understanding of the inverse addition–subtraction

relation on computation prowess is that one can use additions to solve subtraction sums if

the numbers are close to each other. The efficiency of this computation strategy seems

obvious in problems such as 71−69, which could lead to calculation errors if the children

were trying to use the written algorithm. Counting up from 69 is a quick and easy approach

to this subtraction. Torbeyns, Smedt, Stassens, Ghesquière and Verschaffel (2009) reviewed

the literature on the use of addition to solve subtractions, which they called indirect

addition, and found that this is considered by many researchers as a useful computational

approach (e.g., Fuson, 1986; Beishuizen, 1997; Brissiaud, 1994). However, the evidence

that the ability to perform indirect additions improves children’s computational success is

meager. Torbeyns et al. (2009) carried out three studies to investigate the use of indirect

Teaching children how to include the inversion principle

addition to solve subtractions, one with adults and two with children. Their study with

adults revealed that most of the participants used indirect addition to solve subtractions

when it was advantageous to do so. However, second to fourth grade children adopted

indirect addition to solve subtractions much less often than the researchers expected, even

when the numbers in the problems favored its use and the interviewer instructed them to

think of alternative strategies.

Fuson (1986) and Fuson and Willis (1988) considered their teaching of indirect addition

to solve subtraction problems successful, but they had no control group in either study, and

so their conclusion that the children’s subtraction skills improved due to the use of indirect

addition were not substantiated properly. Klein, Beishuizen and Treffers (1998), who taught

children to use indirect addition with the support of the empty number line, also concluded

that they had demonstrated the success of this method, but this was not the only method

that the children had been taught, and so it is difficult to attribute the children’s progress in

computation skills only to the use of indirect addition. Unfortunately, the study did not

include a control group that received instruction on direct subtraction and thus the evidence

for the importance of indirect addition is not robust.

The third example of the use of the inverse relation between addition and subtraction to

solve computations comes from oral arithmetic. Nunes, Schliemann and Carraher (1993), in

their descriptions of oral arithmetic procedures used by street vendors in Brazil, report two

different ways in which this inverse relation was used to solve computations. Problems

about change were typically solved through indirect addition: for example, when

calculating the change to be given to someone who spent 80 Cruzeiros and paid with a

500 note, one child said: “Eighty, ninety, one hundred. Four twenty” (Nunes et al., 1993, p.

25), solving the subtraction through indirect addition. In another example, this inverse

relation was used successfully in a different way. A child solving the problem 243−75 said:

“You just give me the 200. I’ll give you 25 back. Plus the 43 that you have, the hundred and

forty-three, that’s 168” (Nunes et al., 1993, p. 41). In this latter example, the child makes the

move of subtracting 143 from 243, simplifying the problem, which becomes 100−75, and

then adds 143 back to find the answer. This method is not peculiar to children who work as

street vendors. In schools in England, children are taught that instead of subtracting 9 from

another number, they can subtract 10 and then add 1. If the children understand the inverse

relation, they know that subtracting n+1 and then adding 1 gives the same result as

subtracting n.

The case that, in principle, knowledge of the inverse relation could have an important

role in computational effectiveness is plausible but, so far, the evidence that this is true of

children’s computations is not at all strong. The connection between understanding the

inverse relation and computation skills may be restricted to additions and subtractions that

are not easy for children and that cannot be solved by memorization. Torbeyns et al. (2009)

point out that the use of indirect addition to solve a subtraction, which they refer to as

“compensation”, may tax young children’s working memory. So, children may stay away

from this approach unless they have external aids to support their memory. In the

intervention studies that appear to have succeeded, children had finger patterns (Fuson,

1986; Fuson & Willis, 1988) or the empty number line (Klein et al., 1998) as external aids

that could reduce working memory demands during calculation. It is possible that the

Brazilian street vendors had such fluency in the use of the inverse relation that it did not tax

their working memory, and so they adopted this calculation strategy spontaneously. This

would be in line with the view proposed by Case, Kurland, and Goldberg (1982) who

suggest that greater fluency increases processing space because it increases efficiency in the

use of mental resources.

In conclusion, it seems to be a smart move to use the inverse relation between addition

and subtraction in order to simplify some difficult computations. It may be necessary for

understanding multi-digit addition and subtraction with regrouping. Although it may confer

an advantage in some problems, it is not so easily adopted by children even after teaching.

Comparative research in which teaching relies exclusively on mental resources versus

teaching that uses external tools could shed light on why school children did not adopt the

method so easily in certain studies.

2 The role of the inverse addition–subtraction relation in relational calculus

The role of inversion in children’s relational calculus has been entirely neglected, even in

cases where some discussion of this possible link might have helped the researchers’

theoretical analysis. An example is the argument by Booth (1981) that “there may be two

“systems” of mathematics coexisting in the secondary school classroom: the formal taught

system, and a system of child methods which are based upon a ‘counting’, ‘adding-on’, or

‘building-up’ approach…The difficulties which some children appear to experience in

mathematics is suggested to be due in part to these children’s non-initiation into the formal

taught system” (Booth, 1981, p. 29). Her objection is not that the “child-methods” are

incorrect but that, in the context of solving problems, the children may be unable to convert

their “add-on” method to the formal representation of the problem solution. In problems in

which it would be advantageous to use a calculator, thinking in terms of “adding on” rather

than subtracting could be an obstacle.

Booth illustrates her point with an example. The 10–13-year-old children in her study

were asked: “The Green family have to drive 261 miles to get from London to Leeds. After

driving 87 miles they stop for lunch. How do you work out how far they still have to

drive?” The children were offered different alternatives for the answer; among these were:

261−87, 261+87, 87−261, and 87+174. The problem is a typical missing addend problem:

87+?=261. In order to decide how to work out its solution, presumably using a calculator

rather than “child methods”, some relational calculus is required to arrive at 261−87.

The rate of correct responses was 60% for the 10-11-year olds, 60% for the 11–12 year-

olds, and 67% for the 12–13-year-olds (Hart, Brown, Kerslake, Kuchermann &

Ruddock, 1985); these rates show little improvement across this age range. The

expression 87+174 would be a good representation of the child’s thinking if the child

had used indirect addition but its choice decreased from 5% for the younger participants

to 2% for the oldest; it seems that the large gap between the values in the problem did not

favor the use of indirect addition.

Booth (1981) reports the answer of a 13-year-old, who first attempted to work out the

solution by indirect addition, and then chose 87+228 as the answer (the numbers were

different in this version of the problem). When asked whether this is really what he had done,

he said “No, I built it up” and when asked which expression showed how to work out the

solution he answered that he did not know “the sign for adding on” (Booth, 1981, p 32).

The point that we wish to make is that what may be advantageous in terms of

computation skill may not be advantageous when one has a calculator to work out the

solution to an applied problem. In order to take full advantage of the calculator, one needs

to carry out the necessary relational calculation. In this case, the children should use their

understanding of the inverse relation between addition and subtraction in order to convert a

problem that seems to be about addition—how many more miles do the Greens need to

travel to get to Leeds—into a subtraction.

Teaching children how to include the inversion principle

We think that children do live with two “systems” of mathematics, as Booth suggests,

but that the difference between them is not that one system is characterized by “child

methods” and the other by formal representations. When students need to solve a

computation, they can use a variety of properties of the operation to arrive at the answer

efficiently and accurately. It is an advantage to solve 71−69 by counting on: this is about

the relations between numbers—i.e., about numerical calculation.

When students plan to use a calculator to solve a problem, they need to carry out

the relational calculation before they enter the numerical expression in the calculator.

In the Greens problem, knowledge of the inverse relation should help students arrive

at 261−87 as the operation to enter in the calculator. This is a different system,

because it is about the relations between quantities in the problem and not about the

relations between numbers; in the terminology used by Vergnaud (1979), the difficulty

is the relational calculus rather than numerical calculus. Thompson (1993) emphasized

the difference between numerical and relational calculation in much the same way:

“Quantitative reasoning is the analysis of a situation into a quantitative structure—a

network of quantities and quantitative relationships…A prominent characteristic of

reasoning quantitatively is that numbers and numeric relationships are of secondary

importance, and do not enter into the primary analysis of a situation. What is important

is relationships among quantities” (p. 165).

Pre-school children usually make the appropriate moves to solve problems that describe

a situation in which a quantity increases and the problem is solved by addition or the

quantity decreases and the problem is solved by subtraction (Carpenter & Moser, 1982;

Riley, Greeno, & Heller, 1983). Problem difficulty increases significantly when the story is

about a quantity that increases but it is most appropriately solved by a subtraction (or the

reverse). Vergnaud (1982), for example, reports approximately twice as many correct

responses (about 50% correct) by French pre-school children (aged about 6 years) to a

problem that involves no relational calculus, because the quantity increases and the problem

is solved by addition, than to a problem in which the quantity decreases and the solution is

obtained by addition (about 20% correct responses). In the latter, the children were told that

Bertrand lost 7 marbles in a game and had 3 marbles at the end; they were asked how many

marbles he had before the game. It would be to the children’s advantage to realize that they

can revert to the initial state, before Bertrand lost the marbles, by adding 7 back to

Bertrand’s marbles; this would be particularly useful if the numbers in the problem were

large and the children could use a calculator to solve the problem.

Thus, it might be important for pre-school children to realize that they can use different

strategies when solving problems without a calculator; when using a calculator, they need

to think which expression to enter into the calculator and why. Vergnaud (1979) suggests

that “solving a problem by choosing the right calculation is a very strong criterion for the

acquisition of concepts (p. 264)”; it reveals the compositions of relations that the children

are able to carry out mentally.

3 Interventions to teach children about the inverse relation

The few studies that aimed to teach children about the inverse relation have all focused on

its use in numerical calculus. No intervention studies on the use of inversion in relational

calculus have been done. Nevertheless, these intervention studies raise some points to

consider in the design of interventions to improve children’s use of the inversion principle

in relational calculation.

Siegler and Stern (1998) conducted a study with German second graders, aged between

8.4 and 9.6 years, who participated in eight sessions in which they were asked to solve

computations. The aim of the study was to investigate in detail how children changed their

solution of problems as they encountered them repeatedly in the experimental situation. In

the pre-test (session 1), the children were presented with 20 sums, half of the form a+b−b

(inversion problems) and half of the form a+b−c (control problems). In the practice

sessions, the children were randomly assigned (a) either to a block practice group, in which four of the practice sessions contained only inversion problems (sessions 5 and 7 contained both inversion and control problems), (b) or to a mixed-problems group, in which all seven sessions contained a mixture of problems.

The eighth and final session was a post-test. It included inversion generalization (a+

b−a) questions, in which the inversion reasoning was appropriate, and questions of the

form a+b+b, which are superficially similar to the inversion questions because of the

repetition of a number, but to which the inversion reasoning should not be applied.

Siegler and Stern predicted that the children in the block group would be able to pick up

the pattern of the inverse problems easily, because it was applicable to all the problems in

four sessions, whereas the mixed group saw only half of the problems with this pattern in

each session. They also predicted that the block group would use the inversion reasoning

more frequently, would make more appropriate generalizations, but also make more

inappropriate generalizations. Thus Siegler and Stern expected that children in the block

group would achieve surface learning, i.e., identify superficial aspects of the problem, but

not necessarily understand the inverse relation. Deep learning should result in understand-

ing the inverse relation and not applying it to inappropriate questions. Siegler and Stern

found that all these predictions were supported in the overall analysis of the sessions.

However, in sessions 5 and 7, in which both groups only had half of the problems of the

form a+b–b, the use of inversion to find the solution in these problems did not differ

between the two groups.

The block group did in fact overgeneralize the inversion solution to questions in

which this strategy was not appropriate: 63% of the children in the block group

preferred the inversion solution (i.e., used it at least 67% of the time) when it was

appropriate for the question and 63% also preferred it when it was not. The comparable

figures for the mixed group were: 20% preferred it when it was appropriate for the

question and 13% when it was not. When the two criteria were combined (i.e., using

inversion most of the time when it was appropriate and rarely when it was not) the two

groups did not differ: this ideal pattern was demonstrated by 33% of the children in the

mixed group and 31% in the block group. The apparent advantage of the block group

in the overall use of inversion across sessions seemed to disappear in the sessions in

which they had to make a choice. The differences were most likely task differences

rather than differences between the groups.

A second intervention study was carried out by Nunes, Bryant, Hallett, Bell and Evans

(2009) in the UK. They worked with two different age levels, 8 and 5 years, in two studies.

The aim of the first study, with 8 year olds, was to compare two ways of teaching the

children about inversion, by visual or by oral means. The design was a pre-test,

intervention, and post-test. In the pre- and post-test, the children were asked questions

that could be solved by inversion (a+b−b=?), control questions (a+b−c=?) and transfer

questions (a+b=c; what is c−b?), which were not part of the training. All questions were

presented orally and symbolically (e.g., 17+8−8).

During the intervention, the visual group observed a researcher performing trans-

formations on rows of bricks. The transformations were visible but the row of bricks was

partially covered, so that the children could not answer by counting. Some of the questions

involved exact inversion (a+b−b) whereas others required the children to use inversion and

compensation at the same time [e.g., a+b−(b+1)]. These questions were included because

the researchers wanted to keep the children attending rather than repeating always the initial

number in the problem. Feedback was given by allowing the children to count the bricks.

The oral group answered the same questions but these were presented only orally; after they

gave their answer, they were allowed to enter the question in a calculator and check the

result. This gave the children the opportunity to repeat the question and thus a second

chance to note the sequence of numbers and operations in the question. The control group

solved the same additions and subtractions presented to the intervention children but not as

part of the same question: instead of answering the inverse questions, they were asked to

compute a+b=? and later c−b=?

Both intervention groups made significantly more progress than the control group in the

inversion questions; only the visual group made significantly more progress than the

control group in the transfer questions.

In the study with 5 year olds, Nunes et al. (2009) adopted the visual method for the

intervention. The control group worked on a task related to the structure of the decimal

system. The intervention with 5 year olds was moderately successful: the children in the

intervention group in one of the two schools made significant progress compared to the

control group but there was no sign of improvement in the second school. The authors

interpreted this result as demonstrating that it is possible to improve 5 year olds’

understanding of inversion but could not account for the differences in results between

schools.1

These studies suggest that it is not easy to teach children how to use knowledge of the

inverse relation to simplify numerical calculations. Visual methods in which the children

can obtain feedback seem more promising than those in which the children are presented

with computations only symbolically. The modest gains in the intervention by Siegler and

Stern (1998) suggest that methods in which no feedback is given are not very effective. A

puzzling issue raised by the Siegler and Stern (1998) study is whether block or mixed

presentation of different types of problem is better for learning.

Why should it be so difficult to teach children how to use inversion reasoning in

numerical computations? Torbeyns et al. (2009) hypothesized that the use of inversion

reasoning places high working memory demands on children. Their hypothesis is consistent

with Sweller, van Merrienboer and Paas’s (1998) application of cognitive load theory to

mathematics learning. Sweller et al. divided cognitive tasks into two kinds, low-element

interactivity and high-element interactivity, and suggested that the former can be learned

serially (i.e., in blocks) whereas the latter requires that several elements in the task should

be manipulated simultaneously in order for learners to attain understanding. This would

suggest that, when teaching about the inverse relation, it is best to mix items that can be

solved by inverse reasoning with those that cannot, rather than use serial presentation of the

items, even though mixing different types of item places greater working memory demands

on the learner.

1

Lai, Baroody and Johnson (2008) report an intervention study in which they taught the children about the inverse relation; however, it was taught as an aim in itself, not as a tool to solve computations. For this reason, it is not reviewed here.

In a high-element interactivity task, learners should be able to develop a schema that

helps them understand the task and thereby reduces memory load. If a high-interactivity

task is cut into sections that obscure the relationships between elements, learners will not

attain understanding. This analysis suggests that, although it may be possible to teach

children about the inverse relation using blocks of questions of the same type, this

teaching might lead to surface learning. The study by Siegler and Stern (1998) showed

that this was indeed the consequence of block learning. But can children learn about the

inverse relation if they are taught by designs that challenge their memory capacity? We

addressed this question in our study about children’s use of inversion in the context of

relational calculus.

The study

The study reported here compared two ways of helping children to use their knowledge of

inversion in relational calculus. The design was a pre-test, intervention, post-test design,

with two intervention groups and one control group. Both intervention groups received the

same amount of teaching about relational calculus, but the block group solved all the

problems that did not require relational calculations in 1 day and all those that did on

another day, whereas the mixed group solved the same problems mixed during both training

sessions. The control group solved computations using a number line, i.e., worked on

numerical calculus. The control group would thus work on a numerical task, be exposed to

the same pre- and post-tests, and have the same level of familiarity with the experimenters.

We hypothesized that teaching children about numerical calculus does not improve their

relational calculus whereas explicit teaching about relational calculus is effective, which

means that this control group offers a good baseline for analyzing the results of the

intervention. Thus, we predicted that both groups taught about relational calculus would

differ significantly from the control group in solving applied problems that require using the

inverse relation at post-test.

5 Method

5.1 Participants

The children were recruited from two grade levels (2nd, mean age=7.20 years, and 3rd,

mean age=8.06 years) and randomly allocated to one of three groups. They all participated

in a pre-test, two training sessions, an immediate post-test, and a delayed post-test, given

about 8 weeks later. We recruited an equal number of children in grades 2 and 3 and

randomly assigned them to each group. However, due to a faulty connection of the

equipment which the children used to enter the answers with the computer, occasionally

and randomly some of the responses made by some of the children were not recorded. The

missing data resulted in 15 children being in the mixed group and 14 in each of the other

two groups. The children’s age did not differ across groups at pre-test according to a one-

way analysis of variance (F2,57 =0.33; ns). A chi-square test did not show a significant

association between group membership and grade level (chi-square=2.69; ns), but because

the distribution was rather asymmetrical we decided to control for grade level when making

group comparisons.

Teaching children how to include the inversion principle

5.2 The pre- and post-tests

Three researchers were involved in this study. Two researchers carried out the pre- and

immediate post-tests, as well as the intervention, with the same children. The delayed post-

test was carried out by a third researcher, blind to the children’s group membership and who

was introduced to the children as the first researcher’s friend. In the pre-test and the

immediate post-test, the children were presented with 12 story problems and were asked to

use a calculator to find the answer; they were not asked for a numerical answer. In the

delayed post-test, there were 24 problems and the procedure was the same. The children

were given a point for entering into the calculator each arithmetic expression that would

lead to the right answer. For example, in the problem “Donald grows sunflowers in his

garden. He gives Daisy 2 sunflowers to take home with her. Now Donald has 5 sunflowers.

How many sunflowers were in Donald’s garden before?”, the correct calculation could be

either 2+5= or 5+2=. If the child were to enter 5−2= or 2−5=, using a subtraction rather

than an addition, this was considered incorrect.

Because the children were inexperienced with large number arithmetic, the pre-test

involved only problems with numbers up to 11 (including the result of additions). Although

in principle, the difficulty of the numerical calculus might not affect the children’s

performance, because the answer we required was to enter in the calculator an operation

that would lead to finding the answer, we took into account that Baroody and Lai (2007)

noted that children’s performance in a mathematical task can suffer interference if the

numbers are too large and unfamiliar to them (see also Dowker, 1997). However, during the

teaching sessions, the children were exposed to small numbers at first and moved on to

larger numbers later, so they became more familiar with large numbers during the study and

it was possible to use large numbers in the post-tests. The immediate post-test contained

only problems with larger numbers, and the delayed post-test contained problems with

small and with large numbers. These were the same story problems used in the pre- and

post-tests. The larger numbers used in the teaching sessions and the post-tests included

values in all the decades over 10 and under 60.

The pre-test, immediate post-test, and delayed post-test included an equal number of

change story problems (i.e., problems in which quantities change) of three types:

(1)

(2)

(3)

result-unknown or direct problems, which required no relational calculation;

start-unknown problems, which were all inverse problems and required relational

calculation;

change-unknown problems, which are similar to direct problems if the story is about a

decreasing quantity because the children hear a story about a quantity that decreases

and can use subtraction to find the answer, but are similar to inverse problems if the

story is about an increasing quantity because the operation they must enter in the

calculator to solve the problem is a subtraction.

5.3 The teaching sessions

Two researchers carried out the intervention sessions, which took place over 2 days. On the

first day, the children participated in the pre-test and the first intervention session; on the

second day, they participated in the second intervention session and the immediate post-test.

Both researchers were trained to run the different groups. They received a package which

contained the record sheets on which the children were pre-assigned to the different groups.

On two consecutive days of work, they carried out the same procedure (i.e., mixed, block,

or control group procedure), moving on to a different procedure in the next 2 days. This

was found more suitable during the training phase than attempting to follow different

scripts for the intervention on the same day.

The sessions were presented to the children as a math challenge game, with levels that

the children had to conquer. For all three groups, each set of 10 questions was separated

from the subsequent set by an animated slide that announced that the children were

progressing to the next game level. For all three groups, the problems were presented on a

computer screen and feedback was given on the screen by “consulting the computer”,

represented by a computer image on the screen that acted as an action button: when the

children clicked on it, the answer appeared on the screen. This resource was used to make

the procedure as similar as possible across groups and make the experience more enjoyable

for the children.

5.4 Procedure for the control group

During the teaching sessions, the control group was presented with arithmetic operations on

a computer screen and asked to say what the answer was. The initial operations were with

small numbers and the children were able to answer these questions either by saying the

answer by memory or by counting. After the first set of small number problems, they were

presented with computations with larger numbers and with a number line to help them solve

the problems. Before they started this level in the “maths challenge”, they were given three

examples of how to use a number line to solve the computations. When the first example

appeared on the screen (15+24), the experimenter explained that the numbers now would

be larger but the children would not have to solve the computations mentally, and that

they could use a number line, which the experimenter provided. The number line had

the tens in different colors to make it easier to count them; the children were asked to

count in decades as the experimenter pointed to these on the number line. Although it

is common to use number lines to solve computations in primary schools in England,

the experimenter asked each child whether (s)he knew how to use it and went through

one addition (15+24) and two subtractions (53 −37 and 53−39) with each child. If the

child counted in ones, the experimenter showed that the child could count in tens and

then in ones, as this would save time. For example, when adding 24, the child could

move 10 up, then 20, and then 4 to find the answer; this would always be done while

demonstrating on the number line. The experimenter would then write the child’s

answer and they would “consult the computer” by clicking on the computer image on

the screen, and the answer would appear on the screen next to the arithmetic

expression, providing the children with feedback.

For the subtraction examples, if the children moved down the number line from the

value of the minuend to find the answer, the experimenter wrote the answer and then asked

them to show the two numbers on the number line; the children were then asked if they

could find the answer in another way. If they did not spontaneously count up from the

smaller number to find the difference, the experimenter suggested that they could try doing

so. If the child did not know how to do this, the experimenter demonstrated it, wrote the

answer down, and then they “consulted the computer” to find the answer. If the child solved

the subtraction using the counting up procedure, the child was asked whether there was

another way of finding the answer, and the use of counting down was demonstrated, if

necessary. For the second subtraction example (53−39), the children were asked which

procedure would be faster.

After this initial demonstration, the arithmetic expressions were presented on the screen

and the children were asked to provide the answer using the number line. The experimenter

continued to encourage them to use the number line in different ways. The experimenter

wrote the child’s answer after each trial and they then checked each answer by consulting

the computer.

5.5 Procedure for the intervention groups

During the teaching sessions, the intervention groups solved result-unknown, direct

problems and start-unknown, inverse problems; change-unknown problems were not

included in the training to provide an indication of whether the taught groups had achieved

deep learning of the use of inversion in relational calculation.

The children in the intervention groups were presented with a story problem verbally

and also represented as a cartoon (three drawings in series) on a computer screen. Figure 1

illustrates a missing minuend problem. The text which appears in the figure was read to the

children; there was no text on the screen. 2

The calculations required to solve these problems were the same ones presented to the

control group but, in contrast to the control group, they did not have to perform numerical

calculations. The children were asked to show how the solution could be worked out using

a calculator. The numerical expression that they entered in the calculator and its result

appeared on the computer screen; for example, if the child entered 39−12 into the

calculator, the expression 39−12=27 appeared on the screen. The children then obtained

feedback by “consulting a computer”: clicking on an action button (a computer image)

made the right expression(s) appear on the screen: in this example, the computer image

would show 12+39=51 and 39+12=51, one above the other. If the child had made the

correct choice, (s)he was asked to explain how (s)he knew that this was the right sum to do.

If the child had entered a different operation, the experimenter and the child discussed the

problem.

The children made almost no errors in the direct problems. Most mistakes in the inverse

problems resulted from entering the wrong operation, only occasionally from an error in

typing the numbers. In all inverse problems, entering the wrong operation would produce

an answer that could be considered unreasonable for the problem: i.e., the value for the start

was larger than the result when it should have been smaller or smaller than the result when

it should have been larger. For example, in the problem in Fig. 1, if the child entered a

subtraction, the answer would be a number smaller than 39. The pictures on the screen were

used as a support for the discussion of the child’s answer. The experimenter would note that

in the final slide the postman has 39 letters; in the middle slide, he was delivering 12 letters;

the child would then be asked whether he should have more or fewer letters before

delivering them. The children in general acknowledged that he should have more letters in

his bag in the first slide. The experimenter then asked how one could find out how many

letters were in the bag before he delivered some. If the child did not have an idea, the

experimenter would suggest that perhaps one could imagine that we would be putting the

letters back in his bag. This was generally sufficient for the children to think of using

addition to solve the problem. It was expected that these conversations would help the

children build a scheme of the situation, which would help them reason that, if they wanted

2

The characters in the story were common cartoon characters at the time. Permission was obtained for use of these in the research. For copyright reasons, a different illustration is used in Fig. 1.

Fig. 1 An illustration of how the problems were presented in the pre- and post-test. The screen showed only the pictures; the stories were read to the children by the researcher

to know what the situation was before the transformation, they would need to undo the

transformation. The same discussion pattern was used with both intervention groups.

The two intervention groups differed only in how the problems were ordered: the block

group solved all the 40 direct problems in one session and all the 40 inverse problems in

another session, whereas direct and inverse problems were randomly mixed in both sessions

for the mixed group. For each problem type, there were 20 examples with single digit

numbers and 20 examples with two-digit numbers (over 10 and below 60). In each session,

the children solved the 20 examples with small numbers in the first half of the session and

the 20 with larger numbers at the end of the session. It was reasoned that the children would

feel a greater need to use the calculator if the numbers were too large for their mental

computation skills.

6 Results

Our prediction was that the children who were taught about relational calculations would

make significantly more progress than the children taught numerical calculations in start-

unknown problems, which require relational calculation. The change unknown problems

provide an interesting case for testing the flexibility of the children’s understanding: if the

quantity increases (a+?=b), the operation entered in the calculator should be a subtraction,

which is the inverse, but if the quantity decreases (a−?=b), the operation to be entered is

not the inverse. If the children had learned a rule that they applied without reasoning—that

if the problem is not a result-unknown one they must use the inverse operation—they

would make little progress in these problems as a result of instruction. Although they were

not taught during the intervention about change-unknown problems, we expected them to

be able to use the understanding gained through the discussion of start-unknown problems

to solve change-unknown problems rather than use the inverse operation without thinking.

We did not expect the groups to differ in the result-unknown, direct problems. For reasons

given earlier on, we had no specific prediction regarding the comparison between the mixed

and the block group.

Table 1 presents the total scores by group and by testing occasion. The children’s pre-test

scores were subjected to an analysis of covariance (ANCOVA) to assess the comparability

of the groups; in this analysis, grade level was used as a covariate, groups (mixed vs. block

vs. control) was a between participants factor and the dependent variable was the children’s

total score. Grade level was a significant covariate (F1,42 =4.05; p=0.05) and the group

membership was not significant (F2,39 =0.67; ns). This means that the groups were

comparable at pre-test with respect to their general scores in the task.

Table 2 presents the pre-test scores for each group by problem type. We were surprised

that the children’s performance in the change-unknown problems was so low because the

subtraction, change unknown problems could have been solved by relying on surface cues.

However, as Vergnaud (1982) points out, change unknown problems can be surprisingly

difficult for children. The pre-test scores were subjected to an ANCOVA with grade level as

a covariate, group as a between participants factor (mixed, block, control) and repeated

measures on problem type (result vs. change vs. start-unknown). The analysis revealed that

the covariate was significant (F2,38 =4.05; p=0.05) and there was a significant effect of

problem type (F2,38 =9.70; p<0.001); group membership and the interactions were not

significant. Although this confirms that the groups were comparable at pre-test, there were

differences in favor of the taught groups in the start-unknown problems, so we decided to

control for pre-test scores when comparing the groups at the post-tests.

Comparison between post-test scores A repeated measures analysis of variance was not

appropriate to compare the groups’ performance at the post-tests because we had different

predictions for the effect of teaching on the different problem types. Thus, the groups’

performance must be compared by problem type. The analyses of the immediate and

delayed post-test were carried out by separate ANCOVAs, controlling for the effects of

grade level and of pre-test in the problems of the same type. We ran separate analyses for

each problem type because the covariate differed between problem types: it was the

children’s pre-test score in the problems of the same type.

We predicted that there would be no differences between the groups in the result-

unknown problems after training. We predicted significant differences between the

intervention groups and the control group in the start-unknown problems, which were

included in the intervention, and also in the change-unknown problems, as a consequence

of an improvement in relational calculation in the training.

Immediate post-test In all the analyses reported here, we used the control group as a

baseline. Our hypothesis was that the teaching of numerical calculations would not have an

effect on their ability to choose the correct operation to be entered in a calculator, although

they might show a slight improvement across occasions due to the familiarization with the

task. Thus all the planned comparisons were between the control and the intervention

Table 1 Means and standard deviations by group at the different testing occasions

Pre-test (max=12)

Group

Mixed (n=15)

Block (n=14)

Control (n=14)

Mean

5.67

4.64

5.14

SD

1.91

2.06

1.61

Immediate post-test (max=12)

Mean

7.60

7.43

5.79

SD

3.16

2.71

2.01

Delayed post-test (max=24)

Mean

18.47

16.00

13.43

SD

4.09

4.96

4.50

Table 2 Means and standard deviations by group and by type of problem at pre-test (maximum score: 4)

Result-unknown

Group

Mixed

Block

Control

Mean

3.47

3.00

3.21

SD

0.64

1.04

0.81

Change-unknown

Mean

0.73

0.36

0.93

SD

0.88

0.50

0.83

Start-unknown

Mean

1.47

1.28

1.00

SD

1.24

1.27

1.11

groups. Table 3 presents the means and the standard error of the mean for each problem

type by group. Surprisingly, the intervention groups made relatively less progress at the

immediate post-test in the start-unknown problems, about which they received teaching,

than in the untaught change-unknown problems.

The analysis of covariance with result-unknown problems showed that grade level was

not a significant covariate but the pre-test scores were F1,42 =5.95; p=0.02. As predicted for

the result-unknown problems, the difference between the groups was not significant.

With the change-unknown problems, grade level and pre-test scores were not significant

covariates. The overall group effect was significant (F2,38 =3.50; p=0.04). The planned

contrasts indicated that the mixed group did not differ significantly from the control group

but the block group did (p=0.012; Cohen’s d=1.01).

With the start-unknown problems, grade was a significant covariate (F1,42 =4.35; p=

0.02) but the pre-test scores were not. The overall group effect was also not significant. In

summary, the results of the immediate post-test were to some extent unexpected: the taught

groups differed from the control group in the change unknown, untaught problems but not

on the start unknown, taught problems.

Delayed post-test The means and standard errors of the means are presented in Table 4.

In the analysis of the result-unknown problems, grade was not a significant covariate but

the pre-test scores were F1,42 =7.83; p=0.008. The overall group difference was significant

(F1,42 =3.73; p=0.03). Planned contrasts showed that the mixed group differed significantly

from the control group (p=0.01; Cohen’s d=0.19) but the block group did not. Although

this difference is statistically significant, the effect size is small.

In the analysis of the change-unknown problems, the covariates were not significant.

The overall group difference was significant F1,42 =4.64; p=0.02. The planned contrasts

showed that both intervention groups differed significantly from the control group (p<

0.05). Cohen’s d effect sizes were: for the mixed versus control group, d=0.99 and for the

block versus control group, d=0.83; both these effect sizes are large.

Table 3 Adjusted means (controlling for pre-test scores) and standard error of the mean by group and by type of problem at the immediate post-test (maximum score: 4)

Result-unknown

Group

Mixed

Block

Control

Mean

3.45

3.25

3.26

SE

0.24

0.25

0.24

Change-unknown

Mean

2.09

2.73

1.29

SE

.36

.39

.38

Start-unknown

Mean

1.85

1.79

1.08

SE

0.32

0.34

0.34

Table 4 Adjusted Means (controlling for pre-test scores) and standard error of the mean by group and by

problem type at the delayed post-test (maximum score: 8)

Result-unknown

Group

Mixed

Block

Control

Mean

7.74

7.61

7.17

SE

0.15

0.16

0.16

Change-unknown

Mean

4.76

4.37

2.31

SE

0.58

0.63

0.62

Start-unknown

Mean

5.64

4.72

3.58

SE

0.60

0.63

0.63

In the analysis of the start-unknown problems, grade was a significant covariate (F1,42 =

5.23; p=0.03) but the pre-test scores were not. The overall group effect only approached

significance (F1,42 =2.81; p=0.07). However, the planned contrasts showed that the mixed

group differed significantly from the control group (p=0.02) but the block group did not.

Cohen’s d effect size for the comparison between the mixed group and the control group

was equal to 0.81, which is again a large effect size.

In summary, the analyses of the delayed post-test showed a significant effect of the

intervention on the change-unknown problems. The effect was stronger for the mixed

group, which differed significantly from the control group in both problem types and

produced a larger effect size.

7 Discussion

The starting point for this intervention study was the distinction between numerical and

relational calculation (Vergnaud, 1979; Thompson, 1993; Bryant & Nunes, 2009).

Numerical calculations depend on children’s understanding of relations between numbers

whereas relational calculations are about relations between quantities. The two kinds of

calculations are relatively independent of each other and are probably carried out

separately, but they are linked because our decisions about what type of computation to

do to solve a problem often depend on our understanding of the relation between the

quantities involved in the problem. In some problems, as in result-unknown problems,

these relations are transparent and obvious, and there is very little relational calculus to

be done. People’s success in solving other problems, like the start-unknown problems,

depends more heavily on their ability to work out the relations between the quantities

involved in the problems.

Recent research indicates that children’s ability to carry out the sort of relational calculus

that is needed to solve problems is a better predictor of their later mathematical

achievement than their numerical calculation skills. Nunes, Bryant, Sylva and Barros

(2009) found that a short test of mathematical reasoning about quantitative relations given

to a large number of 8 year olds was a strong predictor of their achievement in learning

mathematics at school over the next 5 years, even after stringent controls for the effects of

differences in their IQ.

The central concern of the present study was how children understand the connections

between quantities and transformations in additive problems. There are different ways for

children to approach start-unknown problems, but if they are using a calculator they need

to convert problems of the form of ?+a=b into b−a=? In this context, they would have to

convert a problem that on the surface is an addition into a subtraction, which they can do

easily if they understand that subtraction is the inverse of addition. This link between

relational calculus and children’s understanding of the inversion principle has often been

recognized, but has never before been the subject of a systematic intervention study. The

intervention study that we have just reported is therefore the first to show that it is possible

to improve children’s use of the inversion principle to carry out the relational calculus

needed to solve start-unknown story problems. The experimenter and the child discussed

how it was possible to know how many letters, for example, the postman had in his bag

before he delivered some, and the children were able to realize that they should imagine

that they had been added back to the bag. In the delayed post-test, the two intervention

groups, who were taught about the underlying relations in start-unknown problems,

improved in their ability to solve both change-unknown problems as a result of this

intervention more than the group taught about numerical calculations. The improvement in

the change-unknown problems was seen as a test of the children’s deep learning, because

half of these could be solved on the basis of surface cues but the other half could not. The

improvement in this untaught type of problem can also be explained by a theory that draws

on schemas as the explanation for how children reason, such as Piaget’s (1950), or more

recently, that of Fuchs et al. (2003): if children develop a schema of problems that involve

additive transformations, they can use this schema to solve start-unknown or change-

unknown problems.

The comparisons between each of the taught groups and the control group in the start-

unknown problems suggests that teaching the children by mixing the different types of

problems is more effective than teaching them in blocks of the same type: only the mixed

group differed significantly from the control group in the start-unknown problems. It also

suggests that learning in the mixed type of problems may become more solid over time,

because the statistical effect of the intervention, which was not statistically significant in the

immediate post-test, became significant in the delayed post-test, 8 weeks later. This last

result suggests that the relational teaching given to the children in the mixed group led to

some cognitive reorganization over time. We conclude that discussing the inverse relation

between addition and subtraction, as it was done in this study, is an effective way of

promoting relational reasoning in state-transformation-state problems.

Our intervention was carried out in the context of story problems but it is possible that

other interventions could succeed also: if children were presented with the symbolic string

?+128=343 and asked to solve this by using a calculator, they would have to carry out the

relational calculation in order to use the calculator. We did not choose this method for the

intervention because we thought that children might easily learn a rule to solve these

problems (e.g., move the number to the other side and change the sign) whereas it would be

more difficult to formulate such a rule if the questions were about story problems. However,

the form of instruction that we used in this study could be extended to the use of symbolic

expressions and a comparison between learning from reasoning about quantities in story

problems and learning from reasoning about symbolic string might prove very instructive.

We realize that this study should be seen as a starting point for further comparisons rather

than the answer to so many important questions about the difference between numerical and

relational calculations.

Acknowledgment The authors are grateful to the Economic and Social Research Centre, Teaching and Learning Research Programme for grant #L139251015 which made this research possible. We are very thankful to the schools, teachers, and children without whose generous participation, no research would be possible.