Neuropsychologia 47 (2009) 604–608

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To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving Roland H. Grabner a,∗,1 , Daniel Ansari b,∗∗,1 , Karl Koschutnig c , Gernot Reishofer d , Franz Ebner c , Christa Neuper e a

Research on Learning and Instruction, Institute for Behavioral Sciences, Swiss Federal Institute of Technology (ETH) Zurich, Universitaetsstrasse 6, CH-8092 Zurich, Switzerland Numerical Cognition Laboratory, Department of Psychology, University of Western Ontario, London, Ontario N6A 5C2, Canada Division of Neuroradiology, Department of Radiology, Medical University of Graz, Austria d Division of MR Physics, Department of Radiology, Medical University of Graz, Austria e Section of Applied Neuropsychology, Institute of Psychology, University of Graz, Austria b c

a r t i c l e

i n f o

Article history: Received 30 July 2008 Received in revised form 24 September 2008 Accepted 15 October 2008 Available online 21 October 2008 Keywords: fMRI Arithmetic Angular gyrus Strategy Problem solving

a b s t r a c t While there is consistent evidence from neuropsychological and brain imaging studies for an association between the left angular gyrus and mental arithmetic, its specific role in calculation has remained poorly understood. It has been speculated that the angular gyrus mediates the retrieval of arithmetic facts during problem solving, but this hypothesis has not been directly tested. In the present functional Magnetic Resonance Imaging study comprising 28 adults, we used trial-by-trial strategy self-reports to identify brain regions underpinning different strategies in arithmetic problem solving. Analyses revealed stronger activation of the left angular gyrus while solving arithmetic problems for which participants reported fact retrieval whereas the application of procedural strategies was accompanied by widespread activation in a fronto-parietal network. These data directly link the left angular gyrus with arithmetic fact retrieval and show that strategy self-reports can be used to predict differential patterns of brain activation. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Early investigations of brain-damaged patients revealed that lesions to areas of the parietal cortex, in particular the left angular gyrus (lAG), cause deficits in mental calculation (Gerstmann, 1940). Subsequent neuropsychological and neuroimaging studies have corroborated the strong link between the lAG and arithmetic problem solving, but little is known about the particular processes this brain region supports during problem solving (Ansari, 2008). Since the lAG has been found to be implicated in language processing (Price, 2000) it has been speculated that it mediates the retrieval of verbally stored arithmetic facts (such as the multiplication table) from long-term memory (Dehaene, Piazza, Pinel, & Cohen, 2003). This assumption is supported by evidence showing that the lAG is more strongly activated during exact compared to approximate

∗ Corresponding author. Tel.: +41 44 63 25053; fax: +41 44 63 21219. ∗∗ Corresponding author. Tel.: +1 519 661 2111x80548; fax: +1 519 661 3961. E-mail addresses: [email protected] (R.H. Grabner), [email protected] (D. Ansari). 1 These authors contributed equally to this work. 0028-3932/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.neuropsychologia.2008.10.013

arithmetic (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999), and shows greater activation during computation of problems with a relatively small compared to large problem size (Stanescu-Cosson et al., 2000) as well as of multiplication compared to subtraction problems (Lee, 2000). Furthermore, it has been revealed that the lAG increases in its activation after the training of arithmetic facts (Delazer et al., 2003). However, the interpretation that the lAG mediates the retrieval of arithmetic facts lacks direct empirical evidence since the effect of using arithmetic fact retrieval on the activation of the lAG has not been systematically compared with the neural correlates of less efficient procedural (calculation) strategies. In addition, the role of the lAG in fact retrieval has been disputed based on the results of some neuropsychological studies. For example, Van Harskamp, Rudge, and Cipolotti (2002) reported data from a patient with lesions in the left supramarginal and angular gyri who displayed no impairments in multiplication fact retrieval. The present study builds on the well-established finding that children and adults apply a variety of strategies in arithmetic problem solving and that their strategy use varies on a trial-by-trial basis (Siegler, Adolph, & Lemaire, 1996). In order to evaluate whether the lAG is indeed linked to arithmetic fact retrieval, we presented

R.H. Grabner et al. / Neuropsychologia 47 (2009) 604–608

healthy adults with arithmetic problems of all four operations during functional Magnetic Resonance Imaging (fMRI). Afterwards, they were required to solve half of the problems again in a paperand-pencil test and to indicate after every problem which strategy they used. Based on the strategy self-reports in the post-scan test, we sorted the individual fMRI data into problems for which participants reported fact retrieval (“retrieval problems”) and problems that were solved using procedural strategies (“procedural problems”). In consideration of the aforementioned studies suggesting that the lAG supports arithmetic fact retrieval, it is hypothesized that retrieval problems will be associated with greater lAG activation than procedural problems. This finding would not only provide first direct evidence for a link between the lAG and arithmetic fact retrieval during calculation but would also validate the often controversial use of self-report measures to glean insights into strategy use in mental arithmetic (Kirk & Ashcraft, 2001; Smith-Chant & LeFevre, 2003). 2. Methods 2.1. Sample Twenty-eight male adults (age between 22 and 33 years; M = 26.86, SD = 3.16) who had been screened using the Berlin Intelligence Structure Test (BIS-T; Jäger, Süß, & Beauducel, 1997) participated in the present investigation. The average IQs were 99.02 (SD = 12.46) for mathematical–numerical intelligence, 104.10 (SD = 5.60) for verbal intelligence, and 101.37 (SD = 7.47) for figural–spatial intelligence. All participants were right-handed and had normal or corrected-to-normal vision. They gave written informed consent and were paid for their participation. The study was approved by the local ethics committee (Medical University of Graz, Austria).

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with a single shot gradient echo EPI sequence (TR = 2000 ms, TE = 30 ms, FA = 90◦ , matrix size = 64 × 64, slice thickness = 3 mm, spatial resolution = 3 mm × 3 mm). In total, 787 functional volumes (first 2 were discarded) with 30 transverse slices (3 mm thickness, 0.75 mm gap) were acquired in descending order. Stimulus presentation was accomplished with the Eloquence system (Invivo Corporation, Orlando, FL), containing an LCD display visible for the participant through a mirror mounted above the head coil. The paradigm was presented using the software package Presentation (Neurobehavioral Systems, Albany, CA). 2.4. Data analysis fMRI data analysis was performed using SPM5 (Wellcome Department of Imaging Neuroscience, London, UK). The functional data of each participant were motion-corrected, spatially normalized into the standard MNI space (Montreal Neurological Institute), and smoothed using a Gaussian kernel of 9 mm FWHM. Based on the strategy self-reports in the paper-and-pencil test, arithmetic problems were divided into retrieval and procedural (transformation, counting, and other) problems for each subject and modelled separately. Only correctly solved problems were included in the analysis. The remaining 80 problems (for which no strategy information was available), incorrectly solved problems, the time intervals during the presentation of the run number, and the six motion parameters were included as regressors of no interest. A high-pass filter with a cut-off frequency of 1/256 Hz was employed to remove low frequency drifts. The analysis for the entire group was performed by computing linear t contrasts for each subject individually which were then entered into a random effects one-sample t-test. In light of a specific a priori hypothesis for greater involvement of the lAG in retrieval compared to procedural strategies and the fact that we are relating post-scan self-report measures to differences in the fMRI signal, we chose a considerably liberal threshold of p < .001 uncorrected, with a spatial extent threshold of 20 voxels, to identify clusters of significant activation differences between retrieval and procedural problems.

3. Results 3.1. Experimental task performance and strategy reports

2.2. Materials and procedure During functional MRI, 160 simple arithmetic problems of all four operations were presented in pseudo-randomized order. These were created following the procedure described in Campbell and Xue (2001). Only problems with integers between 2 and 9 were used; tie problems (e.g., “5 + 5”) were excluded. For each arithmetic operation, 20 small and 20 large problems were selected. In addition and multiplication, small problems were defined as problems with the product smaller than 25, and large problems with products larger than 25. In subtraction and division, small and large problems were defined on the basis of the inverse relationship to addition and multiplication. Operand order was counterbalanced in addition and multiplication; likewise was done for subtraction and division by counterbalancing the size of the subtrahend and result, or divisor and result, respectively. The arithmetic problems were presented in an event-related fMRI design consisting of 8 runs with 20 problems each. Each problem was presented for 2 s, followed by the presentation of the correct result (solution) and a distractor for 2 s. Participants had to indicate the position of the solution by pressing the left-hand button for the left and the right-hand button for the right number on the screen using their index fingers. In half of the problems, the correct solution was presented on the left, in the other half on the right side of the screen. The distractors were created to avoid short-cut strategies such as parity judgement. After the presentation of the response options, an inter-trial-interval of 1–5 s (jittered in 1 s steps across the problems) with a fixation point was presented. Each run started with the number of the run presented on the screen for 3 s followed by a 25 s fixation period. At the end of each run also a fixation period of 25 s was included. Before imaging was performed, the task was demonstrated to participants and any questions were answered. Instructions stressed speed and accuracy. Subsequent to the acquisition of the fMRI data, participants were presented with a paper-and-pencil test including 80 arithmetic problems (10 of each operation and problem size) which were selected from the 160 problems presented during scanning. Participants were required to solve the problems as fast and accurately as possible and to indicate which strategy they used in problem solving by choosing one of four different response options: retrieval (remembering the solution), transformation (referring to related operations or using arithmetic facts from other operations), counting or other (use of strategies not listed). These strategies were explained to the participants at the beginning of the paper-and-pencil test in accordance with the procedure described by Campbell and Xue (2001). Importantly, participants were asked to indicate the strategy applied in the paper-and-pencil test which does not necessarily correspond to the strategy used in the scanner. 2.3. MRI data acquisition Imaging was performed in a 3.0 T Tim Trio system (Siemens Medical Systems, Erlangen, Germany) using an 8-channel head coil. Functional images were obtained

Participants solved 96.55% (SD = 2.29) of the analyzed fMRI problems correctly. From all correctly solved problems, retrieval strategy was reported in 68.76% (SD = 20.38) and a procedural strategy in 31.24% (SD = 20.38) of the problems. Strategy use was moderated by arithmetic operation, F(2.03,54.84) = 9.78), p < .001, 2 = .27, showing stronger reliance on retrieval strategies in multiplication (81.77%) and addition (75.43%) as compared with division (58.90%) and subtraction (58.50%). Retrieval problems were solved significantly faster than the procedural problems in the scanner (median response latencies of 460 ms vs. 521 ms, t(27) = 2.64, p < .05). 3.2. fMRI results The results of the whole-brain contrasts between retrieval and procedural problems are listed in Table 1 and depicted in Fig. 1. Compared with procedural problems, retrieval problems elicited a significantly stronger activation only in the lAG. The reverse contrast (procedural minus retrieval problems) revealed a widespread fronto-parieto-occipital network including the basal ganglia, insulae and cingulate, bilaterally, right frontal cortices, and left inferior and superior parietal areas around the intraparietal sulcus. To preclude alternative interpretations of the stronger lAG activation in retrieval as compared with procedural problems, several control analyses were conducted. To this end, the individual contrast estimates for retrieval problems and procedural problems (both relative to baseline) were extracted from the lAG region revealed by the contrast retrieval minus procedural problems. Given that previous studies have shown that the lAG is more strongly activated in easier as compared to more difficult arithmetic problems (Stanescu-Cosson et al., 2000), we evaluated whether the strategy-related activation difference in the lAG can be accounted for by mere difference in task difficulty. Since procedural problems took significantly longer to be solved than retrieval problems (see above) the reaction time difference between retrieval and proce-

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Table 1 Significant activation clusters (p < .001 uncorrected, extent threshold 20 voxel) from the contrasts between self-reported retrieval and procedural problems. Area

Retrieval > Procedural L angular G Procedural > Retrieval R pallidum, L pallidum, L + R insula, L + R putamen, L + R caudate L precuneus, L inf parietal G, L sup parietal G, L mid occipital G, L sup occipital G R SMA, L SMA, R mid cingulate, L + R ant cingulate, L + R sup medial frontal G R sup frontal G, R precentral G, R mid frontal G R inf frontal G R sup occipital G L calcarine, R calcarine L inf occipital G, L inf temporal G L lingual G

Cluster size

t

Peak coordinates x

y

z

23

4.14

−54

−66

36

3107 800 658 116 147 170 467 52 41

9.01 6.61 6.15 6.10 5.47 5.03 4.82 4.80 4.50

12 −9 9 36 45 30 −3 −51 −15

3 −72 15 −3 9 −66 −75 −63 −42

0 51 51 63 27 45 12 −12 −3

Note: Coordinates are reported in MNI space as given by SPM5 and correspond only approximately to Talairach and Tournoux space (Talairach & Tournoux, 1988). Anatomical labels are based on the AAL (automated anatomical labelling) atlas (Tzourio-Mazoyer et al., 2002). The first label represents the location of the peak activation. Abbreviations: L = left hemisphere; R = right hemisphere; L + R = bilaterally; G = gyrus; inf = inferior, sup = superior, mid = middle; SMA = supplementary motor area.

dural problems was used a measure of task difficulty. Therefore, we calculated an ANCOVA with the within-subjects factor Strategy (retrieval vs. procedural) and the strategy-related response time (RT) difference (calculated as procedural RT minus retrieval RT, referenced to retrieval RT) as covariate. Here, as in the below analyses, we divided the difference between our performance variables of interest by the easiest condition (in this case retrieval). Such a proportional measure is independent of individual differences in general reaction time and, thus, represents a more stringent measure of inter-individual variability in performance. This ANCOVA only revealed a main effect of Strategy, F(1,26) = 13.27, p < .01, 2 = .34, and no significant main effect of the covariate, F(1,26) = 0.41, or interaction, F(1,26) = 0.26. Another related concern may be the effect of problem size on activation of the angular gyrus. Small problems are more frequently solved by retrieval strategies compared to problems with a relatively large problem size (LeFevre, Sadesky, & Bisanz, 1996). Also in the present study, retrieval strategies were more often reported in small than in large problems (79.11% vs. 56.64%; t(27) = 8.68, p < .001). However, as two control analyses revealed, the observed

strategy effect cannot be reduced to the effect of problem size. Including the individual problem size effect (calculated as RT difference between large and small problems, referenced to small problems) as covariate neither eliminates the effect of Strategy, F(1,26) = 6.48, p < .05, 2 = .20, nor does it have a significant main effect, F(1,26) = 0.30, or interacts with strategy, F(1,26) = 0.39. Likewise, controlling for the neurophysiological problem size effect (lAG activation in the contrast large minus small problems) did not change the result pattern (main effect of Strategy: F(1,26) = 5.86, p < .05, 2 = .18, but no effect of the covariate, F(1,26) = 1.40, or interaction, F(1,26) = 1.64). Moreover, we evaluated whether the strategy-related effect might be due to differential strategy use in individuals with different mathematical abilities. Even though mathematical–numerical intelligence does not significantly correlate with retrieval strategy use (r = .29, p = .14) it might impact on the lAG activation (Grabner et al., 2007). An ANCOVA with mathematical–numerical intelligence as covariate, however, revealed a main effect of Strategy, F(1,26) = 5.86, p < .05, 2 = .18, which did not interact with math competence, F(1,26) = 3.57, demonstrating that individual differ-

Fig. 1. Brain activation differences between self-reported retrieval and procedural problems. Significant activation clusters (p < .001 uncorrected, extent threshold 20 voxel) emerging from the contrasts (a) retrieval minus procedural problems and (b) procedural minus retrieval problems are depicted on the standard single-subject volume-rendered brain implemented in SPM5.

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ences in mathematical competence do not confound the effect of self-reported strategy use on the lAG activation. Interestingly, we found that mathematical–numerical intelligence exerts a significant but independent effect on lAG activation, F(1,26) = 5.00, p < .05, 2 = .16, which replicates the finding of Grabner et al. of stronger lAG activation in more as compared with less competent individuals. In the final control analysis we investigated the potential confounding of arithmetic operation and strategy use by extracting the lAG activation contrast estimates for both strategies within each arithmetic operation, separately. However, since not all participants reported both, retrieval and procedural strategies in each of the four operations, this analysis could only be performed for a sub-sample of 13 participants. An ANOVA with the factors Strategy and Arithmetic Operation revealed significant main effects of Strategy, F(1,12) = 12.45, p < .01, 2 = .51, and Arithmetic Operation, F(2.16,25.87) = 3.88, p < .05, 2 = .24, but no interaction, F(3,36) = 1.41. Consistent with the analyses above, retrieval problems were associated with stronger lAG activation than procedural problems. This effect was independent of the operation main effect which showed higher lAG activation in multiplication and subtraction as compared with division and addition. Thus, the effect of strategy on lAG activation does not interact with the type of arithmetic operation. 4. Discussion Data from neuropsychological studies with brain-damaged patients have long implicated the lAG in mental arithmetic (Gerstmann, 1940). Recently, it has been speculated that this region subserves the retrieval of arithmetic facts (Dehaene et al., 2003). However, there has thus far been no direct evidence linking arithmetic fact retrieval with activation of the lAG. In the present study we evaluated the relationship between lAG activation and retrieval by relating strategy self-reports to brain activation during arithmetic problem solving. Analyses of the fMRI data revealed that solving arithmetic problems for which participants indicated the use of fact retrieval (as compared with procedural strategies) is accompanied by a stronger activation the lAG, thereby uncovering a direct link between the lAG and arithmetic fact retrieval. However, activation of the lAG has also been found to be modulated by arithmetic task difficulty and problem size (Stanescu-Cosson et al., 2000) as well as by individual differences in mathematical abilities (Grabner et al., 2007). Given that participants solved retrieval problems faster than procedural problems, and because our sample represented students of different mathematical abilities, it might be argued that these factors confounded the observed effect of self-reported strategy use on brain activation. Contrary to these potential alternative explanations of the data our control analyses demonstrated that arithmetic problem solving strategy predicts unique variance in lAG activation over and above task difficulty and individual differences in mathematical competence. An unresolved issue in research on the neurophysiological basis of mental calculation lies in the differentiation between effects of problem solving strategies and effects of arithmetic operations. It is a well-established behavioral finding that multiplication problems are solved more frequently by fact retrieval than problems of other operations (Campbell & Xue, 2001). This behavioral result pattern, which also emerged in the present study, coincides with the results of several neuroimaging studies showing stronger lAG activation in multiplication (as compared with other) problems (Dehaene et al., 2003). Consequently, it is unclear to what extent lAG activation is moderated by problem solving strategy and arithmetic operation. We tried to disentangle strategy and operation effects in an additional control analysis, which revealed that both factors moderate

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lAG activation independently of each other. However, future studies are needed which address this issue more elaborately, for instance, by experimentally varying strategy use in different operations. The present results indicate that trial-level strategy self-reports predict differential brain activation patterns. Arithmetic strategy self-reports have been controversially discussed in the literature (Kirk & Ashcraft, 2001; Smith-Chant & LeFevre, 2003). In particular, it has been argued that people change their solution strategies when asked to describe them, that they may be generally unable to report them accurately, and that experimental conditions (e.g., instructions emphasizing speed or accuracy) bias self-reports and solution procedures. In the present study, the self-reported strategy data and the fMRI data were collected independently, and subjects were instructed not to recall the strategies they used in the scanner but to report those used while completing the post-scan test. Despite this independence of measurements a strong brainbehavior association was found. Thus, our findings not only suggest that self-reports are a valid instrument for assessing the neurocognitive processes underlying arithmetic problem solving but also that the applied solution procedures to arithmetic problems display a certain degree of temporal stability. Future studies have to show whether similar findings can be obtained when the trial-bytrial strategy assessment is administered during problem solving in the scanner. Such an approach might also yield stronger activation differences between strategies than were observed in the present study. The observed association between arithmetic strategies and brain activation can also serve to better understand the arithmetic deficits in patients with parietal brain lesions. There is considerable behavioral evidence that not only children but also educated adults use a variety of strategies during arithmetic problem solving (e.g., Campbell & Xue, 2001). If, as the present study suggests, the different problem solving strategies are related to specific brain networks, it is crucial to also assess the intra-individual variability in strategy use in lesion studies. Consistent with the notion that procedural strategies are more effortful than fact retrieval, we found that procedural (compared to retrieval) problems engaged an extensive network of frontal and parietal cortices as well as the basal ganglia. Interestingly, in a recent neuroimaging study on developmental changes of brain activation during mental calculation, a very similar network of regions was observed to be negatively correlated with age (Rivera, Reiss, Eckert, & Menon, 2005). Younger children more strongly activated regions found to be related to procedural problem solving strategies than their older peers. Positive correlations between age and brain activity, in contrast, were found in the left temporo-parietal cortices (including supramarginal gyrus extending to the lAG). In light of the present data, it could be speculated that these dynamic age-related activation changes reflect a shift away from effortful procedural strategies towards a predominant use of fact retrieval which seem to be mediated by regions of the left temporo-parietal cortex. 5. Conclusion The present data revealed that the self-reported use of fact retrieval strategies during arithmetic problem solving is linked with lAG activation, providing first direct evidence that the lAG mediates the retrieval of arithmetic facts. In addition, the present findings demonstrated that self-reported use of procedural strategies is associated with widespread fronto-parietal activation. These findings demonstrate the utility of self-reports as predictive measures of brain activation and have implications for interpreting data on the neural networks involved in arithmetic problems solving.

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