Children's Understanding of the Relation between Addition and Subtraction: Inversion, Identity, and...

Children’s Understanding of the Relation between Additionand Subtraction: Inversion, Identity, and Decomposition

Peter Bryant, Clare Christie, and Alison Rendu

University of Oxford, Oxford, United Kingdom

In order to understand addition and subtraction fully, children have to know about therelation between these two operations. We looked at this knowledge in two studies. In onewe asked whether 5- and 6-year-old children understand that addition and subtractioncancel each other out and whether this understanding is based on the identity of the addendand subtrahend or on their quantity. We showed that children at this age use the inversionprinciple even when the addend and subtrahend are the same in quantity but involvedifferent material. In our second study we showed that 6- to 8-year-old children also usethe inversion in combination with decomposition to solvea 1 b 2 (b 1 1) problems. Inboth studies, factor analyses suggested that the children were using different strategies inthe control problems, which require computation, than in the inversion problems, whichdo not. We conclude that young children understand the relations between addition andsubtraction and that this understanding may not be based on their computational skills.© 1999 Academic Press

Key Words:addition; subtraction; arithmetic; inversion; decomposition of number;reversibility; computation; mathematics.

A point often made by Piaget (1952), and not seriously disputed by anyoneelse, is that no one can grasp the nature of addition or of subtraction fully unlesshe or she also understands the inverse relations between these two operations.Piaget himself made the understanding of inversion an essential part of thegroupings that underlie concrete operations, in his theory, but one does not haveto subscribe to this particular theory to appreciate the significance of understand-ing inversion. A child must understand how addition and subtraction cancel eachother out in order to grasp the additive composition of number. The realizationthat if 8 1 4 equals 12 then 122 8 must equal 4 demands an understanding thatone operation, in this case subtraction, cancels the other out. Nor can a childunderstand the ordinal nature of number fully unless he or she realizes thatadding a number and then subtracting the same number, or vice versa, moves thenumber up and down the number scale by exactly the same amount. Inversionmight also be a powerful aid in the often-used informal procedure of decompo-

Address correspondence and reprint requests to Peter Bryant, Department of Experimental Psy-chology, University of Oxford, South Parks Road, Oxford OX1 3UD, United Kingdom.

Journal of Experimental Child Psychology74, 194–212 (1999)Article ID jecp.1999.2517, available online at http://www.idealibrary.com on

0022-0965/99 $30.00Copyright © 1999 by Academic PressAll rights of reproduction in any form reserved.

194

sition (Nunes, Schliemann, & Carraher, 1993) in problems with an addend anda subtrahend that are unequal but close to one another. For example, with aproblem like 521 28 2 27 it is quite easy for an adult to see that the 28 can bedecomposed into 27 and 1 and therefore, through inversion, that the problem canbe reduced to 521 1.

Given the importance of inversion and its central place in Piaget’s theory, it issurprising that very little empirical research indeed has been done on the matterso far. Piaget himself often wrote about the question of inversion but delayedstudying it directly until fairly late in his life. In one of Piaget’s last books, Piagetand Moreau (1977) report a study in which they asked each child to put togethera number of bricks, without revealing how many to the experimenters. Then theexperimenters asked the child to add and subtract certain numbers of bricks tothis amount, and occasionally to multiply it by a certain number as well, andfinally asked the child to say how many bricks he or she had now. From thisfigure the experimenters then worked out (by inversion) what the original amountset aside by the child had been, and having done this they asked the child toexplain how they had been able to work out an amount which they had neverseen. Piaget and Moreau (1977) reported that the younger children were unableto give any explanation, but that some older children (most of them 10 years ofage or older) could account for the experimenters’ inference by appealing to theprinciple of inversion.

This ingenious task does show a change in children’s understanding ofinversion but it does not definitely demonstrate that younger children have nounderstanding at all of this principle. The experimental task was too stringent forthat. To be successful the child had not only to understand the inversion principlebut also to work out that someone else had been using the principle to solve amathematical problem. Children might be able to use the principle correctlythemselves and yet might still find it impossible to realize when someone else hasused it too. We need a direct test of whether young children are able to takeadvantage of the inversion principle in problems that they are solving themselves.

In 1982 Starkey and Gelman reported an attempt to study children’s ability towork out inversion problems directly. They gave 3-, 4-, and 5-year-old childrena certain number of pennies (one to four) to start with and then added and finallysubtracted either one or two pennies at a time. Six of the seven problems that tookthis form were inversion problems in that the addend and subtrahend were equal.In four of these (e.g., 21 1 2 1) the addend and subtrahend were both 1 and inthe other two problems (e.g., 11 2 2 2) the figure was 2. The remaining problemwas a control problem (21 2 2 1) in which the addend and subtrahend wereunequal. The percentage of problems solved by the children was quite high butnot noticeably higher in the inversion than in the control problems, and it istherefore not possible to conclude from this study that the children did use theinversion algorithm to solve the inversion problems. They could have solvedthese rather simple problems by adding and then taking away, rather than by

195ADDITION, SUBTRACTION, AND INVERSION

realizing that, since the addend and subtrahend canceled each other out, nocomputation was needed. Another point is that the control problem too couldhave been solved by inversion, provided that the children also decomposed theaddend (2) into 11 1. The note of caution in Starkey and Gelman’s conclusionabout their results (“even 3-year-olds are capable of providing themselves withdata that might be useful in coming to know explicitly the property of inversion”;p. 105) was entirely appropriate.

This need for caution about young children’s understanding of inversion wasreinforced by two subsequent studies of older children, in both of which latencywas the main dependent variable. Bisanz, LeFevre, and Gilliland (1989) gaveproblems with both an addition and a subtraction to children 6, 7, 9, and 11 yearsof age and to a group of young adults. Some of these were inversion problemsand some not, and each type of problem involved relatively small numbers onsome trials and relatively large ones on others. In all the age groups, some of theparticipants solved the inversion problems more quickly than the standardproblems. The children who used inversion solved the large-number inversionproblems as rapidly as the small-number ones. In contrast, they were a great dealslower with the large-number control problems than with the small-number ones.The actual number of children in this category was small, and did not increase,between the ages of 6 and 9 years; thereafter the number of users rose steeply.

In considering inversion, particularly when concrete material or concreterepresentations like the use of fingers to represent addends and subtrahends areinvolved, one must make a clear distinction between a mere canceling out of aprevious change and an abstract understanding of numerical inversion. If some-one starts with a tower of five bricks, adds three to the top, and then takes thesame three bricks away, it is in principle possible for a child to work out that thetower has returned to its first state without thinking about number. The child mayapply exactly the same reasoning to this problem as to a sequence involving, say,a football shirt that is dirtied by some mud and then cleaned in the wash. In bothcases, something is added and then the same thing is taken away and the statusquo restored. In number inversion problems, however, the actual items that areadded and subtracted may be quite different, and no one can be said to understandinversion unless he or she appreciates that 81 3 2 3 5 8 whatever the identityof the items added and subtracted. We must distinguish then between a low-levelidentity inversion and a more abstract quantitative inversion.

This distinction was first made, though not in these terms, by Bisanz et al.(1989), who observed in the study that we have just described that severalchildren use finger counting in inversion as well as in control (noninversion)problems. Given the verbal problem 51 3 2 3 some of these children wouldrepresent the first number with five fingers, would then represent the addend withthree more fingers, and finally would represent the subtrahend by “simultane-ously collapsing” the three fingers which stood for the addend. The children whoused this strategy, which Bisanz et al. call “negation,” might thus have converted

196 BRYANT, CHRISTIE, AND RENDU

the inversion problem into an identity problem. This was a successful strategy touse in inversion problems in which the identical addend and subtrahend weresmall. Bisanz et al. report that the children using “negation” solved both large-and small-number inversion problems more rapidly than the control problems,but that they were slower with the large-number than with the small-numberinversion problems.

There are two possible reasons why some children fail to use the inversionprinciple in the kinds of problems that we have been describing. One is that theydo not understand the principle. The other is that they understand the principlebut decide for one reason or another not to use it. If the second possibility is theright one, their use of the principle should be greater in some contexts than inothers. Stern (1992), using much the same methods as Bisanz et al. (1989),showed that children between the ages of 7 and 9 years were much more likelyto use the inversion strategy if they had had the previous experience of beinggiven inversion and noninversion trials in separate blocks than if they had onlyexperienced these two kinds of problem mixed together within blocks. Sternmade the convincing point that the difference in the children’s use of theinversion strategy after the two different kinds of experience is evidence that theyoften fail to use the inversion principle even though they do have some knowl-edge of it.

To summarize, studies of inversion demonstrate that some young children dounderstand and use the inversion principle and that others may understand theprinciple but fail to use it unless they are encouraged to. These studies alsosuggest that some young children may solve inversion problems on the basis ofidentity rather than of quantity.

However, so far the research leaves four questions unanswered. The first ofthese concerns the interesting and important issue raised by Bisanz et al. (1989)about the inversion of identity or of quantity. We need a direct way to establishwhether a child solves inversion problems on the basis of identity or of quantity.The measure that Bisanz et al. used to distinguish children who inverted identityfrom those who inverted quantity was rather indirect. A child who puts up threefingers when adding 3 and then collapses all three simultaneously when sub-tracting 3 may be treating the three fingers as the same object rather than as anumber, as the authors suggest. But it is also possible that he or she knowsperfectly well that a quantity is involved. When the child moves all three fingersat the same time, he or she may still understand that they represent the number3. The fact that these particular children fared worse in the large-number than inthe small-number inversion problems may simply be due to their dependence onfinger representation, which is much less helpful when problems involve largenumbers.

An alternative way of looking at this question is to present a set of objects, thento add some more objects to the set, and finally to subtract either the identicalobjects or different objects that are the same in number as the original addend.


If children depend on identity they should solve the first type of problem moreoften than the second.

The second question concerns the excellent point made by Stern (1992) thatthere may be a competence–performance gap in the use of the inversion princi-ple. Stern’s comparison of blocked and mixed inversion and control trials was anappropriate but a limited one. Other variations in context need to be explored.One is to juxtapose problems in which concrete material is used and others inwhich it is not. Some time ago Hughes (1981, 1986) demonstrated strikingdifferences in children’s success in addition and subtraction problems that werepresented in concrete form, and in other such problems that were verbal andabstract. The children were helped a great deal by the concrete material. Concretepresentations may also help children to use the inversion principle.

The third question is whether children can use the inversion principle in aflexible manner to solve problems that on the surface are not inversion problems.Inversion is important because additive reasoning depends to some extent on theunderstanding of the inverse relation of addition and subtraction. Therefore, wewould expect children who understand the principle fully to use it in a range ofproblems. We have already mentioned the possibility of decomposing numbersin order to use the inversion principle. Adults can readily see that the problem52 1 27 2 28 may be reformulated as 521 27 2 27 2 1. Can children whounderstand the inversion principle extend it to solve problems of this sort? Anadditional advantage of including inversion/decomposition problems is that theycontrol for another possibility. One danger in a study of inversion is that childrenmight produce false positives in the inversion condition. They may be soconfused by the inversion and control problems that they simply repeat the firstquantity that they hear, which isa. This would lead them to be right in ana 1b 2 b inversion problem but wrong in ana 1 a 2 b control problem (or in anyother control problem). However, the strategy of repeatinga would also leadthem to give incorrect answers in inversion/decompositiona 1 b 2 (b 1 1)problems as much as in control problems. There is no reason why the strategy ofrepeatinga should help children in the inversion/decomposition problems.

Our final question is about the sequence of the addend and the subtrahend.Most studies have concentrated on thea 1 b 2 b (plus before minus) sequenceand have not includeda 2 b 1 b problems (minus before plus). If childrenunderstand the inversion principle, they should use it as readily with onesequence as with the other. This needs to be checked.

We now report two studies of the understanding of the inversion principle by5- and 6-year-old children. In the first study we measured their success withconcrete and abstract presentations. We also compared inversion problems in twoconditions. In one condition identical objects were added to and then subtractedfrom the original set, and in the other condition one group of objects was addedand a different group subtracted. In the second study we looked at the children’sability to use the inversion principle together with decomposition and we also


compared problems in which the addition preceded the subtraction with others inwhich the subtraction came first and the addition followed.

In the studies by Bisanz et al. (1989) and by Stern (1992) the dependentvariable was the children’s speed in solving the problems. The children couldsolve all the control problems as well as the inversion problems, a result thatindicates that they were able to solve the inversion problems by computationwithout the help of the inversion principle. In our two studies the main dependentvariable was success. The children in our studies were younger on the whole thanthose in the studies by Piaget and Moreau (1977), Bisanz et al. (1989), and Stern(1992), and the problems that we gave them involved computations that weintended to be difficult for them. Thus the number of successful answers in theinversion and control trials was an appropriate measure of the use of inversion.

STUDY 1

Method

Participants

Thirty-eight children (18 girls and 20 boys) from two state schools took partin the study. They were divided into two age groups:younger(n 5 20; mean age5 years 8 months,SD 3.05 months) andolder (n 5 18; mean age 6 years 9months,SD 2.80 months).

Procedure

Each child was given 36 trials. There were 6 conditions and 6 trials percondition. In each condition, inversion problems (a 1 b 2 b) were presented in3 of the trials and control problems (a 1 a 2 b) in the other 3. We chosea 1a 2 b for the noninversion problems because we thought it necessary to havecontrol problems in which one value was also represented twice. The correctanswers for individual inversion problems were matched with those of individualcontrol problems. For example, the inversion problem 141 7 2 7 was matchedto the control problem 91 9 2 4. Thus, though the value of the correct answerswas the same in the inversion and the control problems, the values ofa andbwere not.

The six conditions were as follows.Concrete/identical.The children were shown a column of plastic bricks

(Unifix) attached to each other. The middle part of the column was covered bya cloth to prevent the child from counting all the bricks, but both ends wereexposed and the child was told how many bricks there were altogether. Then oneof the two experimenters added some bricks to one end of the column, sayinghow many were being added. Next this experimenter subtracted some bricksfrom the same end. (The bricks that were added were also attached together andso were the bricks that were subtracted. This meant that it took no longer to addand subtract larger quantities than it did smaller ones.) Finally the experimenterasked the child how many bricks there now were in the column. Half the trials


were inversion trials (a 1 b 2 b problems) and the other half were control trials(a 1 a 2 b problems). In the inversion trials exactly the same bricks were addedand subtracted; in the control trials the bricks that were subtracted were alwaysa subset of the actual bricks added (becauseb was always smaller thana).

Concrete/nonidentical.The procedure was exactly the same as in the concrete/identical condition except that the experimenter added bricks to one end of thecolumn and subtracted from the other end, making it clear to the child thatdifferent bricks were being added and taken away.

Invisible/identical.No concrete material was used. The experimenter wentthrough the same actions (i.e., moved her hands in the same way) but said thatshe was dealing with “invisible men” whom the child must imagine but could notsee. Otherwise the procedure was the same as in the concrete/identical condition,so that in the inversion trials the identical men were added to and subtracted fromthe original quantity and in the control trials the men who were subtracted werea subset of the men added.

Invisible/nonidentical.This was the invisible version of the concrete/noniden-tical condition. The experimenter added to one end and subtracted from the otherend of the line of invisible men, making it clear that the men being added andtaken away were different in identity.

Word problems.The children were told stories about Christmas presents undera Christmas tree, but without visual aids or the help of the experimenter’s handmovements. The child was asked, for example, “If there are 11 presents under thetree and 5 more are put there and then you hide 5 somewhere else, how manypresents will be left under the tree?” No mention, one way or the other, was madeof the identity of the addend and subtrahend. As before, the problems took eitherthe a 1 b 2 b (inversion) or thea 1 a 2 b (control) form.

Abstract problems.The children were given inversion and control problems (e.g.,“What is 11 and 5 more, take away 5?”) verbally, with no mention of any concreteentities. In the inversion trials the value ofa ranged from 8 to 15, and ofb from 5 to9. We chose numbers in this range for them to be big enough to cause the child somecomputation difficulties but nevertheless to be reasonably familiar to the children. Asnoted above, the correct answer to each noninversion problem was matched to thatof an inversion problem. In every case the actual values ofa andb were higher in theinversion problem than in the control problem (e.g., the control problem for theinversion problem 121 9 2 9 was 101 10 2 8; the control for the inversionproblem 141 5 2 5 was 81 8 2 2).

The conditions were presented in four separate blocks according to the methodof presentation (concrete, invisible, word problem, abstract). The order of thetrials was randomized within each block and the order of the four blocks wascounterbalanced between children in an incomplete 43 4 Latin square design,which produced four order groups. The design was incomplete because we hadintended to have 20 children (a multiple of 4) in each age group, but 2 of thechildren whom we recruited for the older group dropped out.


In every trial the child was given 30 s to produce an answer. Two experi-menters were present; one asked the child the questions and the other recordedthe child’s answer and (using a stopwatch) the time the child took to produce theanswer.

Results

Number of Correct Answers

Table 1 presents the means for the correct answers in the six conditions and itshows a striking difference between the inversion and the control trials. In all sixconditions the children did much better in the inversion than in the control trialsand this result suggests that many of them did use the inversion principle. Thedifference is so strong that it is hard to resist the conclusion that there is awidespread understanding of inversion among 5- and 6-year-old children.

The first question that we tackled in our statistical analyses was about thecomparative ease with which children solve problems where identical material isadded and taken away and those where the material added and subtracted is not

TABLE 1Mean Number of Correct Answers in All Six Conditions in Study 1

Condition

Concreteidentical

Concretenonidentical

Invisibleidentical

Invisiblenonidentical

Wordproblems Abstract

5-year-old children (n 5 20)

Inversionproblems

Mean 2.40 1.55 1.65 1.20 1.55 1.40SD 1.05 1.23 1.27 1.24 1.28 1.28

Controlproblems

Mean 0.30 0.40 0.50 0.20 0.40 0.90SD 0.57 0.68 0.83 0.52 0.82 0.85


Inversionproblems

Mean 2.67 2.00 2.50 1.83 2.22 2.06SD 0.84 0.14 0.98 1.15 1.17 1.31

Controlproblems

Mean 1.33 1.39 1.56 1.06 1.33 1.61SD 1.32 1.24 1.29 1.00 1.19 1.04

Note.Maximum correct score for each cell is 3.


identical. Preliminary analysis showed no effect of order groups, and so wecollapsed our data across this variable. We carried out a four-way analysis ofvariance in which the main terms were age groups, material (concrete vsinvisible), problem (inversion vs control), and identity (identical vs nonidenticaladdend and subtrahend), with repeated measures on the last three variables. Thusthis analysis dealt with the concrete identical and nonidentical conditions andwith the invisible identical and nonidentical conditions only, and not with theabstract problems and word problems, which were analyzed separately. Resultsreported below as significant were allp , .05 or better.

The analysis produced a significant age group effect (F(1, 36)5 9.83), whichshows that the 6-year-old children did better on the whole than the 5-year-oldchildren. The children’s greater success with concrete material led to a significantmaterial effect (F(1, 36)5 5.64). There was a highly significant problem effect(F(1, 36) 5 62.75), which is consistent with the children using the inversionprinciple. There was also a significant identity effect (F(1, 36)5 25.08), whichshows that children did better in general when identical material was added andsubtracted than when the addition and subtraction were of different material.

There was a significant Problem3 Identity interaction. A Tukey honestlysignificant difference (HSD) post hoc test of this interaction established that theinversion trials were significantly easier than the control trials, both whenaddends and subtrahends were of identical material and when they were not. Thisestablishes that the children use the inversion principle in a genuinely quantita-tive manner, that is, even when different material is added and subtracted. Thisposttest also showed that, though the inversion trials were easier when theaddends and subtrahends were of the identical material, identity made no differ-ence at all in the control trials, presumably because the children tended to resortto computation in these trials. No other effects or interactions were found. Tosummarize, the analysis supports the hypothesis that children of this age use theinversion principle in its full quantitative sense. Although the identity of theaddend and subtrahend helps them to solve an inversion problem, they stillmanage to use the principle of inversion when completely different items areadded and taken away.

We did a separate analysis of variance of the word problem and the abstractproblem conditions to see if the children used the inversion principle in entirelyverbal problems. The analysis had three main terms, age group, material (wordproblems vs abstract problems), and problem (inversion vs control), with re-peated measures on the last two variables. The analysis produced a significantdifference between age groups (F(1, 36)5 6.38) and a highly significant problemterm (F(1, 36) 5 18.71), which establishes that children of this age use theinversion principle even when the presentation is completely verbal and nomention is made of the identity or nonidentity of the material. There were noother significant results. The strong and consistent superiority of the scores in theinversion trials over those in the control trials suggests that the children use


different strategies in the two types of trial. Our hypothesis is that they often usethe inversion principle in inversion trials, and that they resort to computation inthe control trials where the inversion solution is not possible. If the children useone strategy in one set of problems and a different one in the other, there shouldbe stronger correlations (across conditions) among the different inversion scoresand among the different control scores than between the inversion and controlscores.

This was strikingly the case. The 15 correlations among the control scoresranged from .61 to .83, with a median of .75, and the 15 correlations among theinversion scores ranged from .35 to .75, with a median of .62. In contrast, the 36correlations between the inversion and the control scores were generally lower.They ranged from .07 to .62, with a median of .46.

The unevenness of these correlations prompted us to do a rotated factoranalysis, with a Varimax solution, of the correct scores in the inversion andcontrol trials in all six conditions. This produced two factors, the first of whichaccounted for 41.99% of the variance and the second for an additional 32.38%(74.37% in all). The control tasks, all of which required computation, loadedheavily (.80 or above) on the first factor, and yet only one of the inversion scores(the word problem inversion task) had a loading of above .50 (.52) on this factor.The first factor is undoubtedly a Computation factor. In contrast all the inversionproblems and none of the control scores loaded heavily (above .65) on the secondfactor, which can therefore be called the Inversion factor. Thus the correlationsand the factor analysis support the idea that children solved the control problemsby computation and many of the inversion problems by using the inversionprinciple. However, the number of children (38) in this analysis is small for thenumber of variables entered (12) and so this conclusion should be treated withsome caution. It should also be remembered that the fact that two tasks areheavily loaded on the same factor does not necessarily mean that children use thesame strategies to solve the two tasks.

Errors and Qualitative Differences

Although we recorded the latency of each response, it was not possible toproduce a meaningful analysis of the time that it took to produce a correctresponse because many of the children did not produce any correct responses insome of the tasks (particularly in the control tasks). We did look at the numberof trials in which children produced no answer because they ran out of time (30s). These occurred much more frequently in the control than in the inversiontrials. The mean number of trials on which children ran out of time in theinversion trials was 3.26 (out of 18 trials). In the control trials the mean was 7.66(out of 18). However, the distributions of both these scores were so skewed thatit was impossible to compare them with parametric statistics.

We also looked at the number of repetition ofa errors in the control conditions(e.g., giving 10 as the answer to the question 101 10 2 8). If there were a lot


of these, we would be concerned that children had produced the correct answerin the inversion trials for spurious reasons (i.e., simply repeating the first numberin the problem). However, there were very few repetitions ofa in the controltrials. The mean number of times that the children produced these errors in thecontrol trials (again out of 18 trials) was 0.29, which represents only 1.6% of thecontrol trials. We conclude that repetition of the first number in the problem doesnot provide an explanation for the children’s relatively good performance in theinversion trials.

After each trial we asked the child for an explanation, and we also observedwhat the children did during the trials. Only 9 of the 38 children were able toexplain the inversion principle to us explicitly. Five others made ambiguousstatements which might have been appeals to inversion, like “It doesn’t matter ifit’s different.” Although these ambiguous statements were made only in inver-sion trials, it is not clear whether they were about inversion.

Seven children consistently used their fingers, but we only observed a childusing the “negation” strategy noted by Bisanz et al. on one occasion. Somechildren were more consistently successful in the inversion trials than wereothers. Fifteen of the 38 children scored two or three correct (out of three) in allsix inversion tasks. We also noted that 8 of the 9 children who were able toexplain the inversion principle explicitly were among these 15 consistently highscorers. The fact that the other 7 children in the group of 15 high scorers had notbeen able to explain the inversion principle verbally suggests that children maybe able to use the principle without being aware of it at a conscious level.

Our analysis of the errors and of the children’s explanations supports ourhypothesis that many of the children used the inversion principle even thoughseveral of them seemed to be unable to put it into words. The idea that thechildren succeeded in the inversion trials by simply repeatinga seems not to betrue.

STUDY 2

Having found evidence for a widespread use of the inversion principle bychildren as young as 5 and 6 years, we then asked whether children can also solveproblems with a combination of decomposition and the inversion algorithm. Inour second study we gave children problems that took the form ofa 1 b 2 (b 11) or a 1 b 2 (b 2 1).

This study was also designed to answer two subsidiary questions. One wasabout the sequence of addends and subtrahends in inversion trials. Previousstudies have uniformly used thea 1 b 2 b order, but anyone who understandsthe inversion principle should finda 2 b 1 b trials as easy to solve asa 1 b 2b ones. Problems of this sort therefore provide an essential check of the under-standing of inversion. The other question was about the size of the numbers. Inthe studies by Bisanz et al.(1989) and by Stern (1992) one of the main measuresof the use of inversion was the difference between small- and large-number trials.


If children use the inversion principle, the size of the numbers involved shouldaffect the control trials, which require computation, more than the inversiontrials, which require no computation for anyone using the principle. The twoprevious studies were latency studies, and it seemed to us essential to establishwhether the same differences in the effect of number size on the inversion andcontrol trials can also be found when the dependent variable is the number ofcorrect solutions.

Method

Participants

Fifty-five schoolchildren (30 girls and 25 boys) took part in the study. Therewere three age groups: 6-year-olds (N 5 19; mean age 6 years 10 months,SD3.2months), 7-year-olds (N 5 19; mean age 7 years 7 months,SD2.9 months), and8-year-olds (N 5 17; mean age 8 years 9 months,SD 2.9 months).

Procedure

Each child was given 24 trials. Half the trials involved small-number (correctanswer, 10) and half large-number (correct answer. 10) problems. Again thecorrect answers to the individual inversion problems and inversion/decomposi-tion problems were matched to the correct answers to individual control prob-lems. Thus the control for 271 152 15 was 161 162 5. All the children weregiven all three conditions, which were the following.

Plus–minus: inversion a1 b 2 b; controla 1 a 2 b. The procedure here wasthe same as in the concrete nonidentical condition in the first study. In eight trialsthe children were shown a partially covered column of plastic bricks and weretold the number of bricks that it contained. The experimenter added to one endof the column and then subtracted from the other end with an appropriatecommentary on what she was doing. Four of the trials involved inversion (a 1b 2 b) and four involved control problems (a 1 a 2 b). In half of each of thesethe numbers were small and in the other half they were large.

Minus–plus: inversion a2 b 1 b; controla 2 b 1 a. The procedure was thesame as in the plus–minus condition except that the experimenter subtracted firstand added later. Thus the four inversion trials took the form ofa 2 b 1 b andthe four control trials ofa 2 b 1 a.

Inversion/decomposition: a1 b 2 (b 1 1) anda 1 b 2 (b 2 1). Here theprocedure was the same as in the inversion trials in the plus–minus conditionexcept that in four trials the subtrahend was greater by 1 than the addend and inthe other four it was smaller by 1. Half of each of these trials involved smallnumbers and half involved large numbers. For example, one of the small-numbera 1 b 2 (b 1 1) problems was 71 4 2 5, and one of the large-numbera 1 b 2(b 2 1) trials was 241 10 2 9. The control trials for the plus–minus conditionalso served as controls for the trials in both kinds of inversion/decompositionproblems.


In this experiment there were no order groups. The order of the 24 trials wasrandomized for each child. Again two experimenters were present, one whoposed the problem to each child and the other who recorded the child’s responsesand timed each trial.

Results

Number of Correct Answers

The first question that we asked was whether the order of the addend andsubtrahend (a 1 b 2 b/a 2 b 1 b) and the size of the numbers would affect thechildren’s use of the inversion principle. The scores for the plus–minus and theminus–plus conditions are presented in Table 2. The table shows that the

TABLE 2Mean Correct Answers in the Plus–Minus and Minus–Plus Conditions in Study 2

Condition

Plus–Minus Minus–Plus

Small Large Small Large


Inversion problemsMean 1.37 1.05 1.26 1.05SD 0.76 0.91 0.87 0.97

Control problemsMean 0.84 0.05 0.84 0.05SD 0.76 0.23 0.90 0.23









inversion trials were again a great deal easier than the control trials both whensmall numbers and when large numbers were involved. The table also shows thatthe size of the numbers had a much greater effect in the control trials, whichrequire computation, than in the inversion trials. Both of these results support theidea that many of the children used the inversion principle. Finally the tableindicates that the sequence of addend and subtrahend had little effect: Thechildren did as well in thea 2 b 1 b trials as in thea 1 b 2 b ones.

We carried out a four-way analysis of variance of these scores in which themain terms were age group, problem (inversion vs control), addend/subtrahendsequence (plus–minus vs minus–plus), and number size. There was a significantage group difference (F(2, 52)5 3.86), and a subsequent Tukey HSD post hoctest established that the 7- and 8-year-olds both had significantly higher scoresthan the 6-year-olds. However, age group did not interact with any of the othervariables. There was a highly significant problem difference (F(1, 52)5 155.01),which is consistent with the hypothesis that children at all three age levels use theinvariance principle. The number size difference was also highly significant (F(1,52) 5 111.87), as expected. Finally there was a significant Problem3 NumberSize interaction (F(1, 52) 5 36.22). A Tukey HSD post hoc test of thisinteraction established a significant difference between the small-number inver-sion and the small-number control scores and also between the large-numberinversion and the large-number control scores. It also showed a significantdifference between the small- and large-number control scores but not betweenthe small- and large-number inversion scores. This suggests that the children’suse of the inversion principle in the inversion trials overrode the effect ofincreasing the numbers.

Our next question was whether children are able to combine decompositionwith inversion, and we analyzed this by comparing the children’s performance inthe two types of inversion/decomposition trials (a 1 b 2 (b 1 1) anda 1 b 2(b 2 1)) and in the control trials. We reasoned that if the children do better in thedecomposition trials, which can be solved by a combination of inversion anddecomposition, than in the control trials, which require computation, they mustbe taking advantage of the inversion and decomposition principles. Table 3 givesthe mean correct scores and it shows that the children did better in the inversion/decomposition trials than in the control trials in the large-number trials. Therewas no such difference between the inversion/decomposition trials and thecontrol trials in the small-number trials.

We carried out an analysis of variance in which the main terms were the agegroup, problem ((a 1 b 2 (b 1 1), a 1 b 2 (b 2 1)) and control problems), andnumber size, with repeated measures on the last two variables. There was asignificant age group difference (F(2, 52)5 4.42); once again a Tukey HSD posthoc test confirmed that the difference lay between the 6-year-olds and the othertwo age groups. There was also a significant problem difference (F(2, 104)55.23); a Tukey HSD post hoc test established that this was due to the two kinds


of decomposition problems being easier than the control problems. There was asignificant effect of number size (F(1, 52) 5 166.12). Finally, there was asignificant Problem3 Number Size interaction (F(2, 104)5 3.28). A Tukey posthoc test established that when the numbers were large there were significantdifferences between the inversion/decomposition scores on the one hand and thecontrol scores on the other, but that there was no such difference when thenumbers were small. Thus this experiment suggests that children did combineinversion and decomposition to help them solve the large-number problems. It isnot possible to say whether the children used the inversion algorithm or not in thesmall-number inversion/decomposition trials.

We also looked at the correlations between the different scores in this study.There were strong positive correlations between all the small-number scores.However, with the large-number tasks we found that the inversion and theinversion/decomposition scores correlated with each other more highly than theycorrelated with the control scores. The six correlations among the inversion andinversion/decomposition tasks ranged from .24 to .67, with a median of .55. Theeight correlations between the four inversion and inversion/decomposition taskson the one hand and the two control tasks on the other were lower. They rangedfrom 2.01 to .25, with a median of .20. However, for reasons that we cannotexplain, the correlation between the two control tasks was only .16 and nonsig-nificant. We carried out separate factor analyses (rotated with a Varimax solu-tion) for the small- and large-number scores. The small-number factor analysis

TABLE 3Mean Correct Answers in the Inversion/Decomposition Problems in Study 2

Condition

a 1 b 2 (b 2 1) a 1 b 2 (b 1 1) Control

Small Large Small Large Small Large


Mean 0.95 0.26 0.79 0.47 0.84 0.05SD 0.78 0.56 0.71 0.70 0.76 0.23


Mean 1.21 0.37 1.58 0.84 1.16 0.11SD 0.71 0.6 0.51 0.69 0.83 0.32


Mean 1.47 0.82 1.41 1.18 1.53 0.12SD 0.62 0.88 0.71 0.73 0.62 0.33



produced one significant factor only. The analysis of the large-number scoresproduced two significant factors. The inversion and inversion/decompositionscores were highly loaded on the first factor (.66 to .89) but not on the second.Both control scores were highly loaded on the second factor (.52 and .82) but noton the first. The first factor, which we call the Inversion factor, accounted for38.52% of the total variance in the large-number scores. The second factor,which we call the Computation factor, accounted for an additional 23.09%(61.61% in all).

The factor analysis of the large-number scores supports our hypothesis thatchildren in this age range often use different strategies to solve inversionproblems and problems which require computation. The pattern of the factors issimilar to that reported in the first experiment, and it should be noted that thistime the ratio between the number of variables entered (6) and the number ofparticipants (55) was a reasonable one. However, we must remark again that thefact that two tasks are loaded on the same factor does not necessarily mean thatchildren use the same strategy to solve these two tasks.

Error Types and Qualitative Differences

In this experiment we recorded latencies but, for the same reasons as in theearlier study, it proved impossible to subject these to a statistical analysis. Wealso looked at how often the children ran out of time (30 s) and produced noanswer. The figures were roughly comparable to those of the first study. Themean number of trials (out of 8) in which the children ran out of time was 1.36in the inversion trials, 2.31 in the inversion/decomposition trials, and 3.18 in thecontrol trials. Thus the children suffered this kind of failure more often in thecontrol trials than in the other trials. However, again the scores were too skewedto allow parametric analysis.

We looked at the number of repetition ofa errors in the control conditions andin the inversion/decomposition trials. Again we were concerned that a largenumber of these repetitions would suggest that children had produced the correctanswer in the inversion trials for spurious reasons (i.e., simply repeating the firstnumber that they heard in the problem). However, there were very few repeti-tions ofa either in the control trials or in the inversion/decomposition trials. Themean number of times that the children made these errors in the eight controltrials was 0.15, which represents only 1.9% of these trials. The mean number ofthese errors in the eight inversion/decomposition trials was only 0.16 (2%).

Thirteen of the 55 children expressed the inversion principle explicitly in theirjustifications at some time during the experiment. However, we did not find anyinstances of children making ambiguous statements about the inversion problemsin this experiment. Nine of the children used their fingers to count. We did notobserve any instances of the “negation” strategy noted by Bisanz et al. among thefinger counters in this session.

We looked at the consistency of the children in the inversion and the inversion/


decomposition trials. Thirty-three of the 55 children scored 3 or 4 out of 4 in boththe inversion conditions. All 9 of the children who expressed the principle ofinversion explicitly were in this group of 33. None of the finger counters was inthis group.

Only 10 of the 55 children produced scores of 3 or 4 in both inversion/decomposition problems. All of these 10 children were among the 33 childrenwho had done well in the inversion problems, and 8 of the 10 were among the9 children who had expressed the inversion principle explicitly. Thus all but 2 ofthe children who did consistently well in the inversion/decomposition trials haddemonstrated explicit knowledge of the inversion principle. Therefore, explicitknowledge may be needed to solve inversion/decomposition problems consis-tently well, although it seems not to be needed to solve the inversion problems.

DISCUSSION

We have reached three main conclusions. The first is that children as young as5 years understand and frequently use the inversion principle and that they cando so in a genuinely quantitative way. The comparison between the nonidentityinversion scores and the control scores established that many of the children wereable to use the inversion principle even in problems in which identity played nopart. The items added and subtracted were palpably different, and yet the childrenstill did a great deal better when it was possible to use the inversion principle thanwhen it was not.

In both studies many of the children who were consistently successful in theinversion tasks nevertheless seemed unable to express the inversion principleexplicitly. There are two possible reasons why this might be so. One is that theyare explicitly aware of the principle, but cannot put it into words. The other,which has been suggested recently by Siegler and Stern (1998), is that childrenuse the inversion principle at first without being consciously aware of it. Wecannot say which of these two possibilities is the right one, but it is interestingto note that our results also tentatively suggest a different pattern when it comesto the inversion/decomposition problems. We found that all but 2 of the 10children who succeeded consistently well in these problems had shown them-selves able to give an explicit account of the inversion principle. It is thereforepossible that an explicit understanding of inversion is needed if the child is tocombine the principle with the complex process of decomposition.

Our second conclusion is that young children can also use the principleflexibly. They take advantage of it not only in simple inversion problems but ininversion/decomposition problems as well. Their relative success in the inver-sion/decomposition problems suggests that many children use the inversionprinciple actively: They can create an inversion problem out of a problem inwhich the addend and subtrahend are actually different in quantity. It is alsointeresting that they do so by decomposing numbers. There is evidence thatdecomposition is a strategy that is characteristic of informal, out-of-school,


mathematics and is not usually taught at school (Carraher, Carraher, & Schlie-mann, 1985; Nunes et al., 1993).

Our third conclusion concerns the relation between children’s ability to addand subtract and their understanding of the relation between addition and sub-traction. The most surprising result was of an apparent split between the two.This is surprising because it is quite reasonable to expect a relationship betweenthem: It seems quite plausible that the better a child is at adding and subtractingthe more understanding he or she will have of the inverse relation between thetwo operations. Yet the correlations and the factor analyses that we did in bothstudies suggest a distinction between adding and subtracting skills on the onehand and the understanding of the inverse relation between addition and sub-traction on the other hand. The main difference between the inversion and thecontrol problems was that it was possible to use the inversion principle in theformer but not in the latter. In other important respects—in the procedure, in thematerial, in the size of the correct answer—these two sets of problems were thesame as each other, and yet the correlations between them were relatively small.The inversion scores correlated strongly with each other, and the control scoreswith each other, despite great differences in the material and in the procedures inthe various conditions. The apparent split between these two kinds of tasks wasalso suggested by the fact that they loaded on different factors in factor analysesof both studies.

These three conclusions confirm and extend other people’s hypotheses aboutchildren’s understanding of the inversion principle. As far as Piaget is concerned,our results suggest that children do to some extent grasp the principle earlier thaneither his ideas about concrete operations and reversibility or his work oninversion (Piaget & Moreau, 1977) implies. However, the evident split betweenthe children’s use of the inversion principle and their ability to solve problemsthat require computation fits well with Piaget’s ideas. The distinction betweenunderstanding of logical principles and the learning of specific mathematicalprocedures is surely at the heart of Piaget’s approach to the development ofmathematical understanding.

Our evidence that children are more successful in identity than in nonidentityinversion trials confirms the importance of another distinction—the one made byBisanz et al. (1989) between the inversion of identity and the inversion ofquantity. An intriguing point here is that Bisanz et al. worked on different aspectsof identity than we did. Bisanz et al. looked at the incidence of canceling outfinger representations of the added and then subtracted numbers, whereas wevaried the identity or nonidentity of the actual items which were added orsubtracted. The relation between these two responses is potentially interesting.It seems quite likely that the children who are affected by identity in oursense would be more likely to use the finger canceling strategy identified byBisanz et al.

We should like to add that there is still some ambiguity about the effect of


identity in our first study. There are two possible ways of looking at our result.One is to say that in trials, in which the same items were added and subtracted,the children’s correct choices were not quantitative at all. The other is to claimthat the identity of the items in these trials actually prompted the child to thinkabout the quantitative significance of inversions. This latter interpretation fitswell with Stern’s suggestion that the context of the task can have a significantimpact on children’s use of the inversion principle.

Our study has some important educational implications. In England, where theresearch was done, there is currently a great emphasis in teaching mathematicson learning basic skills like adding and subtracting and building knowledge ofnumber facts involving specific additions and subtractions. The one reference toinversion in the National Curriculum is that children “should use the fact thatsubtraction is the inverse of addition” in solving problems. The phrase “use thefact that” suggests that inversion can be a handy shortcut, which is true, butinversion is also a principle which contributes to a proper understanding ofnumber. Our study shows that young school children have an impressive under-standing of that principle.

REFERENCES

Bisanz, J., LeFevre, J-A, & Gilliland, S. (1989, April).Developmental changes in the use of logicalprinciples in mental arithmetic.Poster presented at the biennial meeting of the Society forResearch in Child Development, Kansas City, MO.

Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and inschool.British Journal of Developmental Psychology,3, 21–29.

Hughes, M. (1981). Can preschool children add and subtract?Educational Psychology,3, 207–219.Hughes, M. (1986).Children and number.Oxford: Blackwell.Nunes, T., Schliemann, A.-L., & Carraher, D. (1993)Street mathematics and school mathematics.

New York: Cambridge Univ. Press.Piaget, J. (1952).The child’s conception of number.London: Routledge & Kegan Paul.Piaget, J., & Moreau, A. (1977). L’inversion des operations arithmetiques. In J. Piaget (Ed.),

Recherches sur l’abstraction reflechissante: Vol 1. L’abstraction des relations logico mathema-tiques(pp. 45–62). Paris: Presses Universitaires de France.

Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgeneticanalysis.Journal of Experimental Psychology—General,127,377–397.

Starkey, P., & Gelman, R. (1982). The development of addition and subtraction abilities prior toformal schooling in arithmetic. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.),Addition and subtraction: A cognitive perspective(pp. 99–112). Hillsdale, NJ: Erlbaum.

Stern, E. (1992). Spontaneous use of conceptual mathematical knowledge in elementary schoolchildren.Contemporary Educational Psychology,17, 266–277.

Received July 7, 1998; revised July 12, 1999


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