The scheme of correspondence and its role in children’s mathematics

The scheme of correspondence and its role in children’s mathematics

Terezinha Nunes*, Peter Bryant, Deborah Evans and Daniel Bell

Background.

There are two theories about the origin of children’s understanding of

multiplicative reasoning. One is that multiplicative reasoning has its origin in repeated

addition. The other is that children’s scheme of one-to-many correspondence is the

origin of their multiplicative reasoning, while addition and subtraction originate from

the schemes of joining and separating.

Aim.

The aim of this paper was to assess the evidence for these two theories and

provide new relevant evidence.

Sample.

Two studies were carried out with children from state-supported schools

that served a varied constituency with respect to socio-economic background. Neither

sample had been taught about multiplication or division in school.

Method.

In the first study, the children’s progress in multiplicative reasoning was

assessed after one group received instruction on one-to-many correspondences and

the other on repeated addition. In the second study, the intervention group was shown

how to use correspondences to solve multiplicative reasoning problems and the

control group solved visual non-numerical problems.

Results.

In the first study, with children aged 6–7 years, the correspondence group

made significantly more progress in multiplicative reasoning problems than the

repeated addition group. The second study showed that it is possible to teach young

children (aged 4–5 years) to solve multiplicative reasoning problems. In both studies,

additive or multiplicative reasoning in the pre-test was a specific predictor of post-test

performance on the same type of task.

Conclusions.

The results support the theory that the scheme of one-to-many

correspondences is the origin of children’s multiplicative reasoning. This finding has

important educational implications.

Cardinal numbers are used in primary school mathematics to represent two things:

quantities and relations. Thompson (1993) suggested that ‘a person constitutes a

quantity by conceiving of a quality of an object in such a way that he or she understands

the possibility of measuring it. Quantities, when measured, have numerical value, but we

need not measure them or know their measures to reason about them.’ (pp. 165–166).

So if we think that we could count the number of children in the class who like

mathematics, we would be establishing the quantity ‘children who like maths’. If we say

‘12 children in this class like maths’, the number 12 represents this quantity.

We also use numbers to represent relations between quantities: if we say that ‘there

are 2 more girls than boys in the classroom’, the number two refers to a relation

between two quantities. We could also say that there are two fewer boys than girls in the

classroom. The two sentences express the same relation between these two quantities;

one is simply expressed as the converse of the other.

It is undoubtedly important that children understand quantities and how numbers

represent them. But this is not sufficient for learning mathematics. Mathematics is a

science of relations. According to Thompson (1993), ‘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).

This paper analyses the role that the scheme of correspondence plays in children’s

understanding of relations between quantities. No one would dispute the claim that

one-to-one correspondence is a sine qua non for counting, and philosophers as well as

researchers of children’s development have suggested that knowledge of correspon-

dence relations is at the root of understanding equivalences between sets in the absence

of counting (Decock, 2008; Frydman & Bryant, 1988; Piaget, 1952a). The focus in this

paper is on the scheme of one-to-many correspondence and the role that it plays in

children’s understanding of multiplicative relations. We argue that the scheme of

correspondence provides children with a natural and effective foundation for

understanding relations between quantities, and that schools could build on this

scheme to advance children’s awareness of the very notion of modelling the world

through mathematics.

In the first section, we review briefly evidence that children use their informally

learned knowledge of one-to-many correspondence to solve multiplication and division

problems before they start to be taught about multiplication and division in school.

The knowledge of how to use one-to-many correspondences to solve multiplication and

division problems is informal in the sense that it can be learned outside school (for a

definition of informal knowledge in this sense, see Ginsburg, 1982). In the second

section, we present two empirical studies which show that it is possible to improve

children’s ability to solve multiplication and division problems by teaching them to use

one-to-many correspondence effectively. We suggest that this form of teaching is

effective in brief interventions because it builds on children’s informal knowledge. In the

final section, we discuss the implications of these results for further research and for

education.

The scheme of one-to-many correspondence and multiplicative reasoning

In the last two decades, mathematics educators (e.g. Confrey, 1994; Confrey & Harel,

1994; Steffe, 1994; Thompson, 1994; Vergnaud, 1982, 1983) have described problems

that are solved by addition and subtraction as ‘additive reasoning’ and problems that are

solved by multiplication and division as ‘multiplicative reasoning’ problems.

This categorization focuses on the nature of the relations between quantities rather

Q1 than the specific arithmetic operations that are used to solve problems. It also reflects

the view that the links between addition and subtraction, on the one hand, and between

multiplication and division, on the other, are conceptual and based on the relations

between quantities. In contrast, the links between addition and multiplication and those

between subtraction and division are procedural: you can multiply by carrying out

repeated additions and divide by using repeated subtractions.

The distinction between additive and multiplicative reasoning raises the question of

what is the origin of multiplicative reasoning. Fischbein, Deri, Nello, and Marino (1985)

proposed that children use an intuitive implicit model for each arithmetical operation,

and that the model that they adopt to help them understand multiplication is repeated

addition. This idea has influenced a large number of studies (e.g. Bell, Greer, Grimison, &

Mangan, 1989; Clark & Kamii, 1996; Mulligan & Mitchelmore, 1997) in which the

researchers distinguish between repeated addition and multiplication but nevertheless

hypothesize that the origin of multiplicative reasoning is repeated addition.

The view that multiplicative reasoning stems from repeated addition has been

influential not only in research but also in educational practice. Recommendations are

often made that children should be taught about multiplication starting from the idea of

repeated addition (e.g. The National Strategies, 2009).

Three types of evidence have been advanced to support the claim that the origin of

children’s understanding of multiplication is in repeated addition. First, when children

are asked to make up stories that would be solved by a multiplication operation (e.g.

3 £ 5), they create repeated addition stories more frequently than other types of stories

(Brown, 1981). The second type of evidence is indirect: it is assumed that children hold

repeated addition as the implicit model of multiplication because they believe that

multiplication always ‘makes bigger’ (e.g. Greer, 1992), a belief that leads them astray

when they have to multiply a number by a fraction (e.g. 10 £ 0:5). The third type of

evidence is that children’s strategies for solving multiplication problems can be

described as ‘repeated addition’ (e.g. Anghileri, 1989). However, all three of these

observations are about responses that could result from the children’s school learning.

In all these studies, the children had been taught about multiplication as repeated

addition. We cannot be certain that repeated addition is the implicit model that they

bring to school from their everyday experiences. The use of repeated addition as a

problem-solving strategy may be a consequence of the procedural connections, which

are encouraged by teachers, and which children form between repeated addition and

multiplication, rather than a demonstration that the natural origin of multiplicative

reasoning is in repeated addition. So, the evidence for the claim that children’s implicit

model of multiplication is repeated addition is questionable, despite its undoubted

influence. At best, the evidence suggests that children who are taught about

multiplication as repeated addition use this procedure to solve multiplication problems

and, perhaps from this practice, draw the inference that multiplication makes bigger.

There is an alternative view of the origins of children’s multiplicative reasoning.

It is that their understanding of multiplication is founded, not on addition, but on one-

to-many correspondence. This view rests on the analysis of the relations that define

additive and multiplicative reasoning and also on children’s actions when solving

multiplicative reasoning problems before they receive instruction in school. Additive

reasoning involves understanding part–whole relations. In additive reasoning problems,

one may be asked to find a whole given the parts (a þ b ¼ ?), find a part given the whole

and the other part (a þ ? ¼ b), or find the difference between two wholes (in

comparison problems). The whole may be constituted by static parts (e.g. five boys and

seven girls in the class) or by a transformation (e.g. Connie had five marbles and won

seven marbles in a game; how many marbles does she have now?). In contrast,

multiplicative reasoning is based on the idea of an invariant relationship between two

quantities. This constant relation is called a ratio and can be symbolized as x ¼ f ð yÞ.

This hypothesis, originally proposed by Piaget (1952a) and developed later by

Vergnaud (1983) and by Nunes and Bryant (1996), is supported also by some researchers

in mathematics education (e.g. Thompson, 1994) and professional mathematicians

(Devlin, 2008a,b). These authors recognize the procedural connections between

addition and multiplication, but do not regard them as conceptual connections.

To paraphrase Devlin (2008a), you can walk to work or go by car and you will get to

the same place; but this does not mean that the two are the same thing.

Piaget (1952a) pioneered research on how the scheme of one-to-many

correspondence helps children deal with multiplicative reasoning problems. His

studies investigated children’s understanding of multiplicative equivalences. He asked

children to put one red flower in each of a set of vases; after these were removed,

the children placed one blue flower in each vase. All the flowers were then removed

and only the vases were left on the table; the children were asked to take from a box

the right number of tubes to place one flower in each tube. Piaget reports that many

5-year-olds succeeded in constructing the set of tubes with the same number as the

flowers by placing two tubes in correspondence with each vase. This success, Piaget

argued, was due to the children’s understanding that if set A (flowers) has a 2:1 ratio

to set B (vases), and set C (tubes) also has a 2:1 ratio to set B, then A and C are

equivalent.

Since Piaget’s pioneering work, other researchers have shown that children can

construct equivalent sets in sharing when they need to take ratio into account. Frydman

and Bryant (1988, 1994) asked children to share sweets fairly to two different recipients;

the size of the units that each recipient was to be given was different – for example, one

recipient received units that were three times the size of the units given to the other.

The children succeeded in constructing fair shares by using one-to-many correspon-

dences in the sharing procedure: for each treble unit they gave to recipient A, they gave

three singles to recipient B.

Later research showed that children successfully use the one-to-many correspon-

dence scheme to solve multiplicative reasoning problems before they had been taught

about multiplication and division in school. Kouba (1989) asked young children in the

USA to solve multiplicative reasoning problems such as: at a party, there were six cups

and five marshmallows in each cup; how many marshmallows were there? In a series of

problems, children were asked to supply a missing piece of information, which could be

the product (in this case, the total number of marshmallows) or either of the factors (the

number of groups or the number of elements in the group).

For the children in first and second grade, who had not received instruction on

multiplication and division, the most important factor in predicting the children’s

solutions was which quantity was unknown: the product, the number of groups, or the

number of elements in the group. For example, in the problem above, about the six cups

with five marshmallows in each cup, when the size of the groups was known (i.e. the

number of marshmallows in each cup), the children used correspondence strategies:

they paired objects (or tallies to represent the objects) to something that represented

the cups and counted or added, creating one-to-many correspondences between the

cups and the marshmallows. If they needed to find the total number of marshmallows,

they pointed five times to a cup (or its representation) and counted to five, paused,

and then counted from 6 to 10 as they pointed to the second ‘cup’, until they reached

the solution. Alternatively, they may have added the number of marshmallows as they

pointed to the ‘cup’.

In contrast, when the number of elements in each group was not known, the

children used dealing strategies: they shared out one marshmallow (or its

representation) to each cup, and then another, until they reached the end, and then

counted the number in each cup. Although these actions look quite different, their aims

are the same: to establish one-to-many correspondences between the marshmallows and

the cups.

Kouba’s description of the children’s strategies makes it clear that the method used

for quantifying the response could be counting or adding. The scheme of

correspondence provided the model for the problem solution: the children counted

or added the number of marshmallows as they pointed to each cup. Addition as a form

of counting was observed only when the size of the groups was known: when it was

not, the children used sharing or dealing, which established the correspondence

relations, and then counted the items in each group to determine the value of the

quantities. Kouba’s description leaves it clear that correspondences were used to

represent the relations; counting or addition was procedures used to determine the

quantities.

Kouba observed that 43% of the strategies used by the children, including first,

second, and third graders, were appropriate. Most of the appropriate strategies adopted

by the first and second grade children were based on correspondences: they tended to

use either direct or partial representations (i.e. tallies for one variable and counting or

adding for the other); few used recall of multiplication facts. The recall of number facts

was significantly higher among children in the third grade, who had been taught about

this way of solving problems.

The level of success observed by Kouba among children who had not yet received

instruction was modest in comparison to that observed in two subsequent studies,

where the ratios were easier. Becker (1993) asked kindergarten children in the USA,

aged 4–5 years, to solve problems in which the correspondences were 2:1 or 3:1. The

children were more successful with 2:1 than 3:1 correspondences, a result also reported

by Frydman and Bryant (1988) and Piaget (1952a). The level of success improved with

age. The overall level of correct responses achieved by 5-year-olds was 81%. This is

clearly a high level of success for children who had not received any instruction on

multiplicative reasoning and who were just starting to learn about addition and

subtraction at school. Carpenter, Ansell, Franke, Fennema, and Weisbeck (1993) also

gave multiplicative reasoning problems to US kindergarten children involving

correspondences of 2:1, 3:1, and 4:1. They observed 71% correct responses to these

problems.

These success rates leave no doubt that many young children start school with some

understanding of one-to-many correspondence, which they can use to learn to solve

multiplicative reasoning problems in school. The results do not imply that children who

use one-to-many correspondence to solve multiplicative reasoning problems

consciously recognize that in a multiplicative situation there is a fixed ratio linking

the two variables. Their actions maintain a fixed ratio between the quantities but it is

most likely that this invariance remains, in Vergnaud’s (1997) terminology, as a ‘theorem

in action’.

In summary, there has been for some time a theory that suggests that repeated

addition is the implicit model that children use to make sense of the operation of

multiplication: so repeated addition would be the origin of children’s understanding

of multiplicative relations. Although the influence of this model both in research and

in practice is considerable, the evidence for it is ambiguous. Most of it comes from

studies of children who have already received instruction about multiplication as

repeated addition in school: thus the children might have developed this conception

of multiplication as repeated addition as a consequence of the school instruction

that they received. An alternative view is that children’s implicit model for

multiplicative relations is one-to-many correspondence. Children use one-to-many

correspondences to solve multiplicative reasoning problems before they are taught

about multiplication and division in school. They are in fact developing their

understanding of additive and multiplicative relations at the same time when they

start school.

The effect of learning to use the scheme of one-to-many correspondence on solving

multiplicative reasoning problems

We report here two studies in which we analysed children’s improvement in solving

multiplicative reasoning problems after being taught how to use the scheme of one-to-

many correspondence.

The first study (Park & Nunes, 2001) compared the effectiveness of teaching children

(aged between 6 and 7 years) about repeated addition with teaching them about

correspondences in promoting their later success in multiplicative reasoning problems.

This study has been published but we present here some additional information not

included in its previous publication. The second study aimed at assessing whether it is

possible to improve young children’s performance in multiplicative reasoning when

they are in their first year in school (aged 4–5 years), before they have made much

progress in computation using addition.

STUDY 1

Park and Nunes (2001) reasoned that, if the scheme of one-to-many correspondence

offers children an insight into multiplicative relations whereas repeated addition only

provides them with a procedure to find a quantity, children taught to solve

multiplication problems by using correspondences between two variables would

improve more in solving multiplicative reasoning problems than other children, taught

about multiplication as repeated addition. Children taught about correspondences

would have the opportunity to construct a model of the relations between variables,

which could be used flexibly in solving multiplicative reasoning problems. Children

taught to solve repeated addition problems would be essentially using addition as a

procedure to determine a total quantity but would not have the opportunity to develop

an implicit model of a fixed ratio between two variables.

Participants

The children (N ¼ 42) in this study were in Year 2 in two different schools in

Buckinghamshire (UK); their mean age was 6 years 7 months. According to their

teachers, they had received instruction on addition and subtraction in school but not yet

on multiplication and division.

Design

The children were given a pre-test and a post-test, in which they were asked to solve

additive and multiplicative reasoning problems. Between the pre- and post-tests, they

participated in a brief intervention, during which they solved multiplicative reasoning

problems with the support of the researcher. They were randomly assigned to either a

repeated addition group or to a one-to-many correspondence group. During the

intervention, the children in the two groups were given the same number of problems

(16) with the same numerical descriptions. However, the children in the repeated

addition group were asked to solve one-variable problems about joining sets, whereas

those in the one-to-many correspondence group solved problems that involved two

quantities related by a fixed ratio. For example, the children in the repeated addition

intervention were presented with this problem: ‘Tom has 3 toy cars. Anne has 3 dolls.

How many toys do they have altogether?’. There is one total quantity in this problem,

number of toys, and two parts to it, number of cars and number of dolls. In contrast, the

one-to-many correspondence problem of the same numerical description (2 £ 3) was:

‘Amy’s Mum is making two pots of tomato soup. She wants to put three tomatoes in each

pot of soup. How many tomatoes does she need?’ This problem involves two variables,

number of tomatoes and number of pots of soup, which are related by a fixed ratio:

three tomatoes for one pot of soup.

Procedure

The intervention was carried out in two sessions. In the first session, the problems

were presented to the children orally and with the support of a picture. Figure 1 shows

the relevant picture for the examples above. In the second session, the children had

fewer pictorial cues: for example, instead of there being three bananas on a page, the

number 3 would be next to a banana. Throughout the two sessions, the children were

given blocks that they could use to solve the problems, if they found that they needed

to count.

Figure 1. One example of parallel items presented to the correspondence and to repeated addition

groups. (a) Correspondence group. The picture in the booklet was shown to the child. The experimenter

said: ‘Amy’s Mum is making two pots of tomato soup. She wants to put three tomatoes in each pot of

soup. How many tomatoes does she need?’; (b) Repeated addition group. The picture in the booklet was

shown to the child. The experimenter said: ‘Tom has three toy cars. Anne has three dolls. How many

toys do they have together?

.

The problems used during the intervention always included information about the

unit ratio: in the example above, Amy’s Mum used three tomatoes for each pot of soup

that she made. In the pre- and post-tests, two of the problems did not involve the unit

ratio: for example, the children were asked: ‘Two cupcakes cost 8 pence. You buy

4 cupcakes. How much do you have to pay?’. The inclusion of problems without the unit

ratio was considered an important test of the flexibility of the children’s use of relational

reasoning beyond what they had been taught during the intervention. The problem can

be solved by repeated addition but it is known that children make more mistakes in

problems that do not present the unit ratio than in those in which the unit ratio is

explicit (Hart, 1981; Ricco, 1982).

When the children were not able to solve a problem during the intervention, the

researcher helped them use the blocks to represent the quantities and then asked them

to solve the problem (for the details, see Park & Nunes, 2001).

Results

At pre-test, the mean number of correct responses (out of 8) in the addition problems

was 3.74, SD ¼ 2:51, and in the multiplication problems it was 2.95, SD ¼ 2:18. The

distribution of scores in both sets of problems was not skewed. The difference between

the means for the addition and multiplication problems was statistically significant,

according to a t test for correlated samples (t ¼ 2:75, df ¼ 41, p , :01). These pre-test

results demonstrate two things. First, there were no floor or ceiling effects in either set

of problems. Thus, the children had not received instruction on multiplication but

demonstrated some competence in solving multiplication problems. This competence

is likely to be acquired through informal learning, as they had not received school

instruction about multiplication, and we can hypothesize that pre-test performance will

be a strong predictor of post-test performance. Second, their performance was higher in

the addition problems, about which they had received instruction in school.

However, there was still room for much improvement, as shown by the mean and

standard deviation observed at pre-test in the addition problems and by the absence of a

ceiling effect.

It was predicted that both groups would make significant progress from pre- to

post-tests, but that the children in the correspondence intervention group would make

more progress in the multiplicative reasoning problems than those in the repeated

addition group. In contrast, the children in the correspondence group were expected to

make less progress in additive reasoning problems than the repeated addition group.

The first prediction was supported by a significant difference between the groups in

the post-test as a function of type of intervention. In the post-test, the mean number of

correct responses to multiplicative reasoning problems for the children in the

correspondence intervention group was 6.05 (SD ¼ 1:56) and for the children in the

repeated addition group the mean was 3.76 (SD ¼ 2:34). Both groups made some

progress from pre- to post-tests but the children in the correspondence intervention

group made significantly more progress in multiplicative reasoning problems than those

in the repeated addition group. This difference supports the hypothesis that

correspondence reasoning is a stronger basis for children to learn about multiplicative

reasoning than repeated addition. The stringent controls used in this design (random

assignment to the groups, same number of problems during the intervention) give us

confidence in this conclusion.

Contrary to our expectation, the two groups made similar amounts of progress in

additive reasoning problems: both performed significantly better at post-test than they

had performed at pre-test, and the difference between the groups in the additive

problems was not significant. It is possible that the progress that the children in the

correspondence group made through the intervention was not specific to using

the joining and separating schemes: during the intervention both groups had the

experience of representing quantities with blocks and using them to find the answers.

This experience could have enhanced the children’s general problem-solving skills and,

consequently, led them to use more effectively additive reasoning schemes, which they

grasped better than the correspondence scheme at pre-test, perhaps as a result of their

previous school instruction. This hypothesis about developing general numerical skills

would also explain the progress that the children in the repeated addition group made in

multiplicative reasoning problems; they performed better at post-test through a more

efficient use of their general numerical skills. Their grasp of correspondences was weak

at pre-test and did not change as a consequence of the intervention.

Two further analyses, carried out for this paper, supported the idea that children’s

grasp of correspondences has a specific impact on multiplicative reasoning problems.

The first was an analysis of the children’s progress in problems that did not include the

unit ratio and the second was an analysis of the partial correlations between pre- and

post-tests performance.

The first analysis showed significant differences between the repeated addition and

the correspondence group in problems that did not make the unit ratio explicit. The

groups did not differ significantly at pre-test: their pre-test means (out of 2) were 0.43

and 0.52 problems correct, respectively. At post-test, the repeated addition group

obtained a mean of 0.62 and the correspondence group a mean of 1.14 correct. About

half of the children in the correspondence group who had answered problems without

the unit ratio incorrectly at pre-test solved them correctly at post-test. In the repeated

addition group, only 15% of the children made progress from pre- to post-test on

these items.

This result suggests that the children in the repeated addition group continued to use

additive reasoning when asked to solve multiplicative reasoning problems after their

intervention. In contrast, the children in the correspondence group seemed to be able

to use a different model, based on correspondences, which was more appropriate for

the relations involved in these more complex problems.

In the second analysis, we expected to show, through partial correlations between

additive and multiplicative problems, that additive reasoning is distinct from

multiplicative reasoning. Our assumption, in these partial correlations, was that

variance shared in solutions to additive and multiplicative reasoning problems gives us

a measure of the children’s general numerical skills. We predicted that the partial

correlation between the pre- and post-tests addition tasks would remain significant

after controlling for performance on the pre-test multiplication task. This would

demonstrate a specific connection between additive reasoning measures after

partialling out these general numerical skills. For the same reasons, we also predicted

that the partial correlation between the pre- and post-tests multiplication tasks would

remain significant after controlling for performance on the pre-test addition task.

This would demonstrate a specific connection between multiplicative reasoning

measures after partialling out general numerical skills. However, we predicted that the

partial correlations between performance on the pre-test addition and post-test

multiplication tasks would not be significant after partialling out the effects of pre-test

multiplication performance. Analogously, the correlation between pre-test multipli-

cation and post-test addition tasks was expected to be non-significant after controlling

for pre-test multiplication performance.

In summary, we expected a significant partial correlation within problem type but a

non-significant partial correlation between problem types. This was the pattern that

actually emerged from the analysis. Table 1 displays these results.

Table 1. Bivariate and partial correlations between performance on the pre- and post-tests (N ¼ 42)

Bivariate correlations

Pre-test addition

Pre-test multiplication

Post-test addition

Partial correlations controlling for

pre-test multiplication

Pre-test addition

Partial correlations controlling for

pre-test addition

Pre-test multiplication

*

Pre-test multiplication Post-test addition Post-test multiplication

.70*

.76*

53*

.41*

.71*

.30

.65*

2.16

.01

.65*

Correlations significant at p , :01:

These results support the idea that there are general cognitive and numerical skills

that influence performance both in additive and multiplicative reasoning tasks but that

there is also a specific component in multiplicative reasoning, which is independent of

additive reasoning and vice versa. Children’s informal knowledge of multiplication at

pre-test was a specific predictor of their post-test performance in multiplicative

reasoning problems; children’s pre-test performance on the additive reasoning

problems was also a specific predictor of their performance in the additive reasoning

problems at post-test.

STUDY 2

Our second study investigated whether it is possible to improve younger children’s use

of correspondence reasoning to solve multiplicative relations problems.

Participants

The children (N ¼ 32) were recruited from two schools in Oxford and were in their first

year in school; their mean age was 5 years 4 months (range 4 years 8 months to 5 years

9 months). They had received no teaching about multiplication and division in school.

Design

The children were randomly assigned either to a training group, which received

instruction on multiplicative reasoning, or to a control group, which was taught how to

solve visual analysis problems. These latter problems were similar in structure to the

items used in the British Abilities Scale-II (BAS-II) matrices subtest. Our aims in offering

this training to the control group were: (1) to expose the control group to the same

number of one-to-one sessions with the researcher as the experimental group; (2) to

work with them on non-numerical, reasoning problems, in view of our finding in Study 1

that children can improve in a set of non-trained problems just from learning how to use

blocks to represent stories as a way to solve the problems; and (3) to assess whether the

control group maintained a comparable level of motivation during their own training.

Our predictions were: (1) that the experimental group would make significantly

more progress in multiplicative reasoning problems than the control group due to their

specific experience with one-to-many correspondence during training; (2) that the

experimental group would also make significantly more progress than the control group

in additive reasoning problems due to the general numerical skill that they would

develop during the training; and (3) that the children in the control group would make

significantly more progress in the BAS-II matrices subtest than the experimental group,

due to their training in visual analysis.

The children were given a pre-test before the intervention started, and two post-

tests, one immediately after the teaching had been completed and one about 2–4 weeks

after the teaching had been completed.

At pre-test, they were given 12 multiplicative reasoning problems (6 multiplication

and 6 division problems), 6 additive reasoning problems, and the matrices subtest of the

BAS-II (Elliott, Smith, & McCulloch, 1997), which assesses children’s visual analytical

skills. In this subtest of the BAS, the children are presented with an array of figures

displayed in a matrix (2 £ 2 or 3 £ 3). The figure at the right-bottom corner is missing.

The child is asked to choose the correct one to complete the matrix, from six

alternatives. The missing figure is predictable from the pattern: e.g. three crosses in the

first column, three black circles in the second column, and two white triangles in the

third column; the bottom figure in this column is missing.

The children were given the same measures of additive and multiplicative reasoning

in both post-tests as in the pre-test. Our aim in the analysis of the additive reasoning

problems was twofold: (1) to test whether the children in the experimental group had

learned correspondence procedures in a mechanical and unreflective way or whether

they analysed the relations in the problems before implementing the solution; and (2) to

test whether they would improve more than the children in the control group, who did

not solve numerical problems during the intervention, in the additive reasoning

problems as a consequence of having developed their general numerical skills. If they

had learned to use correspondences mechanically, they should show a decline in

performance in additive reasoning problems by the inappropriate use of one-to-many

correspondence procedures. If they developed their general numerical skills through

the intervention, they should show some progress in additive reasoning.

Procedure

Both interventions consisted of two teaching sessions, administered individually by a

researcher. The children in the intervention group solved five multiplication and five

division problems in each session. The researcher asked the child to construct first a

representation of the situation, using manipulatives to signify the objects mentioned

in the problems; if the child did not spontaneously set the manipulatives in the

appropriate form of one-to-many correspondence, the researcher encouraged the

child to do so. For example, one multiplication problem was: three lorries are

bringing tables to the school; each lorry is carrying four tables; how many tables are

they bringing to school? The children were given cut-out figures of lorries and some

cubes, and were encouraged to show what the problem had indicated. They received

further directives, as required: for example, if the child made the correspondence for

one lorry and stopped, the experimenter suggested showing the correspondence

for the other lorries before answering the question. For division problems, the same

strategy was to be implemented but the question was not about the total: e.g. a boy

has 12 marbles; he wants to put the same number inside each of these two bags;

how many marbles will go in each bag? The children had cut out circles to represent

the bags and cubes to represent the marbles. The use of these simple manipulatives

to represent different objects at different times during the sessions did not cause any

difficulty for the children.

The children in the control group worked individually with the researcher and

solved visual analysis problems for a comparable period (approximately 25–30 min for

each session).

Results

The mean scores and standard deviations for the children in each group at pre-test and

both post-tests are presented in Table 2. The pre-test means show that the children in

this study had a much weaker performance in the multiplicative reasoning problems

than those in Study 1. They were, on average, 1 year younger than those in Study 1, and

also younger than US 1st graders (whose average age seems to be above 6 years).

Table 2. Means (out of 12) on the multiplicative reasoning tasks and standard deviations in the pre- and

post-tests by group

Testing occasion

Intervention group

Mean

SD

N

Multiplicative reasoning (max 12)

Pre-test

Control

Intervention

Immediate post-test

Control

Intervention

Delayed post-test

Control

Intervention

Additive reasoning (max 6)

Pre-test

Control

Intervention

Immediate post-test

Control

Intervention

Delayed post-test

Control

Intervention

1.93

2.23

2.73

7.35

3.00

7.76

3.07

3.18

2.93

4.82

3.47

4.53

1.75

2.11

2.43

2.74

3.27

3.99

1.75

1.78

1.83

0.81

1.64

1.42

15

17

15

17

15

17

15

17

15

17

15

17

Table 2 also shows that the group that received instruction on correspondences

improved their performance considerably in the multiplicative reasoning problems: at

pre-test, 19% of their responses were correct and at the immediate post-test it was 61%,

an improvement of 42%. The intervention group also improved on the additive

reasoning problems: at pre-test, the rate of correct responses was 53% and at post-test

80%, an improvement of 27%. This indicates that they did not simply use

correspondence procedures after the intervention in an unreflective manner.

As Table 2 also shows, there was a difference between the groups in multiplicative

reasoning scores at pre-test. This is a risk in random assignment to treatment groups when

the number of participants is small. So we carried out an analysis of covariance to

compare the performance of the children in the two groups in the post-tests with the pre-

test scores as a covariate. The means on the two post-tests, immediate and delayed, were

the repeated measures in this analysis. The group membership, control versus

experimental, was the independent variable. This analysis showed that the overall effect

of testing occasion was not significant. The effect of the covariate was significant

(F 1;29 ¼ 15:07, p ¼ :001). The difference between intervention groups was significant

(F 1;29 ¼ 8:49, p ¼ :007) and the interaction between testing occasion and intervention

group was also significant (F 2;28 ¼ 8:87, p ¼ :006). Post hoc tests showed that the

difference between the groups was significant at both post-test occasions. Cohen’s d

effect size was calculated using the adjusted means and the pooled standard deviation for

the two groups on each testing occasion. The effect size for the immediate post-test was

1.26 and for the delayed post-test 1.03. So we can conclude that the intervention group

improved significantly more than the control group in multiplicative reasoning problems,

and a large effect size was observed despite the fact that the intervention was so brief.

In order to test whether children improve in problem solving from general

experiences in solving problems, even if they do not involve the same type of relations,

we compared the intervention’s and the control group’s improvement in additive

reasoning problems. We used an analysis of covariance, in which the covariate was the

children’s pre-test performance, to compare the intervention and control groups’

performance in additive reasoning problems. The immediate and delayed post-tests

were treated as repeated measures; the covariate was the pre-test measure; and the

dependent variable was the number of correct responses to the additive reasoning

problems. The covariate was significant (F 1;29 ¼ 6:83, p ¼ :014), the differences

between testing occasion were significant (F 1;29 ¼ 9:14, p ¼ :005) and the difference

between the groups was also significant (F 1;29 ¼ 13:24, p ¼ :001), as predicted. Cohen’s

d was 1.01 for the immediate post-test and 0.67 on the delayed post-test. These results

support the hypothesis that children can improve their general numerical problem-

solving skills through the experience of representing quantities numerically in order to

solve problems.

The same pattern of specific connections between the tasks observed in Study 1,

which were analysed through partial correlations, was observed in this study. The partial

correlations within tasks were significant and those between tasks were not. The partial

correlation between pre- and immediate post-test addition scores, controlling for

performance on the pre-test multiplication task was significant (r ¼ :41, p ¼ :01) and

the partial correlation between pre- and immediate post-test multiplication scores,

controlling for performance on the pre-test addition task was significant (r ¼ :50,

p ¼ :003). In contrast, the partial correlation between pre-test addition scores and post-

test multiplication scores, controlling for pre-test multiplication scores was not

significant (r ¼ :30, p . :05) and the partial correlation between pre-test multiplication

and post-test addition scores, controlling for pre-test addition scores, was not significant

(r ¼ :08, p . :05). The replication of the partial correlations in this study gives further

support to the idea that additive reasoning and multiplicative reasoning tasks are

distinct, apart from a shared variance that can be attributed to general numerical skills in

problem solving.

In order to assess whether the children in the control group had simply not been

interested in participation in the study, we compared their results in the BAS-II matrices

subtest with those obtained by the experimental group on the three testing occasions.

We used an analysis of covariance parallel to those described earlier on, with BAS-II

matrices subtest at pre-test as a covariate and both post-tests as repeated measures. The

group membership was the independent measure. This analysis showed a significant

effect of group membership (F 1;29 ¼ 13:85, p ¼ :001); the control group performed

significantly better than the intervention group in the control task, the BAS-II matrices

subtest, on both post-tests but not at pre-test. We conclude that the control group must

have remained motivated throughout the experiment and gained significantly from their

experience in the cognitive skill that they had practised during the study.

In summary, both groups in this study made gains during the intervention.

These gains were clearly related to the learning experiences that they had during the

teaching period.

We conclude that it is possible to teach young children to use one-to-many

correspondence reasoning to solve multiplicative reasoning problems. The 4- and

5-year-olds that participated in this study showed clear gains in their general numerical

skills and also in their ability to solve multiplicative reasoning problems long before they

would receive teaching about multiplication and division in school. The children were

actually quite young and their knowledge of addition and subtraction was still

developing, thus their ability to solve multiplicative reasoning problems would be

developing at the same time as their understanding of additive reasoning.

The importance of the correspondence scheme for teaching and learning

Adopting one of Piaget’s main hypotheses (see, for example, Piaget, 1950, 1952a,b), we

have argued elsewhere that the origin of children’s understanding of key mathematical

ideas is in their action schemes (Nunes & Bryant, 1996) and have provided both

longitudinal and intervention data to show a causal relation between children’s use of

action schemes to solve mathematical problems at school entry and their later mathematics

achievement (Nunes et al., 2007). One-to-many correspondence was one of the action

schemes included in our measure of children’s logical–mathematical reasoning.

In this previous study, we did not report the results of the longitudinal prediction

obtained from the children’s performance in the correspondence items and so we

report this result here. The outcome measure in our previous study was the children’s

performance in a government-designed and teacher-administered assessment of

mathematics achievement, which is obtained at the end of their second year in school.

The predictors were the children’s age at the time of the achievement test and several

measures obtained at school entry: (1) their general cognitive ability as measured by the

BAS-II, (2) working memory, assessed by means of counting recall, a subtest from the

working memory battery for children (Pickering & Gathercole, 2001), and our

assessment of their logico-mathematical reasoning, which included their understanding

of the inverse relation between addition and subtraction, one-to-many correspondences,

additive composition of number and seriation.

In order to address the significance of children’s ability to use correspondences to

solve multiplicative reasoning problems as a predictor, excluding the other items in our

measure from their score, we analysed how well the correspondence items predicted

the children’s mathematics achievement. There were 12 items that involved

correspondences and we obtained for all the children a score based only on these

12 items. We then ran a fixed order regression to test whether these items still added

significantly to the prediction of the children’s mathematics achievement, after

controlling for their age, BAS-II and working memory performance. This analysis

showed that, after controlling for the other predictors, the children’s performance still

added significantly to the prediction of their mathematics achievement (F 1;46 ¼ 5:05,

p , :05). The beta values showed that this was not a minor contribution; the largest beta

corresponded to the BAS-II scores (b ¼ 0:4, p , :001), followed by the value for the

children’s use of correspondences to solve multiplicative reasoning problems (b ¼ 0:3,

p ¼ :03); the remaining beta values were not significant.

This overview of the evidence on the role of correspondence reasoning for

children’s mathematics learning shows that this is indeed a key understanding

for children’s progress in mathematics. The use of correspondence reasoning allows

children to maintain a fixed ratio between two variables, and thus use a type of

reasoning that differs from part–whole reasoning and can form the basis for

understanding ratio and proportions.

Our current hypothesis is that the representation of ratio – i.e. of the relation

between the quantities in these situations – remains implicit. It is likely that the children

focus their attention on the quantities themselves in such problems and they may not

build an awareness of the relations between quantities. This would be entirely justified

by the types of questions that they are asked ‘how many’, which are about quantities.

For example, in multiplication situations they are asked about the total number of items

(e.g. in the problem mentioned earlier on, they are asked how many tables were brought

to the school) and in division situations they are asked about the quantities in a group

(e.g. how many marbles in each bag) or about the number of groups.

     Children often have some implicit knowledge, which they develop outside school and

which they can use in order to solve school problems. It is also quite commonly the case

that teachers need to develop children’s awareness of this implicit knowledge in order to

teach them something new. For some time now it has been recognized (Bryant & Bradley,

1985; Nunes & Bryant, 2009) that children have a good knowledge of the sounds of

their own language, which they use to speak and to discriminate between words that they

hear. However, this knowledge is implicit and must be rendered explicit to some extent

in order for them to learn to represent sounds using letters and to become literate.

We suggest that the same sort of approach may be needed for children to transform

their implicit understanding of fixed ratios, used in one-to-many correspondence action

schemes, into an understanding of ratio and proportions. This would allow them to

become conscious of the relations that they only deal with implicitly, and help them

build a conception of how relations between quantities are connected to the

mathematics that they use in solving problems. Further research is urgently needed to

test this idea, which may be very fruitful and help bring more success in mathematics to

many children in school.