aNational Institute of Education, Nanyang Technological University, SingaporebCognitive Neuroscience Laboratory, Duke-NUS Graduate Medical School, Singapore
A R T I C L E I N F O
⁎ Corresponding author. Centre for ResearSingapore. Fax: +65 6896 9410.
E-mail address: [email protected] (K.URL: http://acdl.crpp.nie.edu.sg/ (K. Lee).Abbreviations: SE, Symbolic experimenta
Article history:Accepted 10 April 2007Available online 21 April 2007
In this study, we built on previous neuroimaging studies of mathematical cognition andexamined whether the same cognitive processes are engaged by two strategies used inalgebraic problem solving. We focused on symbolic algebra, which uses alphanumericequations to represent problems, and the model method, which uses pictorialrepresentation. Eighteen adults, matched on academic proficiency and competency in thetwomethods, transformed algebraic word problems into equations ormodels, and validatedpresented solutions. Both strategies were associated with activation of areas linked toworking memory and quantitative processing. These included the left frontal gyri, andbilateral activation of the intraparietal sulci. Contrasting the two strategies, the symbolicmethod activated the posterior superior parietal lobules and the precuneus. These findingssuggest that the two strategies are effected using similar processes but impose differentattentional demands.
Mathematical cognition has been studied in a number ofneuroimaging investigations. Although most studies havefocused on the representation of numbers and on arithmeticcomputation (e.g., Dehaene et al., 2003; Menon et al., 2002;Rivera et al., 2005), there has been some recent attempts tostudymore complexmathematical operations. Anderson et al.(2003), for example, found algebraic transformation to besubserved by the left posterior parietal region and the leftdorsal lateral prefrontal cortex. Sohn et al. (2004) founddifferences in prefrontal versus parietal engagement, depend-
ch in Pedagogy and Prac
Lee).
l; SC, Symbolic control; Hrior superior parietal lob
er B.V. All rights reserved
ing on whether algebraic questions were presented in a verbalor symbolic format. In this study, we examined another aspectof algebraic problem solving: differences resulting from theuse of different problem solving strategies.
In school, algebraic problems are often presented as storiesor word problems (see Fig. 1 for an example). Like studentselsewhere, students in Singapore often find these questionsdifficult. To give them better access, primary schoolers (10–12-year olds) are taught a diagrammatic or model method.Students are taught to draw diagrams, normally made up ofrectangles, to represent relationships presented in wordproblems (see Fig. 1). The rectangles represent unknowns.
tice, National Institute of Education, 1 Nanyang Walk, 637616
IPS, Horizontal segment of the intra-parietal sulcus; ME, Modelules
Fig. 1 – A model solution to the question: A cow (C) weighs150 kgmore than a dog (D). A goat (G) weighs 130 kg less thanthe cow. Altogether the three animals weigh 410 kg.What is the weight of the cow?
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Students are expected to solve for the unknowns by analysingthe quantitative relationships between the rectangles.
As students' success with word problems are affected bywhether they understand the questions and whether they cantransform the questions into equations or models (Carpenteret al., 1988; Kintsch and Greeno, 1985; Mayer, 1992; Riley andGreeno, 1988; Stacey and MacGregor, 1999; Verschaffel and DeCorte, 1993), a strategy that requires explicit consideration ofrelevant relationships should promote accuracy. Indeed,Lewis (1989) showed that even college students benefitedfrom training involving the use of pictorial representations.
The present study is part of an effort to examinewhether themodel method assists in the acquisition of formal or symbolicalgebra (which, in Singapore, is taught in secondary or highschool). Previous studies suggest teaching the model methodmay have both positive and negative effects. Findings fromKhng and Lee (submitted) showed that evenwhen instructed touse only symbolic algebra, students from secondary schoolsoften exhibited intrusion errors and used the model method.Although such behaviour could be seen as adaptive in thatstudents were using an alternative heuristic that was moreaccessible,many teachers saw the samebehaviour in anegative
Fig. 2 – Experimen
light. In interviews and in feedback from in-service training,many secondary school teachers viewed the model method aschildish, non-algebraic, and thought it a hindrance to theteaching of symbolic algebra (Ng et al., 2006).
A full answer to whether the model method assists in theacquisition of symbolic algebra will need to address cognitive,motivational, and pedagogical issues. In this study,we focusedon the cognitive issues. We examined whether the modelmethod and symbolic algebra were subserved by similarprocesses in adults with similar behavioural competencyacross the two methods. In terms of surface characteristics,the two methods seem to engage different types of informa-tion. The model method makes use of pictorial and alphanu-meric information in depicting information. Symbolic algebramakes use of alphanumeric information only. Despite suchdifferences, skilledmathematicians consider the twomethodsto be equivalent. The main difference being the way in whichunknowns are represented: as rectangular boxes in the modelmethod and as letters, x or y, in symbolic algebra. Because themodel method has been part of the national curriculum inSingapore for over a decade, traditional programme evaluationtechniques are of little assistance. In this study, we usedfunctional magnetic resonance imaging (fMRI) to examinesimilarities and differences in processes that subserve the twostrategies.
Information processing models of word problem solvingguided the construction of experimental tasks. Most descrip-tions stipulated two stages: problem representation andproblem solution (Bobrow, 1968; Briars and Larkin, 1984;Lewis, 1989; Riley and Greeno, 1988; cf. Koedinger andMacLaren, 2002). In a recent rendition, Mayer and Hegarty(1996) expanded these stages further. They argued thatinformation such as quantitative relationships betweenprotagonists is first extracted from the word problem. Pre-existing knowledge relevant to the problem is then activatedand is integrated with the extracted information. Procedure
tal procedure.
Table 1 – Mean accuracy (%) and reaction times (ms) for allconditions
Conditions Mean accuracy (S.D.) Mean RT (S.D.)
Model experimental 94.52 (6.33) 1223.46 (191.80)Symbolic experimental 96.06 (4.32) 1175.19 (207.09)Model control 99.07 (1.35) 1152.52 (194.09)Symbolic control 94.29 (4.83) 1132.27 (144.90)
Table 2 – Talairach coordinates of activation maxima forthe model and symbolic methods: (SE>>SC) and (ME>>MC)
a=Activations for SE>SC.b=Activations for ME>MC.a,b=Areas activated in both (SE>SC) and (ME>MC). For theseregions, coordinates reflect activation maxima found inthe conjunction analysis.
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needed to compute the solution is then planned. The final stepinvolves computation of values. In this study, we focused onproblem representation. Participants were asked to transforminformation from text to structure using either the modelmethod or symbolic algebra. This was followed by a validationtask in which participants were asked to compare thepresented solution with the one they had in mind (see Fig. 2).
From a functional neuroanatomical perspective, becausequantitative relationships were depicted and needed to beconsidered regardless of problem solving strategy, both meth-odswereexpected to activateareasassociatedwithquantitativeprocessing. In a recent paper, Dehaene andhis colleagues (2003)proposed a triple-code model in which the horizontal segmentof the intra-parietal sulci (HIPS) was involved in cross-modalquantitative processing. Other studies have found the HIPSactivated in magnitude comparison tasks involving differentstimuli (lines, angles, versusArabic numbers, Fias et al., 2003), innumerosity habituation tasks involving dot arrays (Piazza et al.,2004), and inmental arithmetic tasks involving Arabic numbersversus array of dots (Venkatraman et al., 2005).
Findings from Terao et al. (2004) are of direct relevance tothis study. Terao and his colleagues presented participantswith three-protagonist algebraic problems. A generator or basevariable was first defined, e.g., x = A. This was followed byspecification of quantitative relationships between the gener-ator and other protagonists in the problem. Participants wereasked to depict the relationships between the various prota-gonists using either a mental number line or an algebraicequation. In comparison to the generator assignment stage,i.e., x = A, the mental number line condition was associatedwith bilateral activation in the HIPS. The HIPS was alsoactivated in the equation condition but was largely leftlateralised. This finding was surprising as previous studieshad found the HIPS insensitive to modality based differences(e.g., Fias et al., 2003). However, it should be noted that Terao etal.'s finding was based on a preliminary analysis and no directcomparison was made between the two conditions.
Another issue of interest was whether the two methodsdiffered in terms of working memory or attentional require-ment. The model method is taught in primary school becauseit is believed to give students easier access to challengingwordproblems. Whether this is effected by the more concretenature of models (Kho, 1987), reduced attentional require-ment, or some other means is unknown. If symbolic algebradoes indeed have higher working memory or attentionaldemands, it can be expected to result in greater activation infrontal and parietal areas previously found to subserve theseprocesses (e.g., Owen et al., 2005).
According to Kieran (2004), processes central to algebraicproblem solving include analyzing the quantitative relationshipsbetween protagonists and modelling the structure of such
relationships. By asking participants to transform text basedquestions into either equations or models, this study wasdesigned to identify differences in such processes across thetwo strategies. Because the two experimental conditions alsodiffered in solution formats, the data were potentially confound-ed by format specific differences. To assist in interpretation, weincluded two format-specific control conditions. Participantswere presented with explicit verbal descriptions of either (a) therectangles used in constructing models or (b) the alphanumeric
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characters used in symbolic algebra completion (see Fig. 2,model control, MC, and symbolic control, SC, for examples). Theywere asked to use these descriptions to construct appropriatemental representations. Descriptions used in the control condi-tions were designed to produce model or symbolic representa-tions identical in form to those produced in the experimentalconditions. Prior to scanning, participants were given detailedinstructions and practice. Although numbers were included inthe control descriptions, they were presented as nominal labels.Unlike the experimental conditions, no numeric considerationwas required for task completion.
Fig. 3 – Activation map for (ME>MC) and (SE>SC). Overlapping aconjunction between the two contrasts. A threshold of p<0.001,activated. The left side of each transverse slice represents the rig
2. Results
2.1. Behavioural findings
Paired-sample t-tests conducted on data from the experimentalconditions revealed no significant differences in either accuracyor reaction time between the two strategies. Data from thecontrol conditions revealed a reliable difference in accuracywithmoreaccurate responses in themodel than in the symboliccondition (see Table 1 for means and standard deviations).
reas, in a darker shade, reflect activations for theuncorrected, was used to determine whether a voxel washt side of the brain.
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2.2. fMRI findings
2.2.1. Similarities between the model and symbolic methodsTo find common areas activated for both the model and thesymbolic methods, we conducted a conjunction analysisinvolving the model experimental (ME), model control (MC),symbolic experimental (SE), and symbolic control (SC) condi-tions. In this analysis, contrasts between the experimentaland control conditions were first estimated for each method,i.e., (ME>MC), (SE>SC). These two contrasts enabled us toidentify regions specifically activated by different problemsolving strategies by excluding regions involved in processesunrelated to problem solving, such as reading the stimuli.They also exclude processes specific to the generation ofpictorial or text images. We then identified areas that wereactivated by both solutions using a conjunction analysis, i.e.,(ME>MC) and (SE>SC). Based on Nichols et al. (2005), areaswere identified as commonly activated only if each contrib-uting contrast exceeded the statistical threshold ( pb0.001,uncorrected). Reported t values corresponded to the contrastwith the smaller signal difference.
The conjunction analysis revealed activation in the frontalcortex, both left lateralised and along the midline. It alsorevealed bilateral activation in the HIPS (see Table 2 and Fig. 3for details). These findings suggest the experimental condi-tions engaged working memory and magnitude comparisonprocesses more extensively than did the control conditions.
Table 3 – Talairach coordinates of activation maxima inthe symbolic method (SE>>ME)
a Areas activated for the conjunction: (SE>ME) and (SE>MC) and(SE>SC).b Activation for the conjunction: (SE>ME) and (SE>MC) and(SE>SC) was only found in the left hemisphere.
2.2.2. Differences between the model and symbolic methodsWe were interested in whether different neural systemssubserved the model versus symbolic methods. An initialanalysis involving the symbolic and model experimentalconditions – SE>ME – showed that the symbolic method wasassociated with activation in the precuneus and bilateralposterior superior parietal lobules (PSPL). This finding sug-gests the symbolic condition recruited attentional processesmore extensively than did the model method. Also activatedwere various loci in the visual processing area and in the basalganglia. The model condition did not activate any areas aboveor beyond those activated by the symbolic condition (seeTable 3 and Fig. 4 for details).
To identify differences between the model and symbolicmethods that are specific only to the methods and not toprocesses required for generating different solution formats,an interaction analysis of the sort (ME>MC)> (SE>SC) isnecessary. These interactions, however, can arise from bothdifferences in experimental as well as control tasks. Becausebehavioural differences were observed in performance accu-racy between the two control tasks, results from such analysesare likely to be confounded by interactions involving differ-ences in control task activation (see Fig. 3, time course d, for anexample of activation differences resulting from the controlbut not the experimental conditions).
To avoid such confounds, we conducted a conjunctionanalysis involving three contrasts: (SE>ME) and (SE>SC) and(SE>MC). These analyses are more conservative and consid-ered only areas in which activation in each experimentalcondition was greater than those in both control conditions. Asubset of areas found in the initial analysis was foundactivated by the symbolic method (see Table 3 for details).They included the precuneus, left cuneus, and right caudate. Acorresponding analysis was conducted to identify areasassociated with the model method: (ME>SE) and (ME>SC)and (ME>SC). However, no area was activated only by themodel method.
3. Discussion
This study was motivated by queries regarding cognitiveprocesses that subserved two methods for representingalgebraic word problems. Both the symbolic and modelmethods require more extensive magnitude comparison andworking memory engagement than do the control conditions.This is evident by findings from a conjunction analysisshowing bilateral activation in the HIPS and left lateralisedactivation in the frontal gyri respectively. Although bothmethods draw on attentional resources, the symbolic methodis more demanding. Activation in the precuneus and PSPL,both associated with attentional processes, were found to bestronger in the symbolic than in the model condition.
3.1. Similarities between the model and symbolic methods
As expected, the HIPS were activated bilaterally in both themodel and symbolic conditions. This region has previouslybeen associated with a mental number line (Dehaene et al.,2003) and is active in both symbolic and non-symbolic
Fig. 4 – Activation map for (SE>ME) and the conjunction between (SE>SC) and (SE>MC). Overlapping areas, in a darkershade, reflect the conjunction between (SE>ME) and (SE>SC) and (SE>MC). A threshold of p<0.001, uncorrected, was usedto determine whether a voxel was activated. The left side of each transverse slice represents the right side of the brain.
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numeric processing tasks (Fias et al., 2003; Piazza et al., 2004).In the present study, HIPS activation is likely related toparticipants engaging in magnitude comparison to help verifywhich protagonist possessed more target objects.
It is possible HIPS activation merely reflects exposure tonumbers. Eger et al. (2003) showed mere presentation ofnumbers activated HIPS. However, we deem this explanationless likely. Exposure to numbers was controlled both inquestion presentation and in the response verification phaseof the procedure. The same questions were used in the twoexperimental conditions. Furthermore, HIPS activation wasfound in contrasts in which activation from the controlconditions were subtracted from the experimental conditions.Because the same amount of numeric stimuli was presented
in the experimental and control conditions, the resultingdifference in HIPS activation cannot be attributed to differen-tial exposure to numeric stimuli.
Greater activation in the frontal gyri is indicative of greaterworking memory or executive involvement in the experimen-tal than in the control conditions. Similar areaswere identifiedin two working memory meta-analyses (Owen et al., 2005;Wager and Smith, 2003). The area labelled as dorsolateralprefrontal in Owen et al. overlaps with the middle frontal areafound in the present study and was characterised by Owenet al. as being involved in reorganising material into pre-existing knowledge structures: a process that seems central tothe transformation of information embedded in word pro-blems to either models or equations. Wager and Smith (2003)
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found evidence suggesting that this area was closely associ-ated with continuous updating. Their study also found leftlateralised activation ismost frequently associatedwith verbalworking memory tasks. This interpretation is consistent withfindings from the behavioural literature showing that verbalworking memory tasks predicted reliably individual differ-ences in algebraic problem solving (Lee et al., 2004).
Also activated in the experimental conditions were theright precuneus and the anterior insula. Although theprecuneus hasmost commonly been associated with selectiveattention, in Wager and Smith's (2003) meta-analysis, it wasactivated in all three studied executive functions: updating,remembrance of order information, and manipulation. Thesefindings suggest the precuneus may subserve the attentionalfunctions of executive functioning and are consistent withtheories of working memory that emphasise close linkagesbetween working memory and attention (e.g., Baddeley andLogie, 1999; Cowan, 1999). The insula is commonly associatedwith motivational and affective aspects of learning andmemory. In this study, there are few reasons to suspect thatthe experimental condition engaged affective processing anymore so than did the control conditions. Recent reviews on thefunctions of the insula suggest a broader role that includesverbal workingmemory and selective visual attention (Augus-tine, 1996; Bamiou et al., 2003).
3.2. Differences between the model and symbolic methods
Regarding differences, the symbolic condition was found toactivate the caudate more so than did the model condition.According to the ACT-R model (Anderson et al., 2003), activationin thebasal gangliamayreflect retrieval of proceduralmemory. Inthecontextof this study, this findingsuggests that constructionofalgebraic equations is more reliant on procedural retrieval. Inaddition to the caudate, the PSPL and an area of the precuneus tothe left of that found in the similarity conjunctionwere activatedin the symbolic algebraic condition. In addition to its role inimagery and episodic memory retrieval (Fletcher et al., 1995;Lundstromet al., 2003), the precuneus has been found associatedwith other processes. Zago and Tzourio-Mazoyer (2002), forexample, found the left precuneus activated in a complexarithmetic task as compared to a visual spatial working memorytask. Behrmannet al. (2004) found goal directednon-spatial shiftsof attention to be subserved by the precuneus, which wasactivated when participants shift their attention between twodimensions of an input. Dehaene and his colleagues (2003)classified both the PSPL and the precuneus as being part of aparietal number processing circuit that contributes to attentionalselection and orientation. In the present study, activation of theprecuneus suggests additional resources devoted to attentionorientation or retrieval of relevant information are required ingenerating algebraic equations fromword problems.
One criticism of this conclusion is that differences may berelated to the solution format in the two conditions, i.e., verbalversus pictorial, rather than algebraic versus model. Althoughsuch format specific differences are intrinsic to the twostrategies, we differentiated these differences versus thosethat are not related to solution format by using format specificcontrols. Findings from a simple subtraction involving the twoexperimental conditions showed that occipital areas were
activated in the algebraic condition. This suggests participantsspent more time viewing the questions in the algebraic con-dition. One concern is that other parietal activation, particularlythose in the PSPL and the precuneus, are artefacts of this dif-ference. Findings from the SE>ME conjunction analysis, inwhich activation from the control conditions was subtracted,suggest activation in the PSPL may indeed be related to theseactivities. Activation in the precuneus, however, remainsreliable.
3.3. Conclusions
Both themodel and symbolic methods activated similar areas inthe frontal gyri andHIPS.Differenceswere found in theprecuneusand caudate regions. These findings suggest that one reason forthe efficacy of the model method is its lower demand onattentional resources. These findings are particularly importantbecause differences were found in spite of behavioural equiva-lence: all participants were screened for accuracy and had to fallwithin a narrow inclusion criterion. Although the efficacy ofpictorial strategies has been demonstrated before (Lewis, 1989),this is the first study to show that their efficacy is not due toparticipants' competency in using the strategies.
Given the extensive overlap in activation across the twomethods, it is tempting to conclude that they engaged similarcognitive processes. This should be reassuring for teacherswho may be concerned that the model method is non-algebraic. For readers interested in possible pedagogicalimplications, we have some caveats. First, though the findingsare not consistent with a view specifying extensive differencesacross the two methods, differences suggestive of differentialengagement of similar processes were found. Second, the datawere collected from adults with similar behavioural compe-tency across themethods. Although we see this as a particularstrength of the study, whether the same findings will hold forchildren or for participants with different competency acrossthe two methods will require further studies. One way toexamine this issue is to use a parametric design in whichparticipants with graded levels of competency on the twomethods are compared. Third, although the study wasmotivated by queries on whether the model method aidedchildren in acquiring symbolic algebra, we were only able toaddress one aspect of this question. A full answer will requireadditional studies addressing associated pedagogical andmotivational concerns. Even amongst the cognitive issues ofconcern,we focused only on problem representation.Whetherdifferences exist in later stages of problem solving – incomputing solutions from algebraic versus model representa-tions – remain to be addressed.
4. Experimental procedures
4.1. Participants
The sample contained 18 right-handed volunteers (10 males,20 to 25 years of age). Potential participants were screened toensure they could achieve more than 90% accuracy on a tasksimilar to those used in experiment trials. Furthermore, weselected only those participants who exhibited less than 5%
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difference in accuracy between the twomethods. This ensuredthat differences in activation were not due to variation inparticipants' competencies in the two methods. To furtherensure that differences were due to intrinsic differencesbetween the two methods and to minimise the impact ofdifferences in speed or stage of acquisition, we used adults inthis first study. All participants gave informed consent andwere treated in accordance to applicable ethical guidelines.
4.2. Design and procedure
The study was based on a 2 (Strategies: model vs. symbolic)×2(Condition: experimental vs. control) within subject design. Intotal, participants completed 144 trials, divided into 6 alternat-ing symbolic or model blocks. Within each block, participantswere presented with both experimental and control trials. Ineach trial, the algebraic problem, in text form,was presented for8s. Instructions presented prior to each block askedparticipantsto create and to hold in mind either a model or a symbolicrepresentation of each problem. Depending on the block, eithera model or a symbolic representation of the problem waspresented at the end of the problem presentation period.Participants were given 3s and were asked to compare andvalidate the presented representation against the one they hadin mind. Their decisions were expressed using a responsebutton. The representation disappeared on key press and wasreplaced by a blank screen for the remainder of a 3s responseperiod (see Fig. 2). Participantswere introduced to theprocedureand were given practice trials prior to entering the scanner.
Half of the presented representations were correct and theremainder incorrect. Participants were told there were 2 possibletypes of errors: interchanged relationship (e.g., specifying agreater-than rather than a less-than relationship betweenprotagonists, J=M+50 rather than J=M−50) and number errors(specifying the wrong numeric magnitude, e.g., J=M−43 ratherthan J=M−50). Two typesof errorwereused tomaintainalertnessand to increase task difficulty. If the participants did not respond,the trial was scored as incorrect. Each participant went through apractice session before they entered the scanner. This sessionexposed participants to the procedure and to the types ofproblems they were going to encounter.
4.3. Imaging protocol
Functional images were collected using a Siemens 3T Allegrasystem. Stimuli were projected onto a screen at the back of themagnet. Participants viewed the screen via an angled mirrorfastened to the head coil. 36 axial slices approximately parallelto the SC–PC line were acquired using an interleaved gradient-echo echo planar imaging sequence (TR=3000 ms, TE=30 ms,pixel matrix=64×64; FOV=192 mm; 3 mm thickness, 0.3 mmgap). High resolution co-planar T2 anatomical images wereacquired in the same orientation. High resolution anatomicalreference imageswere acquired using a 3D-MPRAGE sequence.
4.4. Data analysis
Functional imageswerepre-processedandanalyzedusingBrainVoyagerQXversion 1.26 (Brain Innovation,Maastricht,Holland).Gaussian smoothing kernel of 8mm FWHM was applied in the
spatial domain. A high-pass frequency filter was applied fol-lowing linear trendremoval. The functional imageswere alignedto the co-planar high resolution T2 images. The image stackswere then aligned to high resolution 3D images of the brain.The resulting data set was transformed into Talairach space.
Functional datawere analysed at group level with a generallinear model. Each trial was modelled using nine finiteimpulse response predictors, spanning a total of 27s fromtrial onset: 11s for the actual task and an additional 16s for thehemodynamic response to decay to baseline. A randomeffectsanalysis was used to identify significant differences acrossconditions. Region-of-interest (ROI) based analyses wereperformed on voxels that were identified as reliably activated.Data from the fourth to sixth predictor or time point wereanalysed and revealed similar patterns of findings. Unlessotherwise indicated, all reported findings are based on thefourth time point. A statistical threshold of pb0.001 (uncor-rected) and a cluster size of 27 contiguous voxels were used forthe identification of activated clusters. Reported ROIs aremaxima identified with a bounding cube of 10 mm surround-ing the activation peak for each ROI.
Acknowledgments
This study was supported by a grant from the Centre forResearch in Pedagogy and Practice, National Institute ofEducation, Singapore (CRP22/03KL). We thank Joshua Gohand Hwee Ling Lee for their thoughtful suggestions.
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