Solving subtraction problems flexibly by means ofindirect addition

Lieven Verschaffel1*, Joke Torbeyns1,3, Bert De Smedt2,Greet Peters1 and Pol Ghesquiere21Centre for Instructional Psychology and Technology, K.U. Leuven, Belgium2Centre for Parenting, Child Welfare and Disabilities, K.U. Leuven, Belgium3GROEP T – Leuven Education College, Leuven, Belgium

Background. Subtraction problems of the type a2 b ¼ ? can be flexibly solved byvarious strategies, including the indirect addition (IA) strategy ‘howmuch do I have to addto b to get a?’ Although rational task analyses indicate that IA is highly efficient especiallyon subtractions with small differences between the integers, little research has beendone on the frequency and efficiency of this strategy on different types of subtractions.

Aim. The present contribution aims at deepening our understanding of theoccurrence and development of the IA strategy, by summarizing the major results ofseveral recent and closely related studies done by the authors on this issue, in relationto other relevant research by others.

Results. The review reveals that young adults frequently and efficiently apply IA ondifferent types of symbolic multi-digit subtractions, whereas children hardly do so.However, children’s (efficient) use of IA can be enhanced through instruction that payssystematic attention at this strategy.

Discussion. The review ends with a discussion of the major theoretical,methodological, and educational implications of the results.

The development of basic arithmetic principles related to addition and subtraction, and

of arithmetic strategies that are based on these pieces of conceptual knowledge, is an

intriguing and important element of psychological and educational research in

elementary mathematics. With respect to addition and subtraction, we have, for

instance, the following principles:

(1) The commutativity principle, which says that the order of the addends is irrelevantto their sum, that is, aþ b ¼ bþ a.

(2) The addition–subtraction inversion principle, which involves the immediate

recognition that adding an amount to a collection can be undone by subtracting

the same amount and vice versa (aþ b2 b ¼ a or a2 bþ b ¼ a).

* Correspondence should be addressed to Professor Lieven Verschaffel, Center for Instructional Psychology and Technology,Katholieke Universiteit Leuven, Vesaliusstraat 2, B-3000 Leuven, Belgium (e-mail: lieven.verschaffel@ped.kuleuven.be).

TheBritishPsychologicalSociety

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Understanding Number Development and Difficulties, 51–63

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DOI:10.1348/97818543370009X12583699332456

(3) The addition–subtraction complement principle, referring to the insight that if

aþ b ¼ c, then c2 b ¼ a and c2 a ¼ b, or that the difference c2 b ¼ ? can be

efficiently determined by considering what should be added to b to make c.

Previous theorizing and research indicated that an implicit or explicit understanding of

these principles not only plays an important role in the development of children’s

construction of the additive composition of number and additive reasoning, but also

considerably facilitates children’s mental arithmetic by shortening or simplifying the

computational process and freeing up working memory space (Baroody, Torbeyns, &

Verschaffel, 2009; Cowan, 2003). For example, the complement principle grounds a

computation shortcut for solving multi-digit subtractions with a small differencebetween the two integers like 612 59 or 4022 398, by asking how much has to be

added to the smaller integer to make the larger one. In the rest of this article, we will call

this shortcut strategy indirect addition (IA), while we will use the term direct

subtraction (DS) for the more common strategy for doing subtraction whereby the

smaller number is directly taken away from the larger one (see also Torbeyns,

Ghesquiere, & Verschaffel, 2009).

In contrast to the commutativity and inversion principle and their accompanying

computational shortcut strategies (see Verschaffel, Greer, & De Corte, 2007, for anoverview), the complement principle has only received a very limited amount of

attention from researchers (see also Van den Heuvel-Panhuizen & Treffers, 2009). In this

article, we will present a series of closely related studies that we have done so far

concerning the IA strategy. This line of research tries to integrate, from a theoretical

perspective, cognitive-psychological, and sociocultural elements, and from a

methodological perspective, a variety of research methods, quantitative as well as

qualitative ones, involving both children and adults.

Use of IA in young adults

In our first study on the fluent and adaptive use of IA, 25 university students solved aseries of three-digit subtractions (Torbeyns, Ghesquiere, et al., 2009). We presented

three types of subtractions on the basis of the difference between the two given

numbers. All subtractions consisted of a minuend ranging from 812 to 829. Subtractions

with a small difference between the integers had a subtrahend ranging from 770 to 790

(e.g. 8122 783 ¼ ?), subtractions with a medium difference included a subtrahend

between 470 and 490 (e.g. 8212 475 ¼ ?), and subtractions with a large difference had

a subtrahend ranging from 170 to 190 (e.g. 8132 176 ¼ ?).

Adopting Siegler’s choice/no-choice method (Siegler & Lemaire, 1997), allparticipants solved the series of subtractions individually in one choice and two no-

choice conditions. In the choice condition, participants were instructed to mentally

solve the subtractions with either DS or IA. In the first no-choice condition participants

were instructed to mentally solve all subtractions by means of DS, whereas in the second

no-choice condition they always had to apply IA. In all three conditions, the subtractions

were visually presented until solved, and participants had to verbally report the strategy

used after each trial. The experimenter registered the accuracy and speed of responding

in each condition on a trial-by-trial basis.Study 1 had four main findings. First, about two-thirds (68%) of the participants

applied the IA strategy at least once in the choice condition: 48% alternated between DS

and IA, whereas 20% answered all items with IA. The other participants (32%) always

applied DS.

52 Lieven Verschaffel et al.

Second, concerning strategy distribution, the frequency of IA and DS in the choice

condition did not differ significantly. Participants solved 43% of the subtractions with IA

and 57% with DS.

Third, in the two no-choice conditions participants performed equally accurately on

both strategies, but they were significantly faster with IA than with DS. Furthermore, IA

was executed faster than DS not only when there was a small difference between thetwo numbers, but also on the two other subtraction types where the computational

advantage of solving the problem by IA seemed less straightforward.

Fourth, participants fitted their strategy choices adaptively to both item and

individual strategy performance characteristics. Adaptivity towards item characteristics

was revealed by the fact that, in the choice condition, small-difference subtractions were

answered significantly more frequently with IA than the two other problem types.

Adaptivity to strategy performance characteristics was revealed by a significant

correlation between individual speed differences between IA and DS (in the two no-choice conditions) and the individual frequency of IA use (in the choice condition),

indicating that participants who mastered IA relatively better, used this strategy also

significantly more frequently in the choice condition.

The results of this first study were replicated with a larger group of similar students,

similar items, and an equal choice/no-choice procedure (see also Torbeyns, Ghesquiere,

et al., 2009). The administered problems differed slightly as only subtractions with small

to medium differences between the numbers were presented, divided into four problem

types on the basis of the size of the difference (i.e. a subtrahend in the 400s, 500s, 600s,or 700s). The results of this study were similar to our first study, except that in the no-

choice conditions the IA strategy was executed not only faster (8.8 s), but also more

accurately (87% correct) than the DS strategy (respectively 16.18 s and 79% correct).

Again, IAwas executed more efficiently than DS not only on the subtractions with a very

small difference between the two given numbers, but also on the other problem types.

Development of IA in children

The results of our first study demonstrated that young adults use the IA strategy quite

frequently, efficiently, and adaptively to solve subtractions. So, what about the use of this

strategy by elementary schoolchildren, who are, from the second grade on, intensively

confronted at school with symbolically presented multi-digit subtractions of the form

a2 b ¼ ?, which they are supposed to solve mentally with DS? This was the major

question of our second study (Torbeyns, De Smedt, Ghesquiere, & Verschaffel, 2009a)

involving 71 second-, 71 third-, and 53 fourth-graders, who all had received instructionin two-digit mental addition and subtraction and who were divided into three math

ability groups on the basis of a general mathematics achievement test. Like all other

textbooks that are currently in use in Flanders (Belgium), the textbook used in these

schools only paid explicit and systematic attention to the DS strategy (both in the mental

arithmetic and, from grade 3 on, in the written arithmetic parts of the math

curriculum).1 An interview with the teachers of these children further confirmed that

1 In Flanders, mathematics instruction in the domain of multi-digit arithmetic starts with intensive practice of mental arithmeticstrategies. Children learn to mentally solve multi-digit addition and subtraction in second grade (with numbers up to 100) andthird grade (with numbers up to 1,000). The standard paper-and-pencil algorithms for addition and subtraction are normallyonly introduced by the end of the third grade (i.e. after more than 1 year of intensive practice in solving multi-digit additions andsubtractions by means of mental arithmetic).

Subtraction by indirect addition 53

they had not provided systematic instruction and practice in the IA strategy, although

all teachers stated that they had allowed children to apply alternative strategies as

long as these children could also demonstrate perfect mastery of the explicitly taught

DS strategy.

All children individually completed two tasks designed to evaluate their strategy

competencies in the domain of addition and subtraction up to 100: a spontaneous

strategy use task (SST) and a variability on demand task (VDT). The SST consisted of

different types of two-digit additions and subtractions, including two-digit subtractions

with a small difference between the minuend and the subtrahend (e.g. 412 39 ¼ ?),

which are assumed to trigger the IA strategy. Children were instructed to mentally solve

each item as accurately and as fast as possible with their preferred strategy, and to

verbally report both the answer and the strategy used immediately after solving each

item. As in the first study, the items were left visible during the solution process. The

VDT consisted of the same types of two-digit problems as the SST. Children wereinstructed to mentally solve each item with at least two different strategies, and verbally

report each strategy immediately after they solved the problem. If a child immediately

solved a problem by means of IA, the interviewer asked the child if (s)he could solve the

same problem by means of a second strategy, and afterwards moved to the next problem

anyhow. If the child began by solving the problem with a strategy different from IA, the

experimenter kept asking for another possible strategy until the child either reported

the IA strategy, stated that (s)he did not know any other strategy, or reported five other

alternative solution methods.The study had three main findings. First, the analysis of children’s strategy repertoire

in the SST revealed that less than 10% of the second- and third-graders, and only 15% of

the fourth-graders spontaneously applied the IA strategy at least once to answer the five

small-difference subtractions. Thus, children hardly used the IA strategy, even on items

where this strategy can be considered extremely efficient.

Second, all children reported various strategies for solving the two small-difference

items from the VDT, but only a minority of them reported IA as an alternative strategy,

suggesting that IA was not included in the strategy repertoire of most children. Actually,only 5% of the second-graders, 15% of the third-graders, and 20% of the fourth-graders

mentioned the IA strategy at least once as a possible alternative solution strategy.

Third, despite the low overall occurrence of IA in the two tasks and in the three age

groups, there tended to be an effect of age and mathematical achievement level on the

frequency with which IA was spontaneously used (in the SST) or mentioned as an

alternative solution method (in the VDT): higher math achievers tended to solve more

small-difference items with the IA strategy than low math achievers.

Microgenetic investigation

In our next study (De Smedt, Torbeyns, Stassens, Ghesquiere, & Verschaffel, 2009), we

wanted to perform a more fine-grained analysis of the emergence and development of

IA as an alternative solution method for doing symbolically presented subtractions, by

means of the microgenetic method. As argued and demonstrated by Siegler (2000),this method provides a means for obtaining the kind of fine-grained information that is

essential for understanding the strategy change process. As Study 2 indicated that IA

did not show up quickly and easily among elementary schoolchildren who did not

receive specific instruction in that strategy, we decided to try to accelerate the

54 Lieven Verschaffel et al.

emergence and development of this strategy by providing children with experiences

intended to promote the process of strategy change. This resulted in a microgenetic

investigation with two groups that differed in terms of the intensity of the

instructional support.

Thirty-five third-graders, who had not received instruction in IA and who had not

shown any evidence of mastery of the IA strategy during an initial test, participated in astrong (SI,N ¼ 20) or weak (WI,N ¼ 15) instruction group. The groups did not differ in

mathematical achievement or intellectual ability.

All children were individually administered three test sessions, four practice

sessions, one transfer session, and one retention session. The test sessions, practice

sessions, and retention session each consisted of a series of symbolic subtractions in the

number domain 20–100. In the transfer session, children were offered two tasks: a

symbolic subtraction task in the number domain up to 1,000 and a subtractive word

problem task in the number domain 20–100. In each session, three item types wereincluded: items with a small difference (e.g. 812 79 ¼ ? for numbers up to 100; 8032796 ¼ ? for the symbolic subtraction task with numbers up to 1,000), medium

difference (e.g. 812 43 ¼ ?; 8032 436 ¼ ?), or large difference (e.g. 812 7 ¼ ?;

8032 78 ¼ ?) between the minuend and the subtrahend.

The children from the SI group solved the same series of items as the children from

the WI group. All sessions, except the practice sessions, were exactly the same for both

experimental groups. In the three test sessions, the transfer session, and the retention

sessions, all children were asked to mentally solve all items with their preferred strategy.In the four practice sessions, the SI group was explicitly instructed to mentally solve

each item once with DS and once with IA, while children of the WI group mentally

solved each item twice with their preferred strategy without any further instruction.

In the SI group, the IA strategy was briefly demonstrated at the beginning of each

practice session, and if necessary, support with the execution of the IA strategy

was provided during the practice session. In the WI group, the only extra instruction

during practice sessions was an unusually large number of subtractions with a very small

difference between the two given numbers, compared to children’s regularinstructional practice at school, which typically contains little or no such problems.

In each session, children were asked to mentally solve each item as well and as fast as

possible. Accuracy and speed of responding were registered per child and per item;

children had to verbally report their strategy during (practice session) or immediately

after (test, transfer, and retention sessions) solving each item. The exact sequence of the

different sessions was as follows: test 1, practice 1, practice 2, test 2, practice 3, practice

4, test 3, transfer, and retention. Test, practice, and transfer sessions were separated at

least 2 days in time for each child. One month after the transfer session, children wereoffered the materials from the retention session.

The results of the microgenetic investigation can be summarized as follows. First,

with regard to strategy frequency, the IA strategy was, quite surprisingly, not used once

in the WI group during any of the test or retention sessions. But also in the SI group,

wherein children received intensive practice in the IA strategy, this strategy was used

rather infrequently. Nevertheless, there was a significant increase of frequency of IA use

between test Sessions 1 and 2 (from 0 to 5.8%) and between test Sessions 2 and 3 (from

5.8 to 10.6%), and in the retention session there was only a slight decrease to 10%.Second, from the moment they started to use the IA strategy, children from the SI

group applied it with at least the same accuracy and speed as the systematically taught

and intensively practised DS strategy. They even used it somewhat more accurately and

Subtraction by indirect addition 55

more quickly, although only the greater accuracy in favour of IA reached significance.

This greater efficiency of IA was found for all three types of subtractions, thus also for

those with a medium and large difference, for which the computational advantage of IA

is less clear.

Third, in the transfer session, we observed, surprisingly, more IA strategies than in

the test and retention sessions, and even in the WI group, IA was used. On thesymbolic subtractions up to 1,000, the WI group applied the IA strategy on 3.3%,

whereas the SI group used it on 12.5%. On the word problems, the WI group (13.3%)

again used IA less frequently than the SI group (30.8%). While the former difference

was not statistically significant, the latter was. As far as the efficiency data for the

transfer session are concerned, the symbolic subtractions up to 1,000 were solved

more (but not significantly) accurately with the IA strategy than with the systematically

taught and intensively practised DS strategy, and significantly faster with the former

than with the latter. The word problems were solved as accurately and fast with IA aswith DS.

Finally, in the SI group, there was no clear relationship between children’s general

mathematical achievement and the moment they started to apply IA and the frequency

with which they applied it. This conclusion should be interpreted with caution,

due to the relatively small size of the sample and to the relatively rare occurrence of

IA in that sample.

In sum, the most striking finding of this study was – once again – children’s great

difficulty with integrating IA in their strategy repertoire for doing symbolically presentedsubtractions, even in the SI group wherein IA was explicitly presented as a convenient

alternative solution strategy. Two other findings caught our attention. First, the word

problems from the transfer test elicited – in both groups – substantiallymore IA strategies

than the symbolically presented items from the other sessions, suggesting that the use of

IA to solve symbolically presented subtraction problems is within the zone of proximal

development (Vygotsky, 1978) of many children from the first years of elementary

school. Second, even thoughwe included an IA evoking intervention in the SI group, this

did not seem to result in a good mastery of the IA strategy, which stresses the need for amore powerful instructional intervention than the one implemented in the SI group of

our microgenetic investigation. In the two following sections, we will discuss these

findings in more detail relying on empirical evidence coming from other studies.

The difference between bare number problems and context problems

As stated above, the word problems from the transfer test of our microgeneticinvestigation elicited substantially more IA strategies than the symbolically presented

items from the other sessions. In retrospect, this was not surprising, as there is some

evidence to suggest that the way subtraction problems are presented has a serious

impact on children’s strategy choices. We will briefly describe two older studies that

suggest that the IA strategy is indeed quite a popular and successful strategy among

children who are confronted with subtraction word problems, even when the IA

strategy had not been explicitly and systematically taught to them. These studies also

show that certain task variables, such as the nature of the situation that is described inthe word problem (i.e. whether the subtraction is embedded in a dynamic change

situation or in a static combine or compare situation) and the order of presentation of

the different sets in the problem (i.e. whether the larger or the smaller set is given first)

are important determinants of the frequency with which the IA strategy is applied.

56 Lieven Verschaffel et al.

Verschaffel and De Corte (1990) posed beginning second-graders both addition andsubtraction word problems involving numbers smaller than 20, first as a collectivepaper-and-pencil test and afterwards in the context of an individual interview. Allsubtraction problems had a change (e.g. ‘Pete had 6 apples. He gave 2 apples to Ann.How many apples does Pete have left?’) or a combine structure (e.g. ‘Pete and Ann have6 apples altogether. Ann has 2 apples. How many apples does Pete have?’), and somestarted with the minuend whereas others had the subtrahend as their first number.Moreover, the size of the difference between the minuend and subtrahend wasmanipulated too (i.e. whether this difference between the minuend and subtrahend issmall, as in 112 8 ¼ ?, or large, as in 112 3 ¼ ?).

The results can be summarized as follows. First, IA strategy use was very common(75%), even though the children had not been taught to use it. Second, a considerableimpact of two of the three task variables on children’s strategy choices was observed.There were more IA strategies (83%) for subtraction problems starting with the smallergiven number (the subtrahend) than for problems in which the larger number (theminuend) was given first (67%). (Similar findings were obtained by Brissiaud (1994),who investigated second-grade children’s performance on multi-digit subtraction wordproblems with a so-called ‘equalize’ structure such as ‘Julie has 51 pieces of candy andOdile has 47. Odile wants to have the same number of pieces as Julie. How many piecesof candy must Odile buy?’) But the researchers also observed an interaction betweenorder of presentation of the given numbers (larger or smaller number first) and theunderlying semantic structure (change or combine problem). Essentially, the impact ofthe order of presentation of the given numbers was greater for the combine problems(which do not involve any intrinsic order of events) than for the change problems(wherein there is a natural order of events). Third, although a significant effect of thetask factor ‘large versus small difference between minuend and subtrahend’ wasexpected (namely that problems with a small difference would elicit more IA strategiesthan problems with a small difference), no such effect was found, suggesting that theeffect of this number factor was too small – compared to the other two task variablesinvolved in the study – to influence children’s strategy choice.

In another study that involved a more direct comparison between children’sstrategic performance on subtraction word problems and corresponding symbolicproblems, Van den Heuvel-Panhuizen (1996) found remarkable performance differencesin a group of elementary schoolchildren between subtraction problems presented asword problems or as bare number problems. In contrast to Verschaffel and De Corte(1990), the administered word problems were not change or combine problems butproblems about comparing the number of beads in two jars, finding the right change, orcomparing the heights of two boys. In each pair of subtraction problems, the wordproblem yielded a substantially higher performance score than its correspondingsymbolic version, even though the calculations to be carried out were exactly the same.Although no concrete strategy data are reported, the author argues that most childrenwho succeeded on the word problems applied the IA strategy whereas in thecorresponding symbolic format IA strategies were very rare. According to Van denHeuvel-Panhuizen, the minus sign in the symbolic problems was too strongly connectedto taking away to evoke the use of the IA strategy.

So, the available evidence from the above-mentioned studies suggests that manyelementary schoolchildren who do not show any evidence of IA strategy use in relationto symbolically presented problems of the type a2 b ¼ ? are able to solvecorresponding word problems (at least if their task variables are favourable) insightfullyand efficiently by means of that strategy.

Subtraction by indirect addition 57

Intervention studies on IA

Although half of the children from our microgenetic investigation (Study 3) were

involved in an experimental condition that was aimed at enhancing the discovery anddevelopment of the IA strategy, this intervention could hardly be described as a

powerful environment for learning this sophisticated calculation strategy. This may

explain why the impact of that intervention on children’s use of the IA strategy was so

disappointingly low. In this respect, it is interesting to refer to a few design experiments

that also have investigated this strategy. While in some of these experimental

programmes learning to subtract through IA was the main focus (Fuson & Fuson, 1992;

Fuson & Willis, 1988; Thornton, 1984, 1990), in others it was only a side aspect of a

more comprehensive instructional enterprise (Blote, Klein, & Beishuizen, 2000; Blote,Van der Burg, & Klein, 2001; Klein, Beishuizen, & Treffers, 1998).

First, the teaching experiments of Fuson and colleagues (Fuson & Fuson, 1992;

Fuson & Willis, 1988) and Thornton (1984, 1990) revealed that explicit instruction in

and intensive practice of IA in the number domain 0–20 enabled first- and second-

graders not only to solve small-difference subtractions (such as 112 9 ¼ ?) by means of

this strategy, but also to increase their subtraction performances to a level comparable

with the typically better addition performances. Fuson’s and Thornton’s teaching

experiments also demonstrated that instruction focusing on IA allowed bothmathematically higher-achieving children and their lower-achieving peers to accurately

apply IA on single-digit subtractions. Indeed, almost all average and below-average

achieving first-graders who participated in the teaching experiment of Fuson and Willis

(1988) spontaneously and accurately solved subtractions with IA after instruction in this

strategy.

Second, several Dutch researchers, with close links to the so-called realistic

approach to mathematics education developed, implemented, and evaluated

experimental instructional programmes for mental addition and subtraction withnumbers up to 100 wherein IA played a certain role (Blote et al., 2000, 2001; Klein et al.,

1998; Menne, 2001). Although these programmes basically aimed at the development of

abbreviated and flexible mental calculation with numbers up to 100 with a focus on the

development of the sequential variant of DS and based on the empty number line2 as a

powerful and flexible representational tool, they also revealed that second-graders who

had been taught IA applied this strategy flexibly and efficiently on small-difference

subtractions. In contrast, second-graders who did not receive instruction in IA did not

use this strategy, even on subtractions with a very small difference between the integers.For instance, the intervention study of Klein et al. (1998) showed that second-graders

who had been taught IA on small-difference subtractions in the number domain 20–100,

used this strategy on more than 80% of such subtractions, whereas their peers, who had

not been taught this strategy, did not use it at all.

Finally, we refer to another study (to which we will refer from here on as Study 4)

that was recently done at our centre (Torbeyns, De Smedt, Ghesquiere, & Verschaffel,

2009b), wherein we compared the strategy performance on a paper-and-pencil test of

second- to fourth-graders from two Flemish schools in Belgium that did not provide any

2 The empty number line is a representational tool that can be used to learn how to do elementary additions and subtractions inan insightful and flexible way (Treffers & De Moor, 1990). It takes the form of an empty line, with no indication of a start orend-point, on which children can mark their intermediate solution steps when adding or subtracting numbers. As such, theempty number line fits with and enhances the use of sequential procedures.

58 Lieven Verschaffel et al.

instruction in the IA strategy (¼ DS-oriented schools), with that of comparable children

from a third school in which IA did receive special instructional attention (¼ IA-oriented

school). This difference in instructional approach was not only revealed by a textbook

analysis, but also confirmed by interviews with the teachers, wherein they were

questioned about various aspects of their instructional approach to (mental) addition

and subtraction. According to the textbook analysis and the teacher interviews, first-graders from the IA-oriented school were taught to use IA when the difference between

the two given numbers was small. Although the children from the IA-oriented school

generated somewhat more IA strategies during the test, the frequency of IA remained

extremely low: only 7.5% for the IA-oriented school and 0.1% for the DS-oriented

schools. To interpret these results properly, it is important to stress that, in contrast to

the above-mentioned design experiments wherein the experimental treatment was

realized – or at least strictly controlled – by the researchers, this study did not involve any

intervention from the side of the researchers, but was purely observational in nature.Therefore, the dramatically low frequency of IA strategies in the IA-oriented school was

probably due to the features of the instruction in IA as provided by the textbook and as

implemented by the teachers from that school. Apparently, teaching the insightful and

flexible use of IA properly is more difficult and demanding than it looks at first sight.

Conclusion and discussion

Our own studies on the occurrence and development of the IA strategy and the

international research on this topic in general have yielded a number of interesting

contrasts, which demand further research and reflection.

First, whereas young adults use IA frequently, efficiently, and adaptively to solve

symbolically presented multi-digit subtractions (Study 1), IA is almost completely absent

in the strategy repertoire of 6- to 9-year olds. Even when confronted with problems for

which the computational advantage seems overwhelming, or with an explicit invitationto demonstrate strategy variety, or with some instruction in that strategy (see Studies

2–4), the frequency of IA remains dramatically low. Apparently, there is something that

is characteristic of children and/or of their environment that prevents them from

developing and applying IA. At the same time, children who (begin to) use IA

immediately demonstrate relatively high levels of accuracy and speed, compared to the

efficiency of the DS strategy (Study 3).

Second, we point to the finding that, whereas the IA strategy apparently originates

and develops so slowly and laboriously in the vast majority of children (Studies 2–4),other shortcut strategies for doing addition and subtraction, such as disregarding

addend order when doing addition or recognizing that adding an amount to a collection

can be undone by subtracting the same amount, seem to develop much earlier and more

easily (see Baroody et al., 2009; Cowan, 2003). Thus, compared to other shortcut

strategies for doing addition and subtraction, IA seems particularly difficult to develop.

Third, whereas children seem to experience great difficulties with applying IA on

symbolically presented subtractions, they demonstrate relatively good mastery and

frequent use of the IA strategy when confronted with arithmetically isomorphic wordproblems, as shown in the second part of the transfer test of our microgenetic

investigation (Study 3), which elicited – in both experimental groups – substantially

more IA strategies than the symbolically presented items from the test, retention, and

transfer tasks. Based on the findings of an older study by Verschaffel and De Corte

Subtraction by indirect addition 59

(1990), we argue that children’s IA strategy use on subtraction word problems in that

microgenetic investigation would probably have been even larger if we had not used

word problems that always had the larger given number as their first number, but if we

would have, instead, included word problems starting with the smaller number such as

‘Pete has 58 EURO. How many more EURO does he need in order to have 92 EURO in

his money box?’.More research is needed to unravel why so many elementary schoolchildren stick so

strongly and stubbornly to the DS strategy and move so slowly and reluctantly in the

direction of IA strategy use when confronted with symbolically presented subtractions.

In our opinion, there are four possible factors that might have contributed to this

finding. A first factor relates to the role of conceptual knowledge of the mathematical

principle that underlies the meaningful use of the IA strategy, namely the addition–

subtraction–complement principle. Recent research on children’s understanding of

subtraction-related principles suggests that the complement principle develops laterthan other subtraction-related principles (such as the inversion principle) and is most

presumably not yet well-understood by children of that age (Baroody et al., 2009).

A second factor relates to the limited cognitive processing potentials of children of that

age (such as processing speed or working memory capacity, Demetriou, 2004), or to

their metacognitive capacities (Flavell, 1982), such as their limited capacity to suppress

or inhibit a dominant response – in our case their difficulty with suppressing or

inhibiting the subtraction operation when confronted with a problem that contains the

minus sign, which is strongly associated with the (direct) subtraction operation. A thirdfactor, somewhat related to the previous one, might be that the rare appearance and

slow development of IA is due to children’s more restricted specific attentional

resources directly caused by the fact that they are still struggling with the mastery of the

DS strategy, as Siegler’s (Shrager & Siegler, 1998) SCADS model of cognitive strategy

change would predict. Applied to our findings, it could be argued that the children were

still focused on improving the procedural fluency of the DS strategy and, therefore, had

no attentional resources available yet for the discovery of alternative strategies like the

IA strategy. A final factor refers to the effect of the culture and practice of children’sprior mathematics education. As explained above, all children in our experiment had

received systematic instruction and intensive practice in the DS strategy and little or no

instruction in the IA strategy (see also Van den Heuvel-Panhuizen & Treffers, 2009).

They had been immersed in a classroom practice and culture (Yackel & Cobb, 1996) that

values routine mastery of one single (taught) strategy rather than flexible use of various

(self-invented) strategies. In other words, these children had been exposed to classroom

practices and norms that were non-supportive for the appearance and use of IA and that

might even have hindered its appearance. From the above list of explanations, there isprobably no single explanation for the rarity of the IA strategy; the phenomenon is

better conceived as the result of the complex interaction of various factors instead.

We believe that our further search for a deep understanding of the origin and

development must avoid a dichotomous view, wherein a child either has the IA strategy

or not, and shift to a view that describes the child’s mastery of that strategy in terms of a

space defined by several dimensions of theoretical and practical significance. One

dimension could be the various ways of showing mastery of a strategy or principle

(evaluation, application, justification); another dimension could be the various types oftasks and contexts wherein this mastery can be demonstrated (non-symbolic – which

could be further divided into material and verbal – versus symbolic, and IA-favouring

versus IA-unfavouring; see Bisanz, Watchorn, Piatt, & Sherman, 2009).

60 Lieven Verschaffel et al.

Besides descriptive studies that investigate (the development of) students’ cognitive

skills and processes under given instructional circumstances, we also need more design

experiments that study the emergence and the further development of the IA strategy in

an innovative instructional environment. Certainly this should be a (more) powerful

environment than the so-called strong instruction used in our microgenetic

investigation (Study 3). Based on our reflection on the studies conducted, we suggestthe following basic features of this experimental environment aimed at the fluent and

flexible use of IA (De Smedt et al., 2009): (a) working at the development of integrated

knowledge of of the procedural aspects of IA (i.e. knowing how to execute the IA

strategy), its conceptual underpinnings (i.e. understanding the addition–subtraction

complement principle that explains why it is allowed to solve a subtraction problem by

means of IA), and the related conditional knowledge (i.e. knowing when the IA strategy

can be most efficiently applied and why); (b) taking maximal profit from the fact that

quite a number of children, who seem very reluctant to apply IA for solving symbolicallypresented subtractions, are able to solve mathematically similar problems presented in a

verbal and/or pictorial form by means of IA, and (c) the creation of a classroom practice

and culture that is directly aimed at the development of positive beliefs and attitudes

towards variety and flexibility in strategy use (instead of beliefs and attitudes that only

value perfect mastery of one routine procedure taught by the teacher). Based on their

own experiences and didactical insights, Van den Heuvel-Panhuizen and Treffers (2009)

recently added the following recommendations:

. widening the interpretation of subtraction (subtraction as taking away and

subtraction as determining the difference);

. embedding the teaching of IA as part of a more general view on clever strategies,

including also other strategies that are helpful for doing subtraction, such as

compensating;

. using schematic representations such as the empty number line;

. asking questions such as ‘About how much left?’ that may trigger the use of IA; and

. countermanding the opinion among teachers and parents that children get confusedby IA.

Carefully conducted design experiments, comparing the effectiveness of the above-

mentioned characteristics of instruction in IA, will deepen our insights in the acquisition

and further development of IA as an alternative solution method for solving symbolically

presented subtractions of the form a2 b ¼ ? in children at different ages and of

different mathematical achievement levels.

Acknowledgements

Bert De Smedt is a postdoctoral fellow of the Research Foundation Flanders, Belgium. This

research was supported by grant GOA 2006/01 “Developing adaptive expertise in mathematics

education” from the Research Fund K. U. Leuven, Belgium.

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