Published on

27-Dec-2016View

212Download

1

Transcript

mievx 37t 32,

Subtraction by addition

y vaaticaarchstimthisubtr

the subtrahend. These ndings challenge typical special education classroom practices, which only focus

exibilitmathemwafforwel, &ulatedheir un

mathematical learning disabilities (MLD), however, the feasibility

Baroody, 2003; Kilpatrick et al., 2001; Peltenburg, van den Heuvel-Panhuizen, & Robitzsch, 2012; Verschaffel et al., 2007). While thisdiscussion remains to be lively, more scientic evidence is needed.

Mental subtraction is one mathematical subdomain in whichstrategy variety and exibility can be stimulated. When solving

make 81 (e.g., 79 1 80, 80 1 81, so the answer is 1 1 2).ion on such prob-lation process byolution efciency,ch lead faster to aLipowsky, 2009;rast, for problemshe difference, suchdoes not lead to

fewer and/or smaller calculation steps. For these problems thedirect subtraction strategy seems to be more efcient.

* Corresponding author. Tel.: 32 16 32 62 47; fax: 32 16 32 62 74.

1 In the present study, we categorise the variety of subtraction strategies basedon the main operation that is used, i.e., either subtraction or addition. Differentcategorisations are used by other researchers (e.g., Beishuizen, 1993; Blte et al.,2001; Buys, 2001; Peltenburg et al., 2012), such as focussing on the manipulationof the numbers during problem solving, which leads to a classication into jump,split and varying strategies.

Contents lists available at ScienceDirect

Learning and Instruction

evier .com/locate/ learninstruc

Learning and Instruction 30 (2014) 1e8E-mail address: greet.peters@ppw.kuleuven.be (G. Peters).and suitability of strategy variety and exibility remains an issue ofcontinued debate in many countries. Some researchers, curriculumdevelopers, and policy makers argue that it is better for thesechildren to develop mastery and condence in only one way orstrategy to solve problems (e.g., Geary, 2003; Milo & Ruijssenaars,2003; National Mathematics Advisory Panel, 2008). Others claimthat the development of strategy variety and exibility should beeducational goals for all students, including those with MLD (e.g.,

The use of the complementary addition operatlems can thus considerably facilitate the calcureducing computational effort and increasing si.e., fewer and/or smaller calculation steps, whicorrect answer (e.g., Heinze, Marschick, &Verschaffel, Bryant, & Torbeyns, 2012). In contwith a relatively small subtrahend compared to tas 81 2, the subtraction by addition strategyrelations and/or the properties of operations. For children with efciently by determining how much needs to be added to 79 to1. Introduction

In the last decades, variety and use have become major aims ofFreudenthal, 1991; Kilpatrick, SVerschaffel, Torbeyns, De Smedt, Luachieve these goals children are stimuse a variety of strategies based on t0959-4752/$ e see front matter 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.learninstruc.2013.11.001on the routine mastery of the direct subtraction strategy. 2013 Elsevier Ltd. All rights reserved.

y in childrens strategyatics education (e.g.,

d, & Findell, 2001;Van Dooren, 2007). Toto discover and exiblyderstanding of number

subtractions such as 81 43, the most commonly taught solutionstrategy1 is the direct subtraction strategy, in which the smallernumber (43) is subtracted from the larger number (81) (e.g.,81 43 (81 40) 3 41 1 2 38). However, for problemswith a relatively large subtrahend compared to the difference, suchas 81 79, subtraction by addition appears to be a more cleverstrategy (e.g., Torbeyns, De Smedt, Stassens, Ghesquire, &Verschaffel, 2009). With this strategy, one can solve 81 79 veryStrategy useStrategy choiceMathematical learning disabilitiesmethods to infer strategy use patterns, and found that both groups of children switch between thetraditionally taught direct subtraction strategy and subtraction by addition, based on the relative size ofSubtraction by addition in children withdisabilities

Greet Peters a,*, Bert De Smedt b, Joke Torbeyns a, LaCentre for Instructional Psychology and Technology, KU Leuven, Dekenstraat 2, Postbob Parenting and Special Education Research Unit, KU Leuven, Leopold Vanderkelenstraa

a r t i c l e i n f o

Article history:Received 26 March 2013Received in revised form30 October 2013Accepted 3 November 2013

Keywords:

a b s t r a c t

In the last decades, strategFor children with mathemshould be set. Some resestrategy, others advocatechildren. To contribute tomentally solve two-digit s

journal homepage: www.elsAll rights reserved.athematical learning

en Verschaffel a, Pol Ghesquire b

73, 3000 Leuven, BelgiumPostbox 3765, 3000 Leuven, Belgium

riability and exibility have become major aims in mathematics education.l learning disabilities (MLD) it is unclear whether the same goals can anders and policy makers advise to teach MLD children only one solutionulating the exible use of various strategies, as for typically developingdebate, we compared the use of the subtraction by addition strategy toactions in children with and without MLD. We used non-verbal research

ndPrevious work on childrens and adults use of subtraction byaddition in elementary subtraction indicated that children hardlyuse the subtraction by addition strategy spontaneously, not even onproblems such as 81 79 (e.g., Blte, Van der Burg, & Klein, 2001;De Smedt, Torbeyns, Stassens, Ghesquire, & Verschaffel, 2010;Heinze et al., 2009; Selter, Prediger, Nhrenbrger, & Hussmann,2012; Torbeyns, De Smedt, Ghesquire, et al., 2009). Adults, onthe other hand, seem to solve symbolically presented subtractionsefciently and exibly by means of subtraction by addition(Torbeyns, Ghesquire, & Verschaffel, 2009). These available studiesrelied on verbal protocol data to infer strategy use. A closer in-spection of the speed data in the study by De Smedt et al. (2010)suggested that children sometimes used subtraction by additioneven though they reported a direct subtraction strategy. If thesechildren only used direct subtraction, an increase in reaction timesshould have been observed from items with relatively small sub-trahends (81 7) over items with medium-sized subtrahends(8143) to items with relatively large subtrahends (8179), sincesubtracting a larger subtrahend requires more and/or largercalculation steps (Peters, De Smedt, Torbeyns, Ghesquire, &Verschaffel, 2010). This reaction time pattern was not found in DeSmedt et al. (2010): Problems with a relatively large subtrahendwere solved signicantly faster than problems with a medium-sized subtrahend, which suggests that the actual use of the sub-traction by addition strategy might be larger than revealed by thechildrens verbal protocols. In a recent study, Peters, De Smedt,Torbeyns, Ghesquire, and Verschaffel (2013) therefore used twonon-verbal methods to infer the use of the subtraction by additionstrategy in typically developing children solving symbolically pre-sented subtractions: regression analyses and a format manipula-tion. They concluded that children, like adults, switched betweendirect subtraction and subtraction by addition to solve two-digitsubtraction problems, based on the relative size of the subtra-hend: The children used direct subtraction when the subtrahendwas relatively small compared to the difference (as in 83 4), andsubtraction by addition when the subtrahend was relatively large(as in 83 79).

So far, the use of the subtraction by addition strategy has notbeen explored in children with MLD, except for the study byPeltenburg et al. (2012). They showed that Dutch special educationchildren (aged 8e12, with a mathematics level similar to the end ofGrade 2) do report the use of this strategy, and this mostly onproblems with a relatively large subtrahend and crossing the tens(e.g., 61 59); the subtraction by addition strategy was reported inmore than 50% of problems of this type. These authors also foundthat the subtraction by addition strategy was reported more oftenon word problems compared to symbolically presented sub-tractions (i.e., 70% on adding-onword problems and 25% on taking-away word problems versus only 8% on subtractions presented inthe M S . form). While the design of Peltenburg et al.s studyincluded various potentially interesting numerical task features(such as size of the subtrahend, crossing the tens and closeness ofminuend/subtrahend to a ten), they did not deepen the interactionbetween type of problem (i.e., word problems vs. symbolicallypresented problems) and these number characteristics (forexample, large vs. medium subtrahend e they did not includeproblems with relatively small subtrahends). In this regard, it isimportant to point out that Peters et al. (2013) observed thattypically developing children, when confronted with symbolicallypresented two-digit subtraction problems, switch between thedirect subtraction strategy and the subtraction by addition strategydepending on number characteristics: Direct subtraction was usedwhen the subtrahend was relatively small (as in 83 4), subtrac-tion by addition when the subtrahend was relatively large (as in

G. Peters et al. / Learning a283 79). Against this background, we extended the work byPeltenburg et al. (2012) in children with MLD, by investigating therole of the numbers in symbolically presented subtraction prob-lems, also including problems with relatively small subtrahends.Furthermore, we veried whether children with MLD show similarpatterns of exible strategy use as their typically developing peers.

Moreover, it might be that the number of verbal reports ofsubtraction by addition on the symbolically presented problems inthe study of Peltenburg et al. (2012) was an underestimation. Asargued by Peters et al. (2013), children may hide the use of thesubtraction by addition strategy because they think it is not valuedor allowed to use other strategies to solve symbolically presentedproblems than the one(s) taught in the mathematics lessons (e.g.,Yackel & Cobb, 1996). Furthermore, the subtraction by additionstrategy can be executed very fast and quasi-automatic, and ittherefore might be that children have difculties in explaining howthey found their answer: They may not have been aware of, orconfused by, the steps they performed while calculating andtherefore reported a strategy they knew how to explain (e.g.,Cooney & Ladd, 1992; Kirk & Ashcraft, 2001). These problems mightbe particularly prominent in children with MLD (see Milo, 2003;Thevenot, Castel, Fanget, & Fayol, 2010). We therefore used twonon-verbal methods to answer our research questions: regressionanalyses in which reaction times were predicted based on differenttask characteristics, and a method in which speed was contrastedbetween problems presented in different presentation formats.

2. The present study

Extending the study by Peltenburg et al. (2012), we investigatedwhether children with MLD switch between direct subtraction andsubtraction by addition based on number characteristics whensolving only symbolically presented two-digit subtraction prob-lems, and compared their strategy use patterns with those oftypically developing peers. Since verbal self-reports might be lesssuited to identify the subtraction by addition strategy, especially inchildren with MLD, two non-verbal methods were used.

First, we used the reaction times for problems presented in thestandard subtraction format to calculate three linear regressionmodels (see Peters et al., 2013; Woods, Resnick, & Groen, 1975).Thesemodels represented three different strategy use patterns. Therst model, the DS-Model, represents the consistent use of thedirect subtraction strategy. When children consistently use thisstrategy, the reaction times should be best predicted by the size ofthe subtrahend (S), because it takes longer to subtract a largernumber from the minuend (e.g., 83 e 79 .) than to subtract asmaller number (e.g., 83 4 .). The secondmodel, the SBA-Model,starts from the same idea but represents the consistent use of thesubtraction by addition strategy: If children only use subtraction byaddition, reaction times should be best predicted by the size of thedifference (D), because it takes more time to determine how muchneeds to be added to get at a given number when the differencebetween both numbers is large (Howmuch needs to be added to 4to have 83?) than when it is small (How much needs to be addedto 79 to have 83?). The third model, the Switch-Model, representsswitching between both strategies based on the relative magnitudeof the subtrahend (S< D vs. S> D), and reaction times in this modelare best predicted by the minimum of subtrahend and difference(min[D, S]): For problems with the subtrahend smaller than thedifference (e.g., 83 4 . and 84 38 .), problems can be moreeasily solved by means of the direct subtraction strategy, andtherefore reaction times for these problems are expected to in-crease with the size of the subtrahend. In contrast, problems withthe subtrahend larger than the difference (e.g., 83 79 . and84 46 .) can be more easily solved by means of the subtraction

Instruction 30 (2014) 1e8by addition strategy, and therefore reaction times for these

and Iproblems are expected to increase with the size of the difference.Peters et al. (2013) showed that for typically developing childrenthe Switch-Model provided the best t to their reaction times. Forthe present study, we wondered whether children with MLD showthe same strategy use pattern as their typically developing peers:Do both children with and without MLD switch between directsubtraction and subtraction by addition based on the magnitude ofthe subtrahend when solving two-digit subtraction problems(Research Question 1)?

Second, we expected that, besides the magnitude of the sub-trahend, the numerical distance between subtrahend and differ-ence would have an inuence on strategy selection as well. Forproblems with a large numerical distance between S and D (such as83 4 or 83 e 79), the computational gain in using one strategycompared to the other is very clear. However, for problems with asmall numerical distance between S and D (such as 84 38 or84 46) there is no clear computational advantage for one of thetwo strategies. This interaction was found in the study of Peterset al. (2013) with typically developing children solving symboli-cally presented problems. For the present study, we thus wonderedwhether we could replicate these ndings in children with MLD(Research Question 2). To answer this question, we divided thesubtractions into four problem types, based on the combination ofthe magnitude of S (S < D vs. S > D) and the numerical distancebetween S and D (small vs. large) (see Peters et al., 2013). We thencompared reaction times for these four problem types when pre-sented in two different presentation formats: the traditional sub-traction format (M S .) and the (unusual) addition format(S . M) (see also Campbell, 2008; Peters et al., 2010). Based onCampbell (2008), we expected speed differences between the twopresentation formats because the subtraction by addition strategycan be performedmore easily when the problem is presented in theaddition format, while a time-consuming mental re-representationis needed when the same problem is presented in the subtractionformat (and vice versa for the direct subtraction strategy on prob-lems in the addition format). So, if children with and without MLDare switching between direct subtraction and subtraction byaddition depending on the combination of the magnitude of S andthe numerical distance between S and D, then we should nd aninteraction between the magnitude of S, the presentation format,and the numerical distance between S and D. For the large-distanceproblems (such as 83 4 or 83 79), we expected children toselect direct subtraction when S < D, and subtraction by additionwhen S> D. This means that large-distance S< D problems (such as83 4 .) will lead to faster response times in the subtractionformat compared to the addition format (4 . 83) and, similarly,large-distance S > D problems (such as 83 79 .) will be solvedfaster in the addition format (79 . 83) compared to the sameproblems in subtraction format. For the small-distance problems(such as 84 38 or 84 46), we expected no signicantmagnitude format interaction, because such a small distancedoes not yield a clear computational advantage for either directsubtraction or subtraction by addition.

3. Method

3.1. Participants

In this study, we included both children with and without MLD.In the MLD group, participants were 44 children with MLD (31males; mean age 12 years and 5 months [SD 6 months]) from thenal year of special education for children with specic learningdisorders, coming from eight different Flemish special schools ofType 8 (Belgium). In order to get enrolled in such a special school

G. Peters et al. / Learningfor specic learning disorders, children have to take a series ofmandatory standardised tests administered by a multi-disciplinaryteam including experienced psychologists. These tests have toshow a signicant impairment in mathematics and/or reading (i.e.,below percentile 10) and normal general intellectual ability(IQ > 85). We excluded children with MLD from participation ifthey were additionally diagnosed with dyslexia (i.e., readingperformance < 10th percentile).

We included a control group of 23 typically developing childrenfrom the nal year of two mainstream elementary schools (13males; mean age 11 years and 9 months [SD 3 months]). None ofthem was diagnosed with a learning disorder, and they all per-formed better than the 35th percentile on a standardisedcurriculum-based math test (Deloof, 2005). This control group wassignicantly younger than the MLD group, t(65) 6.67, p < .01,but we preferred using a group of children matched on curriculumtargets instead of age in order to avoid including children who hadtaken algebra (which starts in the rst year of Flemish mainstreamsecondary education), which might enhance knowledge about theadditionesubtraction complement principle and from thatknowledge about the relation between the direct subtraction andthe subtraction by addition strategy. (However, we have repeatedall analyses with age included, but this did not alter our ndings.)

All 67 participants completed a short paper-and-pencil pre-test(see Peters et al., 2013), to check whether they were able to solvetwo-digit subtraction problems presented in the (unusual) S .Mformat.

3.2. Materials and procedure

Materials and procedure were exactly the same as in Peters et al.(2013). All 67 participants were asked to mentally solve 64 sub-traction problems, which were presented horizontally in the mid-dle of a computer screen. All problems had a two-digit minuendlarger than 30 and required borrowing. Half of the problems werepresented in the traditional subtraction format (e.g., 83 4 .), theother half were composed by transforming these problems intotheir corresponding addition format (i.e., 4 . 83). The two for-mats were presented in a mixed order.

All 64 problems could be categorised into four problem types,based on the combination of themagnitude of S (SD) andthe numerical distance between S and D (small or large). For thesmall-distance problems, S and D differed by less than 10, while forthe large-distance problems S and D were differing by at least 10and either S or D was a one-digit number. This resulted in thefollowing categorisation: (a) large-distance S < D problems, withsubtrahends smaller than 10 (e.g., 83 4 . and 8 . 34); (b)large-distance S> D problems, with differences smaller than 10 (e.g.,77 68 . and 37 . 42); (c) small-distance S < D problems (e.g.,92 44 . and 36 . 75); and (d) small-distance S > D problems(e.g., 32 17 . and 29 . 53).

4. Results

Results are presented in two parts. The rst part involves adescriptive overview of the accuracy data, whereas in the secondpart we focus on the reaction times analyses. All analyses werecarried out by means of SAS Version 9.3.

4.1. Accuracy

All 67 children solved 64 problems in the computer task. Forthese analyses, we excluded 42 trials due to incorrect task admin-istration (1%), resulting in a total of 4246 trials.

As could be expected, children with MLD performed signi-

nstruction 30 (2014) 1e8 3cantly worse than the control children, F(1, 65) 26.35, p < .0001,

d 1.34: The mean score was 71% correct on all problems for theMLD children, ranging from 24% to 100%, whereas the control grouphad an average score of 93% correct, ranging from 67% to 100%. Thedescriptive data per problem type shown in Table 1 indicate that inboth groups large-distance problems were solved better thansmall-distance problems, and that problems in subtraction formatwere solved better than problems in addition format for all problemtypes except large-distance S > D problems (such as 83 79).

Most of the 926 incorrect answers represent common errors ofchildren mentally solving subtractions (e.g., Beishuizen, 1993): Er-

of the small-distance problems, although these latter problemswere solved 617 ms faster in subtraction format than in the

Table 2Model specications of the linear regression models for problems presented insubtraction format.

Group Model Model specications R2 Effect estimates

DF F-value p-Value Intercept Parameter

MLD DS-Model 1, 30 12.82 .0012 0.2995SBA-Model 1, 30 3.21 .0834 0.0966Switch-Model 1, 30 26.37 D

Large-distance

Small-distance

Small-distance

Large-distance

(83 4 .) (84 38 .) (84 46 .) (83 79 .)MLD Subtraction 93% (13%) 66% (33%) 65% (34%) 70% (31%)

Addition 81% (24%) 53% (35%) 52% (34%) 86% (26%)Control Subtraction 98% (4%) 91% (14%) 90% (15%) 94% (11%)Addition 96% (11%) 86% (17%) 88% (18%) 97% (5%)It might be that aggregating speed over the various participantshas covered different strategy use patterns at the individual level.Therefore, we additionally predicted the reaction times of eachparticipant individually, using the same three regression models.The frequencies of the best tting model per individual in the MLDgroup were: 14 for whom none of the models tted signicantly, 10for theDS-Model, 0 for the SBA-Model, and 20 for the Switch-model.For the control group, these frequencies were: 3 for whom none ofthe models tted signicantly, 1 for the DS-Model, 0 for the SBA-Model, and 19 for the Switch-model. The model representing aswitch between direct subtraction and subtraction by additionbased on the magnitude of the subtrahend thus provided mostfrequently the best t to the reaction times in each of both groups.

4.2.2. Comparison of reaction times between the two presentationformats

To answer Research Question 2, we included the presentationformat and numerical distance between S and D into the analyses.Themean reaction times per problem type and presentation formatare depicted in Table 3. We performed a 2 2 2 2 repeatedmeasures ANOVA on the reaction time data with magnitude of S(S < D vs. S > D), numerical distance between S and D (small vs.large) and presentation format (subtraction vs. addition) as within-subject factors, and group (MLD vs. control) as a between-subjectsfactor. TukeyeKramer adjustments were used for post-hoccomparisons.

There was no main effect of format, F(1, 65) 1.28, p .26, ormagnitude, F(1, 65) 0.15, p .70, but there was a signicant effectof numerical distance, F(1, 65) 167.35, p < .0001, d 0.87: Large-distance problems, such as 83 4 . or 79 . 83, were solvedfaster (6173 ms) than small-distance problems, such as 84 38 .or 46 . 84 (9653 ms). There was no signicantmagnitude distance interaction, F(1, 260) 2.04, p .15, but theformat distance interaction was signicant, F(1, 260) 3.85,p .05: Post-hoc tests showed that reaction times for large-distance problems did not differ signicantly between subtractionand addition format (6216 ms in subtraction format and 6131 ms inaddition format), t(260) 0.29, p .99, nor did the reaction timesAddition 4678 (1832) 6957 (2880) 6928 (2276) 5741 (1569)

addition format, t(260) 2.09, p .16. Comparing within the twoformats, large-distance problems were always solved signicantly

the difference (answering Research Question 1).Secondly, we tested whether the numerical distance between

subtrahend and difference also had an inuence on strategy choice(Research Question 2). We therefore compared reaction times onproblems presented in the standard subtraction format with itscorresponding addition format (based on Campbell, 2008; Peterset al., 2010, 2013), and predicted to nd signicant differencesbetween these two formats for S < D and S > D large-distanceproblems, but not for the two types of small-distance problems.Our prediction was conrmed: When the numerical distance be-tween the subtrahend and the difference was large, S< D problems(such as 83 4) were solved faster in the subtraction than in theaddition format, while S > D problems (such as 83 e 79) weresolved faster in the addition than in the subtraction format. Whenthe numerical distancewas small, therewere no format effects. Thisthree-way interaction suggests that children switched between thetwo strategies when solving two-digit subtractions, but only whenthe numerical distance between subtrahend and difference is large.The four-way interaction with group showed that the above-mentioned format effects were more pronounced in the MLD

G. Peters et al. / Learning and Ifaster than small-distance problems (subtraction format:t(260) 9.68, p < .0001 and d 0.80; addition format:t(260) 11.86, p < .0001 and d 0.88). The interaction betweenmagnitude and presentation format was signicant as well, F(1,260) 39.26, p < .0001: S > D problems were solved faster in theaddition format compared to the subtraction format (t(260) 2.90,p .02, d 0.21), and S < D problems were solved faster in thesubtraction format (t(260) 4.70, p < .0001, d 0.40).

Most importantly, the magnitude format distance interac-tion was signicant, F(1, 260) 35.58, p < .0001 (see Fig. 1). Thepost-hoc tests revealed that the magnitude format interactionwas only signicant for the large-distance problems, and not for thesmall-distance problems. The large-distance S > D problems in theaddition format (e.g., 79 . 83) were solved faster than the sameproblems in the subtraction format (e.g., 83 79 .), t(260) 5.85,p < .0001, d 0.65, whereas the large-distance S < D problemswere solved faster in the subtraction (e.g., 83 4 .) than in theaddition format (e.g., 4 . 83), t(260)5.41, p< .0001, d 0.67.This result shows that the numerical distance between subtrahendand difference affected strategy selection: Children only switchbetween direct subtraction and subtraction by addition when thenumerical distance between subtrahend and difference is large.

There was also a signicant main effect of group,2 F(1,65) 28.11, p< .0001, d 1.39: ChildrenwithMLD (M 10,270ms)were signicantly slower than the control group (M 5557 ms).The group distance interaction was signicant as well, F(1,260) 4.81, p .03, showing that the speed difference betweenlarge- and small-distance problems was larger in children withMLD (i.e., 4070 ms) than in the control children (i.e., 2889 ms). Thegroup format interaction was marginally signicant, F(1,260) 3.52, p .06. Post-hoc tests revealed that there was nospeed difference between the two formats for the control children(t(260) 0.46, p .97), but children with MLD performed signi-cantly slower in the addition format than in the subtraction format(t(260) 2.57, p .05, d 0.17). Furthermore, thegroup magnitude format interaction was signicant, F(1,260) 10.28, p .0015. Post-hoc tests showed that for the controlchildren the above-mentioned magnitude format interaction didnot reach signicance. For the MLD children, S < D problems weresolved signicantly faster in the subtraction format(t(260)6.94, p< .0001, d 0.55), whereas S> D problems weresolved faster in the addition format, but this latter effect did notreach signicance (t(260) 2.86, p .09).

Finally, the four-way interaction between grade, magnitude,format and distance was signicant as well, F(1, 260) 13.68,p .0003. A visual inspection showed similar reaction time pat-terns as in the above-mentioned magnitude format distanceinteraction for both groups (see Fig. 2), but post-hoc tests revealedthat the above-mentioned effects were only signicant in the MLDgroup.

5. Discussion

Strategy variety and exibility have become important goals inmathematics education in the last 20 years. There is, however, stilldiscussion whether these goals should be set also for children withMLD (e.g., Geary, 1993; Kilpatrick et al., 2001). Some scholars claimthat it is better to teach these children only one way of solvingcertain types of problems. Others disagree, and state that also for

2 As mentioned earlier, we have repeated all analyses with age included, but thisdid not alter the ndings.childrenwith MLD one should aim at the discovery and exible useof a variety of strategies based on understanding of number re-lations and/or the properties of operations. In this study, weinvestigated the use of the subtraction by addition strategy in thedomain of mental multi-digit subtraction in children with andwithout MLD, a strategy that is rarely explicitly taught, especially inspecial education. Most often, children only learn the direct sub-traction strategy, in which the subtrahend is taken away from theminuend in several smaller steps. However, for problems with arelatively large subtrahend compared to the difference, such as83 79, using the addition operation might be more efcient todetermine the difference (Torbeyns, De Smedt, Stassens,Ghesquire, & Verschaffel, 2009).

As in Peters et al. (2010, 2013), we used non-verbal researchmethods to infer strategy use. We rst tted the mean reactiontimes of problems presented in the traditional subtraction formatto three regression models, representing three different strategyuse patterns (based on Woods et al., 1975). The minimum of sub-trahend and difference showed to be the best predictor for bothchildren with and without MLD, which suggests that both groupsswitched between the direct subtraction and the subtraction byaddition strategy based on themagnitude of the subtrahend: Directsubtraction when the subtrahend was smaller than the difference,and subtraction by addition when the subtrahend was larger than

Fig. 1. Graph showing the three-way interaction between magnitude, numerical dis-tance, and format. **p < .01.

nstruction 30 (2014) 1e8 5group compared to the control group, yet this difference might be

dis

ndexplained by the large overall reaction time difference betweenthese two groups. Overall, our results thus replicate the ndings byPeltenburg et al. (2012) that children with MLD do use the sub-traction by addition strategy, suggest that they can do more thanoften is expected from them (see also Peltenburg, 2012 for othermathematical subdomains), and that their mathematics instructiondoes not has to be restricted to focussing on the routine mastery ofthe direct subtraction strategy.

These results are thus an important addition to the discussionabout the feasibility and suitability of teaching strategy variety andexibility in childrenwithMLD, in the sense that they show that wemight have to re-consider the typical mathematics instructionabout mental calculation strategies for children with MLD. Futurestudies should try to examine different instructional settings inwhich the use of the subtraction by addition strategy can bestimulated even further in both children with and without MLD. Inthis respect, we have high hopes for a setting in which the currentfocus on the direct subtraction strategy is supplemented by theteaching of subtraction by addition as an alternative strategy (forsimilar suggestions in other mathematical domains, see Baker,Gersten, & Lee, 2002; Kroesbergen, 2002; Swanson & Hoskyn,1998), and by explicit attention to its conceptual underpinnings(i.e., the inverse relation between addition and subtraction) andwhen to apply this particular strategy, followed by plenty of op-portunities to practice and exibly apply both strategies.

We will now focus on the limitations of the present study andsuggestions for further research. A rst limitation deals with theway in which we selected the large-distance problems. As in thework of Peters et al. in adults (2010) and typically achieving chil-dren (2013), we only chose problems that involved a single-digitnumber either as subtrahend (as in 83 4) or as difference (as in83 79). Based on a rational task analysis, we assumed that ifchildren with MLD did not use subtraction by addition on extremeproblems such as 83 79 4, they certainly would not use it forproblems with a less extreme difference between subtrahend and

Fig. 2. Graphs representing the four-way interaction between magnitude, numerical

G. Peters et al. / Learning a6difference, such as 84 68 16. However, this specic problem setlimits the generalisability of our ndings to subtraction in thenumber domain 20e100 with extremely large subtrahends orextremely small differences. It remains to be determined whethersimilar exible strategy choices between direct subtraction andsubtraction by addition will occur when children with MLD, butalso adults and typically achieving children, are confronted withlarge-distance problems including only two-digit subtrahends anddifferences, such as 84 16 and 84 68.

Secondly, the two non-verbal methods we applied to investigatechildrens strategic behaviour did not allow us to identify strategiesat an item level. However, reliable data on that item level arenecessary, both for scientic and practical reasons. Verbal reportsmight seem very useful to solve this issue, but e as stated in theintroduction e using only verbal reports for investigating strategyuse, and particularly strategies such as subtraction by addition, isquestionable. Confronting verbal strategy reports with a combi-nation of several non-verbal research methods e such as reactiontime data (as we showed in the present study), and eye-movementse might help in further understanding what really happens whenpeople solve a problem.

Thirdly, the individual regression models showed that for 25% ofthe children no individual regression model could be tted. Follow-up analyses showed that these children performed signicantlyworse compared to the others in terms of overall accuracy (i.e., 62%correct, compared to 76% for the children who tted best with theDS-Model and 86% for the Switch-Model; F(2, 64) 11.79, p < .01and both post-hoc tests had p < .05). Moreover, when comparingthe types of errors that weremade in the group for whomnomodelcould be tted with those of the other children, we found that theirerrors were predominantly examples of the well-known erroneoussmaller-from-larger solution strategy (e.g., answering 52 48 16because of splitting both minuend and subtrahend into tens andones and then subtracting for both tens and ones the smallernumber from the larger one; in 39% of all errors of the children forwhom no model could be tted, compared to only 18% of all errorsof children for whom either the DS-Model or the Switch-Modeltted best). The use of this erroneous solution strategy resulted in alack of variability in reaction times between the three problemtypes involving two-digit subtrahends (i.e., small-distance S < D[e.g., 32 15], small-distance S > D [e.g., 32 17], and large-distance S > D problems [e.g., 31 28]), what can explain whynone of the regression models provided a signicant t.

Fourthly, the design of our study does not allow us to drawconclusions about the differences and/or similarities in the devel-opment of the subtraction by addition strategy between typicallydeveloping children and children with MLD. In this respect, futureresearch with a chronological-age/ability-level-match design canbe very interesting (e.g., Brankaer, Ghesquire, & De Smedt, 2011;Torbeyns, Verschaffel, & Ghesquire, 2004). With such a design, itwould be possible to investigate whether children with MLD are

tance, and group. Control children on the left; MLD children on the right. **p < .01.

Instruction 30 (2014) 1e8(only) slower in developing the subtraction by addition strategy(showing a delay; i.e., when children with MLD only differ fromtheir chronological agematched peers, but not fromyounger abilitymatched peers), or whether children with MLD demonstrate adifferent developmental trajectory in their strategy development(i.e., when children with MLD differ from both their chronologicalage matched peers and younger ability matched peers).

Fifthly, the MLD children in the present study were onlyimpaired in mathematics according to the Flemish special educa-tion criteria, they did not meet the criteria for specic readingdifculties. Still, their reading ability might have been signicantlylower than the control group, a possibility that we could not verifysince we were not able to collect standardised reading scores. Thispotential difference in reading ability between the two groups canbe seen as a limitation to the present study. However, the between-group analysis showed that the same strategy use patterns were

and Ifound in both groups. So, even if the MLD group had lower readingability, they still show the same pattern of strategy use as thecontrol group. This suggests that reading ability has little impact onstrategy use, but further empirical validation is needed. Therefore,it could be interesting for future studies to distinguish betweenchildren who have only difculties in mathematics, and childrenwho have both mathematics and reading difculties.

Finally, we did not include any cognitive factors that mightexplain individual differences in strategy use in general, and in thedevelopment of the subtraction by addition strategy in particular.For example, magnitude comparison skills might play an importantrole, because the exible use of the subtraction by addition strategyrequires a comparison process of the numbers in the problem.Arguably, such a process relies on a fast and (quasi-)automaticestimation process rather than a precise and deliberate calculation.Against the background of large individual differences in magni-tude comparison skill between children of different levels ofmathematical ability (e.g., De Smedt, Verschaffel, & Ghesquire,2009; Gilmore, McCarthy, & Spelke, 2007; Holloway & Ansari,2009; Vanbinst, Ghesquire, & De Smedt, 2012), the individualdifferences in the regression models for the problems presented insubtraction format (i.e., 20 MLD children tting best to the Switch-model and 10 to the DS-Model) might be explained by such indi-vidual differences in magnitude comparison skill, but moreresearch is needed to investigate this issue.

Related to the above, individual differences in inhibition andshifting skills might also explain the individual differences instrategy use. These skills are needed to switch between directsubtraction and subtraction by addition on problems presented inthe subtraction format: Children have to be able to inhibit thetaking-away interpretation of subtraction, which is represented bythe minus sign (e.g., Van den Heuvel-Panhuizen & Treffers, 2009).Additionally, they should see the advantage of shifting from directsubtraction to subtraction by addition (or vice versa) based on thenumber characteristics in the problem. Since previous research hasobserved individual differences in inhibition and shifting in chil-dren of varying mathematical ability (e.g., Bull & Scerif, 2001; vander Sluis, de Jong, & van der Leij, 2004), such differences mightalso play a role in individual differences in strategy use. Furtherresearch is needed to shed further light on this issue.

6. Conclusion

The nding that children with MLD switch between directsubtraction and subtraction by addition based on the relative size ofthe subtrahend, similar to typically developing peers, is of greatrelevance for the theory and practice of special mathematics edu-cation. It challenges the typical current classroom practice in spe-cial education, which only focuses on the routine mastery of thedirect subtraction strategy and of mental calculation strategies ingeneral.

Acknowledgements

The authors would like to thank Gwen Leconte for her assistanceduring data collection. Greet Peters is a Research Assistant ofthe Research Foundation Flanders (Belgium). This research waspartially supported by Grant GOA 2012/10 Number sense: Analysisand improvement from the Research Fund KU Leuven, Belgium.

References

Baker, S., Gersten, R., & Lee, D.-S. (2002). A synthesis of empirical research on

G. Peters et al. / Learningteaching mathematics to low-achieving students. The Elementary School Journal,103, 51e73. http://dx.doi.org/10.1086/499715.Baroody, A. J. (2003). The development of adaptive expertise and exibility: theintegration of conceptual and procedural knowledge. In A. J. Baroody, &A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructingadaptive expertise (pp. 1e33). Mahwah, New Jersey: Lawrence ErlbaumAssociates.

Beishuizen, M. (1993). Mental strategies and materials or models for addition andsubtraction up to 100 in Dutch second grades. Journal for Research in Mathe-matics Education, 24, 294e323. http://dx.doi.org/10.2307/749464.

Blte, A.W., Van der Burg, E., & Klein, A. S. (2001). Students exibility in solving two-digit addition and subtraction problems: instruction effects. Journal of Educa-tional Psychology, 93, 627e638. http://dx.doi.org/10.1037//0022-0663.93.3.627.

Brankaer, C., Ghesquire, P., & De Smedt, B. (2011). Numerical magnitude processingin children with mild intellectual disabilities. Research in Developmental Dis-abilities, 32, 2853e2859. http://dx.doi.org/10.1016/j.ridd.2011.05.020.

Bull, R., & Scerif, G. (2001). Executive functioning as a predictor of childrens math-ematics ability: inhibition, switching, and working memory. DevelopmentalNeuropsychology, 19, 273e293. http://dx.doi.org/10.1207/S15326942DN1903_3.

Buys, K. (2001). Hoofdrekenen [Mental arithmetic]. In M. van den Heuvel-Pan-huizen, K. Buys, & A. Treffers (Eds.), Kinderen leren rekenen. Tussendoelen annexleerlijnen. Hele getallen. Bovenbouw basisschool (pp. 37e64). Groningen: Wol-ters-Noordhoff.

Campbell, J. I. D. (2008). Subtraction by addition. Memory & Cognition, 36, 1094e1102. http://dx.doi.org/10.3758/MC.36.6.1094.

Cooney, J. B., & Ladd, S. F. (1992). The inuence of verbal protocol methods onchildrens mental computation. Learning and Individual Differences, 4(3),237e257.

Deloof, G. (2005). Leerlingvolgsysteem. Wiskunde: Toetsen 6. Basisboek [System forfollowing pupils. Arithmetic: Tests 6. Textbook]. Leuven, Belgium: Garant.

De Smedt, B., Torbeyns, J., Stassens, N., Ghesquire, P., & Verschaffel, L. (2010).Frequency, efciency and exibility of indirect addition in two learning envi-ronments. Learning and Instruction, 20, 205e215. http://dx.doi.org/10.1016/j.learninstruc.2009.02.

De Smedt, B., Verschaffel, L., & Ghesquire, P. (2009). The predictive value of nu-merical magnitude comparison for individual differences in mathematicsachievement. Journal of Experimental Child Psychology, 103, 469e479. http://dx.doi.org/10.1016/j.jecp.2009.01.010.

Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Reidel.Geary, D. C. (1993). Mathematical disabilities: cognitive, neuropsychological, and

genetic components. Psychological Bulletin, 114, 345e362.Geary, D. C. (2003). Arithmetical development: Commentary on Chapters 9 through

15 and future directions. In A. J. Baroody, & A. Dowker (Eds.), The development ofarithmetic concepts and skills: Constructing adaptive expertise (pp. 453e464).Mahwah, N.J.: Lawrence Erlbaum Associates.

Gilmore, C. K., McCarthy, S. E., & Spelke, E. S. (2007). Symbolic arithmetic knowledgewithout instruction. Nature, 447, 589e591. http://dx.doi.org/10.1038/nature05850.

Heinze, A., Marschick, F., & Lipowsky, F. (2009). Addition and subtraction of three-digit numbers: adaptive strategy use and the inuence of instruction in Germanthird grade. ZDM Mathematics Education, 41, 591e604. http://dx.doi.org/10.1007/s11858-009-0205-5.

Holloway, I. D., & Ansari, D. (2009). Mapping numerical magnitudes onto symbols:the numerical distance effect and individual differences in childrens mathe-matics achievement. Journal of Experimental Child Psychology, 103, 17e29. http://dx.doi.org/10.1016/j.jecp.2008.04.001.

Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up. Helping children learnmathematics. Washington, DC: National Academy Press.

Kirk, E. P., & Ashcraft, M. H. (2001). Telling stories: the perils and promise of usingverbal reports to study math strategies. Journal of Experimental Psychology:Learning, Memory, and Cognition, 27, 157e175.

Kroesbergen, E. H. (2002). Mathematics education for low-achieving students. Effectsof different instructional principles on multiplication learning. Doetinchem, TheNetherlands: Graviant Educatieve Uitgaven.

Milo, B. F. (2003). Mathematics instruction for special-needs students. Effects ofinstructional variants in addition and subtraction up to 100. Unpublished doctoraldissertation. Leiden, The Netherlands: Leiden University.

Milo, B., & Ruijssenaars, A. J. J. M. (2003). Rekeninstructie op scholen voor speciaalbasisonderwijs, wat is realistisch? Tijdschrift voor orthopedagogiek, 42, 423e435.

National Mathematics Advisory Panel. (2008). Foundations for success: The nalreport of the National Mathematics Advisory Panel. Washington: U.S. Departmentof Education.

Peltenburg, M. (2012). Mathematical potential of special education students. Utrecht,The Netherlands: Utrecht University.

Peltenburg, M., van den Heuvel-Panhuizen, M., & Robitzsch, A. (2012). Special ed-ucation students use of indirect addition in solving subtraction problems up to100 e a proof of the didactical potential of an ignored procedure. EducationalStudies in Mathematics, 79, 351e369. http://dx.doi.org/10.1007/s10649-011-9351-0.

Peters, G., De Smedt, B., Torbeyns, J., Ghesquire, P., & Verschaffel, L. (2010). Adultsuse of subtraction by addition. Acta Psychologica, 135, 323e329. http://dx.doi.org/10.1016/j.actpsy.2010.08.007.

Peters, G., De Smedt, B., Torbeyns, J., Ghesquire, P., & Verschaffel, L. (2013).Childrens use of addition to solve two-digit subtraction problems. BritishJournal of Psychology, 104(4), 495e511. http://dx.doi.org/10.1111/bjop.12003.

nstruction 30 (2014) 1e8 7Selter, C., Prediger, S., Nhrenbrger, M., & Hussmann, S. (2012). Taking away anddetermining the difference e a longitudinal perspective on two models of

subtraction and the inverse relation to addition. Educational Studies in Mathe-matics, 79, 389e408. http://dx.doi.org/10.1007/s10649-011-9305-6.

van der Sluis, S., de Jong, P., & van der Leij, A. (2004). Inhibition and shifting inchildrenwith learning decits in arithmetic and reading. Journal of ExperimentalChild Psychology, 87, 239e266. http://dx.doi.org/10.1016/j.jecp.2003.12.002.

Swanson, H. L., & Hoskyn, M. (1998). Experimental intervention research on studentswith learning disabilities: a meta-analysis of treatment outcomes. Review ofEducationalResearch,68, 277e321.http://dx.doi.org/10.3102/00346543068003277.

Thevenot, C., Castel, C., Fanget, M., & Fayol, M. (2010). Mental subtraction in high-and lower skilled arithmetic problem solvers: verbal report vs. operand-recognition paradigms. Journal of Experimental Psychology: Learning, Memory,and Cognition, 36, 1242e1255. http://dx.doi.org/10.1037/a0020447.

Torbeyns, J., De Smedt, B., Ghesquire, P., & Verschaffel, L. (2009). Acquisition anduse of shortcut strategies by traditionally schooled children. Educational Studiesin Mathematics, 71, 1e17. http://dx.doi.org/10.1007/s10649-008-9155-z.

Torbeyns, J., De Smedt, B., Stassens, N., Ghesquire, P., & Verschaffel, L. (2009).Solving subtraction problems by means of indirect addition. MathematicalThinking and Learning, 11, 79e91. http://dx.doi.org/10.1080/10986060802583998 (Special Issue).

Torbeyns, J., Ghesquire, P., & Verschaffel, L. (2009). Efciency and exibility ofindirect addition in the domain of multi-digit subtraction. Learning and In-struction, 19, 1e12. http://dx.doi.org/10.1016/j.learninstruc.2007.12.002.

Torbeyns, J., Verschaffel, L., & Ghesquire, P. (2004). Strategic aspects of simpleaddition and subtraction: the inuence of mathematical ability. Learning andInstruction, 14, 177e195. http://dx.doi.org/10.1016/j.learninstruc.2004.01.003.

Vanbinst, K., Ghesquire, P., & De Smedt, B. (2012). Numerical magnitude repre-sentations and individual differences in childrens arithmetic strategy use.Mind, Brain, and Education, 6, 129e136. http://dx.doi.org/10.1111/j.1751-228X.2012.01148.x.

Van den Heuvel-Panhuizen, M., & Treffers, A. (2009). Mathe-Didactical reectionson young childrens understanding and application of subtraction-relatedprinciples. Mathematical Thinking and Learning, 11, 102e112. http://dx.doi.org/10.1080/10986060802584046.

Verschaffel, L., Bryant, P., & Torbeyns, J. (2012). Introduction. Educational Studies inMathematics, 79, 327e334. http://dx.doi.org/10.1007/s10649-012-9381-2.

Verschaffel, L., Torbeyns, J., De Smedt, B., Luwel, K., & Van Dooren, W. (2007).Strategic exibility in children with low achievement in mathematics. Educa-tional & Child Psychology, 24, 16e27.

Woods, S. S., Resnick, L. B., & Groen, G. J. (1975). An experimental test of ve processmodels for subtraction. Journal of Educational Psychology, 67, 17e21. http://dx.doi.org/10.1037/h0078666.

Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentations, and au-tonomy in mathematics. Journal for Research in Mathematics Education, 27, 458e477. http://dx.doi.org/10.2307/749877.

G. Peters et al. / Learning and Instruction 30 (2014) 1e88

Subtraction by addition in children with mathematical learning disabilities1 Introduction2 The present study3 Method3.1 Participants3.2 Materials and procedure

4 Results4.1 Accuracy4.2 Reaction time analyses4.2.1 Regression analyses for problems in subtraction format4.2.2 Comparison of reaction times between the two presentation formats

5 Discussion6 ConclusionAcknowledgementsReferences